Recent zbMATH articles in MSC 13https://www.zbmath.org/atom/cc/132022-01-14T13:23:02.489162ZUnknown authorWerkzeugPreface: ``Third international meeting on integer-valued polynomials and problems in commutative algebra'', November 29 -- December 3, 2010https://www.zbmath.org/1475.000922022-01-14T13:23:02.489162Z(no abstract)Decidability of the theory of modules over Prüfer domains with dense value groupshttps://www.zbmath.org/1475.030862022-01-14T13:23:02.489162Z"Gregory, Lorna"https://www.zbmath.org/authors/?q=ai:gregory.lorna"L'Innocente, Sonia"https://www.zbmath.org/authors/?q=ai:linnocente.sonia"Toffalori, Carlo"https://www.zbmath.org/authors/?q=ai:toffalori.carloThe authors define relations in a ring \(R\), called \(\mathrm{DPR}(R)\) (double prime radical relation) and \(\mathrm{PP}(R)\) (Point-Prest relation). They show that if \(R\) is an effectively given Prüfer domain such that each localisation at a maximal ideal has a dense value group, \(\mathrm{DPR}(R)\) and \(\mathrm{PP}(R)\) are recursive, then the theory of \(R\)-modules is decidable. In the case of effectively given Bezout domains with the same algebraic property, the decidability of the theory of \(R\)-modules is equivalent to the fact that the defined relations are recursive.
Reviewer: Mihai Prunescu (Bucharest)Recursively-generated permutations of a binary spacehttps://www.zbmath.org/1475.050032022-01-14T13:23:02.489162Z"Abornev, A. V."https://www.zbmath.org/authors/?q=ai:abornev.a-vSummary: Nonlinear permutations of a vector space \(\mathrm{GF}(2)^m\) of any dimension \(m\ne2^t\), \(t\in\mathbb{N}\), induced by iterations of linear transformation over the ring \(R=\mathbb{Z}_4\) with characteristic polynomial \(F(x)\in R[x]\), \(F(x)\equiv(x\oplus e)^m\pmod 2\), are studied.Nonlinear permutations recursively generated over the Galois ring of characteristic 4https://www.zbmath.org/1475.050042022-01-14T13:23:02.489162Z"Abornev, A. V."https://www.zbmath.org/authors/?q=ai:abornev.a-vSummary: The class of nonlinear permutations \(\pi_F\) of a space \(\mathrm{GF}(2^r)^m\) of any dimension \(m\ge3\) is constructed. Each permutation \(\pi_F\) is recursively generated by the characteristic polynomial \(F(x)\) over the Galois ring \(\mathrm{GR}(2^{2r},4)\). Results of [\textit{A. A. Nechaev} and the author, ibid. 4, No. 2, 81--100 (2013; Zbl 07395968)] are generalized to an arbitrary Galois ring of characteristic 4.On graphs with same metric and upper dimensionhttps://www.zbmath.org/1475.050572022-01-14T13:23:02.489162Z"Pirzada, S."https://www.zbmath.org/authors/?q=ai:pirzada.shariefuddin"Aijaz, M."https://www.zbmath.org/authors/?q=ai:aijaz.mallaA graph theoretic method for securing key fobshttps://www.zbmath.org/1475.050812022-01-14T13:23:02.489162Z"Heydari, Farideh"https://www.zbmath.org/authors/?q=ai:heydari.farideh"Ghahremanian, Alireza"https://www.zbmath.org/authors/?q=ai:ghahremanian.alirezaSummary: Key fobs are small security hardware devices that are used for controlling access to doors, cars and etc. There are many types of these devices, more secure types of them are rolling code. They often use a light weight cryptographic schemas to protect them against the replay attack. In this paper, by the mean of constructing a Hamiltonian graph, we propose a simple to implement and secure method which overrides some drawbacks of traditional ones. Let \(m,n>1\) be two integers, and \({\mathbb{Z}}_n\) be a \({\mathbb{Z}}_m\)-module. Here, we determine the values of \(m\) and \(n\) for which the \({\mathbb{Z}}_n\)-intersection graph of ideals of \({\mathbb{Z}}_m\) is Hamiltonian. Then a suitable sequence will be produced which by some criteria can be used as the authenticator.Bumpless pipe dreams and alternating sign matriceshttps://www.zbmath.org/1475.051722022-01-14T13:23:02.489162Z"Weigandt, Anna"https://www.zbmath.org/authors/?q=ai:weigandt.anna-eSummary: In their work on the infinite flag variety, \textit{T. Lam} et al. [Compos. Math. 157, No. 5, 883--962 (2021; Zbl 07358686)] introduced objects called bumpless pipe dreams and used them to give a formula for double Schubert polynomials. We extend this formula to the setting of K-theory, giving an expression for double Grothendieck polynomials as a sum over a larger class of bumpless pipe dreams. Our proof relies on techniques found in an unpublished manuscript of \textit{A. Lascoux} [``Chern and Yang through ice'', Preprint]. Lascoux showed how to write double Grothendieck polynomials as a sum over alternating sign matrices. We explain how to view the Lam-Lee-Shimozono formula as a disguised special case of Lascoux's alternating sign matrix formula.
\textit{A. Knutson} et al. [J. Reine Angew. Math. 630, 1--31 (2009; Zbl 1169.14033)] gave a tableau formula for vexillary Grothendieck polynomials. We recover this formula by showing vexillary marked bumpless pipe dreams and flagged set-valued tableaux are in weight preserving bijection. Finally, we give a bijection between Hecke bumpless pipe dreams and decreasing tableaux. The restriction of this bijection to Edelman-Greene bumpless pipe dreams solves a problem of Lam, Lee, and Shimozono [loc. cit.].A problem based journey from elementary number theory to an introduction to matrix theory. The president problemshttps://www.zbmath.org/1475.110012022-01-14T13:23:02.489162Z"Berman, Abraham"https://www.zbmath.org/authors/?q=ai:berman.abraham-sPublisher's description: The book is based on lecture notes of a course ``from elementary number theory to an introduction to matrix theory'' given at the Technion to gifted high school students. It is problem based, and covers topics in undergraduate mathematics that can be introduced in high school through solving challenging problems. These topics include number theory, set theory, group theory, matrix theory, and applications to cryptography and search engines.Generalised Iwasawa invariants and the growth of class numbershttps://www.zbmath.org/1475.111982022-01-14T13:23:02.489162Z"Kleine, Sören"https://www.zbmath.org/authors/?q=ai:kleine.sorenLet \(K\) be a number field, let \(p\) be a rational prime and let \(d\) be a positive integer. Let \(\mathbb{K}\) be a \(\mathbb{Z}_{p}^{d}\)-extension of \(K\), that is, a Galois extension with \(\Gamma := \mathrm{Gal}(\mathbb{K}/K)\) topologically isomorphic to a direct product of \(d\) copies of the \(p\)-adic integers. For each positive integer \(n\), let \(\mathbb{K}_{n}\) be the subfield of \(\mathbb{K}\) fixed by \(\Gamma^{p^{n}}\). Let \(A_{n}\) denote the Sylow \(p\)-subgroup of the class group of the ring of integers of \(\mathbb{K}_{n}\) and define \(e_{n}\) by \(|A_{n}|=p^{e_{n}}\). In the case \(d=1\), \textit{K. Iwasawa} [Bull. Am. Math. Soc. 65, 183--226 (1959; Zbl 0089.02402)] showed that the growth of the order of \(A_{n}\) can be described in a very explicit manner: there exist integers \(n_{0},\lambda,\mu \geq 0\) and \(\nu\) such that for every \(n \geq n_{0}\), we have \(e_{n}=\mu p^{n} + \lambda n + \nu\). \textit{A. A. Cuoco} and \textit{P. Monsky} [Math. Ann. 255, 235--258 (1981; Zbl 0437.12003)] generalised this result to include the case \(d \geq 2\). They showed that there exist integers \(m_{0}, l_{0} \geq 0\), called the generalised Iwasawa invariants of \(\mathbb{K}/K\), such that \(e_{n}=(m_{0}p^{n} + l_{0}n + O(1))p^{(d-1)n}\).
In the article under review, the author considers the local behaviour of generalised Iwasawa invariants on the set \(\mathcal{E}^{d}(K)\) of \(\mathbb{Z}_{p}^{d}\)-extensions of \(K\), with respect to a suitable topology. The main result is as follows. Let \(\mathbb{K}/K\) be a \(\mathbb{Z}_{p}^{d}\)-extension. Assume that there exists a prime of \(K\) that is totally ramified in \(\mathbb{K}/K\). Then with respect to a suitable topology on \(\mathcal{E}^{d}(K)\), there exists a neighbourhood \(\mathcal{U} \subseteq \mathcal{E}^{d}(K)\) of \(\mathbb{K}\) such that:
\begin{itemize}
\item[(i)] \(m_{0}(\mathbb{L}/K) \leq m_{0}(\mathbb{K}/K)\) for every \(\mathbb{L} \in \mathcal{U}\), and
\item[(ii)] there exists a constant \(k \in \mathbb{N}\) such that \(l_{0}(\mathbb{L}/K) \leq k\) for each \(\mathbb{L} \in \mathcal{U}\) satisfying \(m_{0}(\mathbb{L}/K) = m_{0}(\mathbb{K}/K)\).
\end{itemize}
Moreover, the author gives a condition that ensures that \(l_{0}(\mathbb{K}/K)\) is locally maximal, that is, \(k=l_{0}(\mathbb{K}/K)\) in (ii). In previous work of the same author [Ann. Math. Qué. 43, No. 2, 305--339 (2019; Zbl 1470.11281)], the same results were proven, but under a strong technical assumption, which is now proven in the article under review.
In the case that \(\mathbb{K}/K\) is a \(\mathbb{Z}_{p}^{2}\)-extension such that exactly one prime \(\mathfrak{p}\) of \(K\) ramifies in \(\mathbb{K}\) and, moreover, \(\mathfrak{p}\) is totally ramified in \(\mathbb{K}/K\), the author proves an asymptotic growth formula for the class numbers of the intermediate fields, which improves the aforementioned results of Cuoco and Monsky in this situation. The author also briefly discusses the impact of generalised Iwasawa invariants on the global boundedness of Iwasawa \(\lambda\)-invariants.
Reviewer: Henri Johnston (Exeter)Finer factorization characterizations of class number 2https://www.zbmath.org/1475.112022022-01-14T13:23:02.489162Z"Chapman, Scott T."https://www.zbmath.org/authors/?q=ai:chapman.scott-thomasIn a previous paper, the author characterized the class number 2 property of algebraic numbers rings by using some factorization tools [Am. Math. Mon. 126, No. 4, 330--339 (2019; Zbl 1443.11229)]. In the paper under review, he adds other characterizations by refining some factorization invariants.
Reviewer: Claude Levesque (Québec)Differential algebraic dependence and Novikov dependencehttps://www.zbmath.org/1475.120082022-01-14T13:23:02.489162Z"Duisengaliyeva, Bibinur"https://www.zbmath.org/authors/?q=ai:duisengalieva.bibinur-a"Umirbaev, Ualbai"https://www.zbmath.org/authors/?q=ai:umirbaev.ualbai-uIn the paper under review the authors introduce an analogue of the Fox derivatives for differential polynomial algebras and use their properties to give a criterion for differential algebraic dependence of a finite system of elements in the field of rational differential function over a differential integral domain. As an application of this criterion, it is shown that if \(k\) is a constructive differential field of characteristic zero, then the differential algebraic dependency of a finite system of elements of a free differential field of rational functions over \(k\) is algorithmically recognizable. In the last part of the paper the authors consider Novikov algebras (a nonassociative algebras \(A = (A, \circ)\) is called a (left) Novikov algebra if \(A\) satisfies the identities \((a\circ b)\circ c - a\circ(b\circ c) = (b\circ a)\circ c - b\circ(a\circ c)\) and \((a\circ b)\circ c = (a\circ c)\circ b\)). Using a representation of free Novikov algebras by differential polynomials, the authors obtain a criterion of Novikov dependence for a finite family of elements of a free Novikov algebra.
Reviewer: Alexander B. Levin (Washington)Effective difference elimination and nullstellensatzhttps://www.zbmath.org/1475.120122022-01-14T13:23:02.489162Z"Ovchinnikov, Alexey"https://www.zbmath.org/authors/?q=ai:ovchinnikov.alexey"Pogudin, Gleb"https://www.zbmath.org/authors/?q=ai:pogudin.gleb-a"Scanlon, Thomas"https://www.zbmath.org/authors/?q=ai:scanlon.thomas-jA sequence \((a_{j})_{j=0}^{\infty}\) of elements of a field \(K\) is said to be a solution of a difference equation with constant coefficients if there is a nonzero polynomial \(F(x_{0},\dots, x_{e})\in K[x_{0},\dots, x_{e}]\) such that for every natural number \(j\), one has \(F(a_{j}, a_{j+1},\dots, a_{j+e}) = 0\). This concept can be naturally generalized to systems of difference equations in several variables.
The paper under review answers the following fundamental questions about sequence solutions of systems of ordinary difference equations:
\begin{itemize}
\item[(i)] Under what conditions does such a system have a sequence solution?
\item[(ii)] Can these solutions be made sufficiently transparent to allow for efficient computation?
\item[(iii)] Given a system of difference equations on \((n+m)\)-tuples of sequences, how does one eliminate some of the variables so as to deduce the consequences of these equations on the first \(n\) variables?
\end{itemize}
As the answers to these questions, the authors prove two strong results the first of which (Theorem 3.1 of the paper) can be viewed as effective difference Nullstellensatz; it reduces the problem of solvability of a system of difference equations to the problem of consistency of certain system of finitely many polynomial equations. The second main result of the paper (Theorem 3.4) is an effective difference elimination theorem; it reduces the question of existing/finding a consequence in the \(\mathbf{x}\)-variables of a system of difference equations in \(\mathbf{x}\) and \(\mathbf{u}\) (\(\mathbf{x}=(x_{1},\dots, x_{m})\) and \(\mathbf{u}=(u_{1},\dots, u_{r})\) are two sets of variables) to a question about a polynomial ideal in a polynomial ring in finitely many variables.
Among other important results of the paper, one has to mention is a version of difference Nullstellensatz over an uncountable algebraically closed inversive difference field \(K\). It is shown that if \(F\) is a finite subset of the ring of difference polynomials \(K\{x_{1},\dots, x_{n}\}\), then the following statements are equivalent:
\begin{itemize}
\item[(i)] The system \(F=0\) has a solution in \(K^{\mathbb{Z}}\);
\item[(ii)] \(F=0\) has a solution in \(K^{\mathbb{N}}\);
\item[(iii)] \(F=0\) has finite partial solutions of length \(l\) for sufficiently large \(l\);
\item[(iv)] The difference ideal \(J\) generated by \(F\) in \(K\{x_{1},\dots, x_{n}\}\) does not contain \(1\);
\item[(v)] The reflexive closure of \(J\) in the inversive closure of \(K\{x_{1},\dots, x_{n}\}\) does not contain \(1\);
\item[(vi)] \(F=0\) has a solution in some difference \(K\)-algebra.
\end{itemize}
The paper also contains a number of examples that illustrate applications of the obtained results and counterexamples that show one cannot have a coefficient-independent effective strong Nullstellensatz for systems of difference equations.
Reviewer: Alexander B. Levin (Washington)Computational algebra. Course and exercises with solutionshttps://www.zbmath.org/1475.130012022-01-14T13:23:02.489162Z"Yengui, Ihsen"https://www.zbmath.org/authors/?q=ai:yengui.ihsenThe book under review addresses a wide range of topics in commutative algebra and algebraic geometry along with their applications in cryptography. In addition, it contains 124 exercises with detailed solutions and all these make the book a good reference for a graduate course in computational algebraic geometry.
The first chapter focuses on the theory of Gröbner bases over fields and some kinds of applied rings. Due to the applications of Gröbner bases over rings in e.g. formal verification of paths as well as in coding theory, the computation of these bases over rings have attracted the attention of many researchers in computer algebra. In this chapter, the construction of Gröbner bases over a coherent valuation ring is presented. A ring is called coherent if the syzygy module of a finite sequence of elements in that ring is finitely generated. Moreover, a valuation ring is a ring such that every two elements in the ring are comparable w.r.t. the division. Then, the notion (and construction) of Gröbner bases is extended to polynomial ideals over coherent arithmetical rings (an arithmetical ring is a ring which is locally a valuation ring) by introducing the concept of dynamical Gröbner bases. Finally, Schreyer's construction to compute a Gröbner basis for the syzygy module of a sequence of polynomials over different kinds of rings is studied.
The purpose of the second chapter is to explain the algebraic geometry dictionary (including algebraic varieties, Noether normalization, Hilbert's Nullstellensatz and Hilbert series) related to various applications of Gröbner bases. In particular, the use of Gröbner bases in solving polynomial systems is discussed.
Chapters 3 and 4 deal, respectively, with a quick review of finite fields and with the basic algorithms in cryptography.
In Chapter 5 the concepts of projective varieties, algebraic plan curves, Riemann-Roch theorem, resultants and Bézout theorem are recalled and these concepts are then applied in the last chapter to introduce elliptic curves and elliptic cryptosystems over finite fields.
Reviewer: Amir Hashemi (Isfahan)A note on saturated multiplicatively closed setshttps://www.zbmath.org/1475.130022022-01-14T13:23:02.489162Z"Badie, Mehdi"https://www.zbmath.org/authors/?q=ai:badie.mehdiSummary: In this paper, we introduce and study \(\mathcal{H}_Y\)-s.m.c. and strong \(\mathcal{H}_Y\)-s.m.c. sets and give some connections between them and lattice ideals of \(\mathcal{H}_Y\). Also, we introduce an ideal \(R_S\), for each subset set \(S\) of a ring \(R\). We prove a ring \(R\) is a Gelfand ring if and only if \(R_S\) is an intersection of maximal ideals, for every s.m.c. set \(S\) of \(R\).On Nagata rings with amalgamation propertieshttps://www.zbmath.org/1475.130032022-01-14T13:23:02.489162Z"Molkhasi, Ali"https://www.zbmath.org/authors/?q=ai:molkhasi.ali"Shum, Kar Ping"https://www.zbmath.org/authors/?q=ai:shum.kar-pingAll rings considered in this paper are commutative with identity. Let's start by recalling some classical notions of commutative algebra, necessary to describe the content of the paper. (1) A domain \(R\) is said to be a \(S-\)domain if for each height one prime ideal \(P\) of \(R\), the extended prime ideal \(P[X]\) of the polynomial domain \(R[X]\) is also of height one. (2) A domain \(R\) is catenarian if for each pair \(P\subset Q\) of prime ideals in \(R\), all the saturated chains of prime ideals between \(P\) and \(Q\) have the same length. (3) Let \(R\) and \(S\) be rings, \(J\) an ideal of \(S\) and \(\phi:R\longrightarrow S\) a homomorphism. The subring \(R\bowtie^{\phi}J=\{(a,\phi(a)+j);~a\in A,j\in J\}\) of \(R\times S\) is called the amalgamation of \(R\) with \(S\) along \(J\) with respect to \(\phi\). (4) Let \(R\) be a domain and \(f(X)\in R[X]\). The content of \(f(X)\) is the ideal of \(R\) generated by the coefficients of \(f(X)\). The Nagata ring \(R(X)\) in one variable over \(R\) is the localization of \(R[X]\) at the multiplicative subset \(U=\{f(X)\in R[X];~c(f)=R\}\). We define recursively \(R(X_1,\ldots,X_n)=R(X_1,\ldots,X_{n-1})(X_n)\). Now, the main results of this paper are the characterizations for the amalgamation \(R\bowtie^{\phi}J\) to be Nagata ring, a strong \(S-\)domain and to be catenarian. The paper is also completed by some properties of the Hurwitz series ring over the amalgamation.
Reviewer: Ali Benhissi (Monastir)Some results on 1-absorbing primary and weakly 1-absorbing primary ideals of commutative ringshttps://www.zbmath.org/1475.130042022-01-14T13:23:02.489162Z"Nikandish, Reza"https://www.zbmath.org/authors/?q=ai:nikandish.reza"Nikmehr, Mohammad Javad"https://www.zbmath.org/authors/?q=ai:nikmehr.mohammad-javad"Yassine, Ali"https://www.zbmath.org/authors/?q=ai:yassine.ali-aSummary: Let \(R\) be a commutative ring with identity. A proper ideal \(I\) of \(R\) is called \(1\)-absorbing primary [\textit{A. Badawi} and \textit{E. Y. Celikel}, J. Algebra Appl. 19, No. 6, Article ID 2050111, 12 p. (2020; Zbl 1440.13008)] if for all nonunit \(a,b,c \in R\) such that \(abc \in I\), then either \(ab \in I\) or \(c \in \sqrt{I}\). The concept of \(1\)-absorbing primary ideals in a polynomial ring, in a PID and in idealization of a module is studied. Moreover, we introduce weakly \(1\)-absorbing primary ideals which are generalization of weakly prime ideals and \(1\)-absorbing primary ideals. A proper ideal \(I\) of \(R\) is called weakly \(1\)-absorbing primary if for all nonunit \(a,b,c \in R\) such that \(0\neq abc \in I\), then either \(ab \in I\) or \(c \in \sqrt{I}\). Some properties of weakly \(1\)-absorbing primary ideals are investigated. For instance, weakly \(1\)-absorbing primary ideals in decomposable rings are characterized. Among other things, it is proved that if \(I\) is a weakly \(1\)-absorbing primary ideal of a ring \(R\) and \(0 \neq I_1 I_2 I_3 \subseteq I\) for some ideals \(I_1, I_2, I_3\) of \(R\) such that \(I\) is free triple-zero with respect to \(I_1 I_2 I_3\), then \(I_1 I_2 \subseteq I\) or \(I_3\subseteq I\).Avoidance and absorbancehttps://www.zbmath.org/1475.130052022-01-14T13:23:02.489162Z"Tarizadeh, Abolfazl"https://www.zbmath.org/authors/?q=ai:tarizadeh.abolfazl"Chen, Justin"https://www.zbmath.org/authors/?q=ai:chen.justinIn this article, the authors investigate prime avoidance and prime absorbance by generalizing some classic results to radical ideals and infinite families of ideals in commutative rings with \(1 \neq 0\). We recall the prime avoidance lemma states that if an ideal is contained in a finite union of prime ideals, then it is already contained in one of them. The set-theoretic dual result, prime absorbance, if a finite intersection of ideals is contained in a prime ideal, then one of them is already contained in the prime. It is worth noting that both fail for infinite families. With this in mind, the authors prove several results about analogous notions involving radical ideals instead of prime ideals. The authors study compactly packed (C.P.) Rings where every set of primes has the avoidance property and rings in which every set of primes has the absorbance property, called properly zipped (or P.Z.) rings. The authors then provide several interesting characterizations of these rings and especially investigating the role chain conditions play. For example, it is shown that the only rings which are both C.P. and P.Z. rings are precisely the rings with finitely many prime ideals.
Reviewer: Christopher P. Mooney (Fulton)On \(\phi\)-\(1\)-absorbing prime idealshttps://www.zbmath.org/1475.130062022-01-14T13:23:02.489162Z"Yildiz, Eda"https://www.zbmath.org/authors/?q=ai:yildiz.eda"Tekir, Unsal"https://www.zbmath.org/authors/?q=ai:tekir.unsal"Koc, Suat"https://www.zbmath.org/authors/?q=ai:koc.suatSummary: In this paper, we introduce \(\phi\)-\(1\)-absorbing prime ideals in commutative rings. Let \(R\) be a commutative ring with a nonzero identity \(1\neq 0\) and \(\phi :\mathcal{I}(R)\rightarrow \mathcal{I}(R)\cup \{\emptyset \}\) be a function where \(\mathcal{I}(R)\) is the set of all ideals of \(R\). A proper ideal \(I\) of \(R\) is called a \(\phi\)-\(1\)-absorbing prime ideal if for each nonunits \(x,y,z\in R\) with \(xyz\in I-\phi (I)\), then either \(xy\in I\) or \(z\in I\). In addition to give many properties and characterizations of \(\phi\)-\(1\)-absorbing prime ideals, we also determine rings in which every proper ideal is \(\phi\)-\(1\)-absorbing prime.The \(i\)-extended zero-divisor graphs of commutative ringshttps://www.zbmath.org/1475.130072022-01-14T13:23:02.489162Z"Bennis, Driss"https://www.zbmath.org/authors/?q=ai:bennis.driss"El Alaoui, Brahim"https://www.zbmath.org/authors/?q=ai:el-alaoui.brahim"Fahid, Brahim"https://www.zbmath.org/authors/?q=ai:fahid.brahim"Farnik, Michał"https://www.zbmath.org/authors/?q=ai:farnik.michal"L'Hamri, Raja"https://www.zbmath.org/authors/?q=ai:lhamri.rajaSummary: The zero-divisor graphs of commutative rings have been used to build bridges between ring theory and graph theory. Namely, they have been used to characterize many ring properties in terms of graphic ones. However, many results are established only for reduced rings because a zero-divisor graph defined in the classical manner lacks the information on relationship between powers of zero-divisors. The aim of this article is to remedy this situation by introducing a parametrized family of graphs \(\{\overline{\Gamma}_i(R)\}_{i \in \mathbb{N}^{\ast}}\), for a ring \(R\), which reveals more of the relationship between powers of zero-divisors as follows: For each \(i \in \mathbb{N}^{\ast}, \overline{\Gamma}_i(R)\) is the simple graph whose vertex set is the set of non-zero zero-divisors such that two distinct vertices \(x\) and \(y\) are joined by an edge if there exist two positive integers \(n \leq i\) and \(m \leq i\) such that \(x^ny^m=0\) with \(x^n \neq 0\) and \(y^m \neq 0\). Our aim is to study in detail the behavior of the filtration \(\{\overline{\Gamma}_i(R)\}_{i \in \mathbb{N}^{\ast}}\) as well as the relations between its terms. We give answers to several interesting and natural questions that arise in this context. In particular, we characterize girth and diameter of \(\overline{\Gamma}_i(R)\) and give various examples.On a new extension of the zero-divisor graph. IIhttps://www.zbmath.org/1475.130082022-01-14T13:23:02.489162Z"Cherrabi, A."https://www.zbmath.org/authors/?q=ai:cherrabi.azzouz"Essannouni, H."https://www.zbmath.org/authors/?q=ai:essannouni.hassane"Jabbouri, E."https://www.zbmath.org/authors/?q=ai:jabbouri.el-mostafa"Ouadfel, A."https://www.zbmath.org/authors/?q=ai:ouadfel.aliSummary: Let \(R\) be a commutative ring with nonzero identity and \(Z(R)\) be its set of zero-divisors. The new extension of the zero-divisor graph \(\widetilde{\Gamma}(R)\) with vertices \(Z(R)^{\star}\), and two distinct vertices \(x\) and \(y\) are adjacent if and only if \(xy=0\) or \(x+y\in Z(R)\). In this article, we study, in the general case, the graph \(\widetilde{\Gamma}(R)\). For any commutative ring \(R\), we provide sufficient and necessary conditions for \(\widetilde{\Gamma}(R)\) and \(\widetilde{\Gamma}(R[x_1,\ldots , x_n])\) to be complete. At last, we present some other properties of this new extension of the zero-divisor graph.
For Part I, see [the authors, Algebra Colloq. 27, No. 3, 469--476 (2020; Zbl 1442.13019)].Categorial properties of compressed zero-divisor graphs of finite commutative ringshttps://www.zbmath.org/1475.130092022-01-14T13:23:02.489162Z"Đurić, Alen"https://www.zbmath.org/authors/?q=ai:duric.alen"Jevđenić, Sara"https://www.zbmath.org/authors/?q=ai:jevdenic.sara"Stopar, Nik"https://www.zbmath.org/authors/?q=ai:stopar.nikCut vertices in comaximal graph of a commutative Artinian ringhttps://www.zbmath.org/1475.130102022-01-14T13:23:02.489162Z"Esmaili, Kyuoomars"https://www.zbmath.org/authors/?q=ai:esmaili.kyuoomars"Samei, Karim"https://www.zbmath.org/authors/?q=ai:samei.karimSummary: Let \(R\) be a commutative Artinian ring with \(|{\text{Max}}(R)|=n \ge 2\). We show the comaximal graph of \(R\) has no cut-sets with more than one vertex. It has exactly a cut vertex if and only if \(R \simeq F\times \mathbb{Z}_2 \times \cdots \times \mathbb{Z}_2\), where \(F\) is a field, \( \vert F \vert > 2\) and \(n \ge 3\). It has \(n\) cut vertices if and only if \(R\) is a Boolean ring.Regularity and \(a\)-invariant of Cameron-Walker graphshttps://www.zbmath.org/1475.130112022-01-14T13:23:02.489162Z"Hibi, Takayuki"https://www.zbmath.org/authors/?q=ai:hibi.takayuki"Kimura, Kyouko"https://www.zbmath.org/authors/?q=ai:kimura.kyouko"Matsuda, Kazunori"https://www.zbmath.org/authors/?q=ai:matsuda.kazunori"Tsuchiya, Akiyoshi"https://www.zbmath.org/authors/?q=ai:tsuchiya.akiyoshiSuppose that \(S=K[x_1, \dots, x_n]\) is the polynomial ring over the field \(K\) with standard grading and \(I\) is a homogeneous ideal of \(S\) with \(\dim (S/I)=d\). It is well known that the Hilbert series \(H(S/I,\lambda)\) of \(S/I\) is of the form \[H(S/I, \lambda)=\frac{h(S/I,\lambda)}{(1-\lambda)^d},\] where \(h(S/I, \lambda)\) is a polynomial in \(\mathbb{Z}[\lambda]\). It is known that for every square-free monomial ideal \(I\) of \(S\), \[a(S/I):=\deg h(S/I, \lambda)- \dim (S/I)\leq 0, \tag{1}\] and \[\deg h(S/I,\lambda)-\mathrm{reg} \ (S/I) \leq \dim (S/I) -\mathrm{depth} \ (S/I). \tag{2}\] On the other hand, A Cameron-Walker graph is a simple graph which is neither a star graph nor a star triangle and has the same matching number and induced matching number. A combinatorial structure for such graphs are introduced in [\textit{K. Cameron} and \textit{T. Walker}, Discrete Math. 299, 49--55 (2005; Zbl 1073.05054)]. In fact \(G\) is a Cameron-Walker graph if \(G\) consists of a connected bipartite graph with vertex partition \([n]\cup [m]\) such that at least one leaf edge attached to each vertex \(i\in [n]\) and that there may be possibly some pendant triangles attached to each vertex \(j\in[m]\).
In the paper under review, the equalities in (1) and (2) are studied when \(I\) is the edge ideal of a Cameron-Walker graph. In fact it is shown that when \(I\) is the edge ideal of a Cameron-Walker graph, then \(a(S/I)=0\) and so \(\deg h(S/I, \lambda)= \dim (S/I)\). Also, this value is given in terms of the number of leaf edges and pendant triangles. So inequality (2) changes into \(\mathrm{reg} \ (S/I) \geq \mathrm{depth} \ (S/I)\). Since both of these invariants have combinatorial expression for these kinds of graphs, the authors can specify the Cameron-Walker graphs satisfying the equality in (2). They have also given an example of Cameron-Walker graph for which the inequality (2) is strict. As a corollary they have shown that for each \(2\leq e \leq d\) there are some Cameron-Walker graphs with \(\dim (S/I) =d\) and \(\mathrm{depth} \ (S/I)=e\). In addition, it is shown that when \(I\) is the edge ideal of the following graphs, equality in (2) holds:
\begin{itemize}
\item[\(\bullet\)] a star graph \(S_n\) with \(n\geq 1\);
\item[\(\bullet\)] a path graph \(P_n\) with \(n\geq 2\);
\item[\(\bullet\)] a cycle graph \(C_n\) with \(n\geq 3\);
\item[\(\bullet\)] a broom graph \(B(n, 3)\) with \(n\geq 5\).
\end{itemize}
Although in [\textit{T. Hibi} et al., Electron. J. Comb. 26 (1) 1.22 (2019; Zbl 1440.13060)] it is shown that for given integers \(r, s \geq 1\), there exists a finite simple graph \(G\) such that \(\mathrm{reg} \ (S/I) = r\) and \(\deg h(S/I, \lambda) = s\), but in this paper the authors have shown that for Cameron-Walker graphs we always have \(\deg h(S/I, \lambda)\geq \mathrm{reg} \ (S/I) \) and the equality is precisely characterized. As the last but not least result, they have introduced some Cameron-Walker graphs with specified values for \(\dim (S/I), \mathrm{reg} \ (S/I)\) and \(\mathrm{depth} \ (S/I)\). The paper is well organized and the results are well explained with several illustrative examples.
Reviewer: Fahimeh Khosh-Ahang Ghasr (Ilam)On the compressed essential graph of a module over a commutative ringhttps://www.zbmath.org/1475.130122022-01-14T13:23:02.489162Z"Payrovi, S. H."https://www.zbmath.org/authors/?q=ai:payrovi.s-h"Babaei, S."https://www.zbmath.org/authors/?q=ai:babaei.sakineh|babaei.sadra"Sengelen Sevim, E."https://www.zbmath.org/authors/?q=ai:sengelen-sevim.eIdeal-based \(k\)-zero-divisor hypergraph of commutative ringshttps://www.zbmath.org/1475.130132022-01-14T13:23:02.489162Z"Selvakumar, K."https://www.zbmath.org/authors/?q=ai:selvakumar.krishnan"Subajini, M."https://www.zbmath.org/authors/?q=ai:subajini.manoharanA note on \(\phi\)-\(\lambda\)-rings and \(\phi\)-\(\Delta\)-ringshttps://www.zbmath.org/1475.130142022-01-14T13:23:02.489162Z"Kumar, Rahul"https://www.zbmath.org/authors/?q=ai:kumar.rahul"Gaur, Atul"https://www.zbmath.org/authors/?q=ai:gaur.atulSummary: Let \(\mathcal{H}\) denotes the set of all commutative rings \(R\) in which the set of all nilpotent elements, denoted by \(\mathrm{Nil}(R)\), is a prime ideal of \(R\) and is comparable to every ideal of \(R\). Let \(R\in\mathcal{H}\) be a ring and \(T(R)\) be its total quotient ring. Then there is a ring homomorphism \(\phi:T(R) \rightarrow R_{\mathrm{Nil}(R)}\) defined as \(\phi (r/s)=r/s\) for all \(r\in R\) and for all non-zero-divisors \(s\in R\). A ring \(R\in\mathcal{H}\) is said to be a \(\phi\)-\(\lambda\)-ring if the set of all rings between \(\phi(R)\) and \(T(\phi(R))\) is linearly ordered by inclusion. If \(R_1+R_2\) is a ring between \(\phi(R)\) and \(T(\phi(R))\) for each pair of rings \(R_1, R_2\) between \(\phi(R)\) and \(T(\phi(R))\), then \(R\) is said to be a \(\phi\)-\(\Delta\)-ring. Let \(R\in\mathcal{H}\) be a \(\phi-\lambda\)-ring and \(T\in\mathcal{H}\) be a ring properly containing \(R\) such that \(\mathrm{Nil}(T)=\mathrm{Nil}(R)\). We show that if all but finitely many intermediate rings between \(R\) and \(T\) are \(\phi\)-\(\lambda\)-rings (resp., \(\phi\)-\(\Delta\)-rings), then all the intermediate rings are \(\phi\)-\(\lambda\)-rings (resp., \(\phi\)-\(\Delta\)-rings under some conditions). Moreover, the pair \((R, T)\) is a residually algebraic pair. Two new ring theoretic properties, namely, \(\phi\)-\(\lambda\)-property of rings and \(\phi\)-\(\Delta\)-property of rings are introduced and studied.Amalgamated algebras issued from \(\phi\)-chained rings and \(\phi\)-pseudo-valuation ringshttps://www.zbmath.org/1475.130152022-01-14T13:23:02.489162Z"El Khalfi, Abdelhaq"https://www.zbmath.org/authors/?q=ai:el-khalfi.abdelhaq"Kim, Hwankoo"https://www.zbmath.org/authors/?q=ai:kim.hwankoo"Mahdou, Najib"https://www.zbmath.org/authors/?q=ai:mahdou.najibSummary: Let \(f : A \longrightarrow B\) be a ring homomorphism and let \(J\) be an ideal of \(B\). In this article, we study the possible transfer of the properties of being a \(\phi\)-ring, a \(\phi\)-chained ring, and a \(\phi \)-pseudo-valuation ring to the amalgamated algebra \(A\bowtie^fJ\). Our aim is to provide examples of new classes of commutative rings satisfying the aforementioned properties.On property \((\mathcal{A})\) of the amalgamated duplication of a ring along an idealhttps://www.zbmath.org/1475.130162022-01-14T13:23:02.489162Z"Arssi, Youssef"https://www.zbmath.org/authors/?q=ai:arssi.youssef"Bouchiba, Samir"https://www.zbmath.org/authors/?q=ai:bouchiba.samirSummary: The main purpose of this paper is to totally characterize when the amalgamated duplication \(R \bowtie I\) of a ring \(R\) along an ideal \(I\) is an \(\mathcal{A}\)-ring as well as an \(\mathcal{SA}\)-ring. In this regard, we prove that \(R \bowtie I\) is an \(\mathcal{SA}\)-ring if and only if \(R\) is an \(\mathcal{SA}\)-ring and \(I\) is contained in the set of zero divisors \(Z(R)\) of \(R\). As to the Property \((\mathcal{A})\) of \(R \bowtie I\), it turns out that its characterization involves a new concept that we introduce in [6] and that we term the Property \((\mathcal{A})\) of a module \(M\) along an ideal \(I\). In fact, we prove that \(R \bowtie I\) is an \(\mathcal{A}\)-ring if and only if \(R\) is an \(\mathcal{A}\)-ring, \(I\) is an \(\mathcal{A}\)-module along itself and if \(p\) is a prime ideal of \(R\) such that \(p \subseteq Z_R (I) \cup Z^I (R)\), then either \(p \subseteq Z_R (I)\) or \(p \subseteq Z^I (R)\), where \(Z^I (R) := \{ a \in R : a + I \subseteq Z(R)\}\).On semi-\(n\)-absorbing submoduleshttps://www.zbmath.org/1475.130172022-01-14T13:23:02.489162Z"Khoshdel, Shamsolmolouk"https://www.zbmath.org/authors/?q=ai:khoshdel.shamsolmolouk"Maani-Shirazi, Mansooreh"https://www.zbmath.org/authors/?q=ai:maani-shirazi.mansoorehSummary: Let \(n\) be a positive integer greater than 1. In this paper, we introduce the concept of semi-\(n\)-absorbing submodules. several results concerning this class of submodules and examples of them are given.On a class of \(\lambda \)-moduleshttps://www.zbmath.org/1475.130182022-01-14T13:23:02.489162Z"Wijayanti, I. E."https://www.zbmath.org/authors/?q=ai:wijayanti.indah-emilia"Ardiyansyah, M."https://www.zbmath.org/authors/?q=ai:ardiyansyah.muhammad"Prasetyo, P. W."https://www.zbmath.org/authors/?q=ai:prasetyo.puguh-wahyuSummary: In [Int. Electron. J. Algebra 15, 173--195 (2014; Zbl 1302.13018)], \textit{P. F. Smith} introduced maps between the lattice of ideals of a commutative ring and the lattice of submodules of an \(R\)-module \(M\), i.e., \( \mu\) and \(\lambda\) mappings. The definitions of the maps were motivated by the definition of multiplication modules. Moreover, some sufficient conditions for the maps to be lattice homomorphisms were studied. We define a class of \(\lambda \)-modules and indicate the properties of this class. We also present sufficient conditions for the module and the ring under which the class \(\lambda\) is a hereditary pretorsion class.\(N\)-fiber-full moduleshttps://www.zbmath.org/1475.130192022-01-14T13:23:02.489162Z"Yu, Hongmiao"https://www.zbmath.org/authors/?q=ai:yu.hongmiaoSummary: Let \(A\) be a Noetherian flat \(K [t]\)-algebra, \(h\) an integer and let \(N\) be a graded \(K [t]\)-module, we introduce and study ``\(N\)-fiber-full up to \(h'' A\)-modules. We prove that an \(A\)-module \(M\) is \(N\)-fiber-full up to \(h\) if and only if \(\operatorname{Ext}_A^i(M, N)\) is flat over \(K [t]\) for all \(i \leq h - 1\). And we show some applications of this result extending the recent result on square-free Gröbner degenerations by \textit{A. Conca} and \textit{M. Varbaro} [Invent. Math. 221, No. 3, 713--730 (2020; Zbl 1451.13076)].A study of strongly Cohen-Macaulay ideals by delta invarianthttps://www.zbmath.org/1475.130202022-01-14T13:23:02.489162Z"Dibaei, Mohammad T."https://www.zbmath.org/authors/?q=ai:dibaei.mohammad-t"Khalatpour, Yaser"https://www.zbmath.org/authors/?q=ai:khalatpour.yaserLet \(R\) be a Cohen-Macaulay local ring, \(I\) a strongly Cohen-Macaulay ideal of \(R\). \textit{C. Huneke} [Adv. Math. 56, 295--318 (1985; Zbl 0585.13006)] showed that \(R/\mathrm{Ann}_R(I)\) is Cohen-Macaulay. In the present paper the authors give an alternative proof of this result using the delta invariant. Moreover, they show that \(R/\mathrm{Ann}_R(I)\) is indeed a maximal Cohen-Macaulay \(R\)-module. Also, the authors prove that there is an ideal \(J\) of \(R\) such that the \(\delta_{R/J}\)-invariant of all Koszul homologies of \(R\) with respect to \(I\) are zero.
Reviewer: Shreedevi K. Masuti (Dharwad)On algebraic aspects of SSC associated to the subdivided prism graphhttps://www.zbmath.org/1475.130212022-01-14T13:23:02.489162Z"Javed, Mehwish"https://www.zbmath.org/authors/?q=ai:javed.mehwish"Kashif, Agha"https://www.zbmath.org/authors/?q=ai:kashif.agha"Javaid, Muhammad"https://www.zbmath.org/authors/?q=ai:javaid.muhammadSummary: In this article, some important combinatorial and algebraic properties of spanning simplicial complex associated to the subdivided prism graph \(P(n,m)\) are presented. The \(f\)-vector of the spanning simplicial complex \(\Delta_s(P(n,m))\) and the Hilbert series for the face ring \(K\big[\Delta_s(P(n,m))\big]\) are computed. Further, the associated primes of the facet ideal \(I_{\mathcal{F}}(\Delta_s(P(n,m)))\) are determined. Finally, the Cohen-Macaulay characterization of the SR-ring of \(\Delta_s(P(n,m))\) is discussed.Existence of birational small Cohen-Macaulay modules over biquadratic extensions in mixed characteristichttps://www.zbmath.org/1475.130222022-01-14T13:23:02.489162Z"Sridhar, Prashanth"https://www.zbmath.org/authors/?q=ai:sridhar.prashanthSummary: Let \(S\) be an unramified regular local ring of mixed characteristic two and \(R\) the integral closure of \(S\) in a biquadratic extension of its quotient field obtained by adjoining roots of sufficiently general square free elements \(f, g \in S\). Let \(S^2\) denote the subring of \(S\) obtained by lifting to \(S\) the image of the Frobenius map on \(S / 2 S\). When at least one of \(f, g \in S^2\), we characterize the Cohen-Macaulayness of \(R\) and show that \(R\) admits a birational small Cohen-Macaulay module. It is noted that \(R\) is not automatically Cohen-Macaulay in case \(f, g \in S^2\) or if \(f, g \notin S^2\).Radicals of principal ideals and the class group of a Dedekind domainhttps://www.zbmath.org/1475.130232022-01-14T13:23:02.489162Z"Spirito, Dario"https://www.zbmath.org/authors/?q=ai:spirito.darioSummary: For a Dedekind domain \(D\), let \(\mathcal{P}(D)\) be the set of ideals of \(D\) that are the radical of a principal ideal. We show that, if \(D\) and \(D'\) are Dedekind domains and there is an order isomorphism between \(\mathcal{P}(D)\) and \(\mathcal{P}(D')\), then the rank of the class groups of \(D\) and \(D'\) is the same.Rationality of equivariant Hilbert series and asymptotic propertieshttps://www.zbmath.org/1475.130242022-01-14T13:23:02.489162Z"Nagel, Uwe"https://www.zbmath.org/authors/?q=ai:nagel.uweSummary: An \(\operatorname{FI} \)- or an \(\operatorname{OI} \)-module \(\mathbf{M}\) over a corresponding noetherian polynomial algebra \(\mathbf{P}\) may be thought of as a sequence of compatible modules \(\mathbf{M}_n\) over a polynomial ring \(\mathbf{P}_n\) whose number of variables depends linearly on \(n\). In order to study invariants of the modules \(\mathbf{M}_n\) in dependence of \(n\), an equivariant Hilbert series is introduced if \(\mathbf{M}\) is graded. If \(\mathbf{M}\) is also finitely generated, it is shown that this series is a rational function. Moreover, if this function is written in reduced form rather precise information about the irreducible factors of the denominator is obtained. This is key for applications. It follows that the Krull dimension of the modules \(\mathbf{M}_n\) grows eventually linearly in \(n\), whereas the multiplicity of \(\mathbf{M}_n\) grows eventually exponentially in \(n\). Moreover, for any fixed degree \(j\), the vector space dimensions of the degree \(j\) components of \(\mathbf{M}_n\) grow eventually polynomially in \(n\). As a consequence, any graded Betti number of \(\mathbf{M}_n\) in a fixed homological degree and a fixed internal degree grows eventually polynomially in \(n\). Furthermore, evidence is obtained to support a conjecture that the Castelnuovo-Mumford regularity and the projective dimension of \(\mathbf{M}_n\) both grow eventually linearly in \(n\). It is also shown that modules \(\mathbf{M}\) whose width \(n\) components \(\mathbf{M}_n\) are eventually Artinian can be characterized by their equivariant Hilbert series. Using regular languages and finite automata, an algorithm for computing equivariant Hilbert series is presented.Finiteness dimensions and cofiniteness of local cohomology moduleshttps://www.zbmath.org/1475.130252022-01-14T13:23:02.489162Z"Vahidi, Alireza"https://www.zbmath.org/authors/?q=ai:vahidi.alireza"Aghapournahr, Moharram"https://www.zbmath.org/authors/?q=ai:aghapournahr.moharram"Renani, Elahe Mahmoudi"https://www.zbmath.org/authors/?q=ai:renani.elahe-mahmoudiLet \(R\) be a commutative Noetherian ring with nonzero identity, \(\mathfrak a\) an ideal of \(R\), \(X\) an \(R\)-module, and \(n\) a nonnegative integer. Set \(\mathrm{Spec}(R)_{\geq n}=\{{\mathfrak p}\in\mathrm{Spec}(R): \dim(R/{\mathfrak p})\geq n\}.\) \textit{K. Bahmanpour} et al. [Commun. Algebra 41, No. 8, 2799--2814 (2013; Zbl 1273.13025)] introduced the notion of the \(n\)-th finiteness dimension of \(X\) with respect to a \(\mathfrak a\), as \(f_{\mathfrak a}^n(X)=\inf\{f_{{\mathfrak a}R_{\mathfrak p}}(X_{\mathfrak p}): {\mathfrak p}\in\mathrm{Spec}(R)_{\geq n}\}\). Note that if \(n=0\), then, by Faltings' local-global principle for the finiteness of local cohomology modules [\textit{F. Faltings}, Math. Ann. 255:1, 45--56 (1981; Zbl 0451.13008)], \(f_{\mathfrak a}^0(X)\) is just the finiteness dimension of \(X\) with respect to a \(\mathfrak a\); that is, \(f_{\mathfrak a}^0(X)=\inf\{i\in {\mathbb N_0}: H_{\mathfrak a}^i(X) \text{ is not a finitely generated } R\text{-module}\}.\) Bahmanpour et al. showed that if \(X\) is finitely generated, then \(f_{\mathfrak a}^1(X)=\inf\{i\in \mathbb N_0: H_{\mathfrak a}^i(X) \text{ is not a minimax } R\text{-module}\},\) and, if \(R\) is a semilocal ring and \(X\) is finitely generated, then \(f_{\mathfrak a}^2(X)=\inf\{i\in \mathbb N_0: H_{\mathfrak a}^i(X) \text{ is not a weakly Laskerian } R\text{-module}\}.\) Recall that an \(R\)-module \(X\) is said to be an \(FD_{<n}\) \(R\)-module, if there exists a finitely generated submodule \(X'\) of \(X\) such that \(\dim_R(X/X')<n\). It is shown in [\textit{A. A. Mehrvarz}, \textit{R. Naghipour}, and \textit{M. Sedghi}, Comm. Algebra 43:11, 4860--4872 (2015; Zbl 1329.13028)] that if \(X\) is finitely generated, then \(f_{\mathfrak a}^n(X)=\inf\{i\in \mathbb N_0: H_{\mathfrak a}^i(X) \text{ is not an FD}_{<n } ~~~~ R\text{-module}\}.\)
In the paper under review, the authors generalize the aforementioned results and they get them in the case that \(X\) is an arbitrary (not necessarily finitely generated) \(R\)-module such that \(\mathrm{Ext}_R^i(R/{\mathfrak a}, X)\) is finitely generated for all \(i\).
Reviewer: Morteza Lotfi Parsa (Hamedan)Cofiniteness with respect to the class of modules in dimension less than a fixed integerhttps://www.zbmath.org/1475.130262022-01-14T13:23:02.489162Z"Vahidi, Alireza"https://www.zbmath.org/authors/?q=ai:vahidi.alireza"Morsali, Saeid"https://www.zbmath.org/authors/?q=ai:morsali.saeidAlthough local cohomology modules can be highly non-finitely generated; many local cohomology modules have been shown to hold nice finiteness properties. One such property is \(I\)-cofiniteness: an \(I\)-torsion \(R\)-module \(X\) is \(I\)-cofinite if \(\mathrm{Ext}^i_R(R/I, X)\) is finitely generated for all \(i\). In this paper, the authors consider a related property: \((\mathrm{FD}_{<n}, I)\)-cofinite: an \(I\)-torsion \(R\)-module \(X\) is \((\mathrm{FD}_{<n}, I)\)-cofinite if \(\mathrm{Ext}^i_R(R/I, X)\) are \(\mathrm{FD}_{<n}\) R-modules for all \(i\), (\(Y\) is an \(\mathrm{FD}_{<n}\) \(R\)-module if there is a finite submodule \(Y'\) such that \(\dim (Y/Y')<n\)).
When \(R\) is a commutative Noetherian ring and \(I\) is an ideal with \(\dim(R/I) \leq n+1\), the authors prove that if \(\mathrm{Hom}_R(R/I,X)\) and \(\mathrm{Ext}^1_R(R/I,X)\) are \(\mathrm{FD}_{<n}\) \(R\)-modules, then \(X\) is \((\mathrm{FD}_{<n}, I)\)-cofinite; the category of \((\mathrm{FD}_{<n}, I)\)-cofinite \(R\) modules is an abelian category; and \(H^i_I(X)\) is \((\mathrm{FD}_{<n}, I)\)-cofinite and the set of primes \(\{P \in \mathrm{Ass}(H^i_I(X)) \mid \dim(R/P) \geq n\}\) is a finite set for all \(i\) when \(\mathrm{Ext}^i_R(R/I,X)\) is an \(\mathrm{FD}_{<n}\) \(R\)-module. Many known results are included as corollaries of the results presented in this paper.
Reviewer: Janet Vassilev (Albuquerque)Tame loci of generalized local cohomology moduleshttps://www.zbmath.org/1475.130272022-01-14T13:23:02.489162Z"Dehghani-Zadeh, Fatemeh"https://www.zbmath.org/authors/?q=ai:dehghani-zadeh.fatemeh"Jahangiri, Maryam"https://www.zbmath.org/authors/?q=ai:jahangiri.maryamSummary: Let \(M\) and \(N\) be two finitely generated graded modules over a standard graded Noetherian ring \(R=\bigoplus_{n\geq 0} R_n\). In this paper we show that if \(R_0\) is semi-local of dimension \(\leq 2\) then, the set \(\operatorname{Ass}_{R_0}\Big(H^i_{R_+}(M,N)_n\Big)\) is asymptotically stable for \(n\rightarrow -\infty\) in some special cases. Also, we study the torsion-freeness of graded generalized local cohomology modules \(H^i_{R_+}(M,N)\). Finally, the tame loci \(T^i(M,N)\) of \((M,N)\) will be considered and some sufficient conditions are proposed for the openness of these sets in the Zariski topology.Quasi-cyclic modules and coregular sequenceshttps://www.zbmath.org/1475.130282022-01-14T13:23:02.489162Z"Hartshorne, Robin"https://www.zbmath.org/authors/?q=ai:hartshorne.robin"Polini, Claudia"https://www.zbmath.org/authors/?q=ai:polini.claudiaLet \(A\) be a ring, \(I\) an ideal, and \(M\) a non-zero \(A\)-module with \(\mathrm{Supp} (M)\subset V(I )\). A sequence \(x_1, \dots, x_r\) of elements of \(I\) is \textit{coregular for \(M\)} if \(x_1M = M\) and \(x_{i+1} (0 :_M (x_1,\dots, x_i )) = 0 :_M (x_1,\dots , x_i )\) for \(1 \leq i \leq r-1\). Recall that an irreducible curve in \(\mathbb{P}^3\) is called \textit{set-theoretic complete intersection} if it is the intersection of two surfaces.
\textit{M. Hellus} [Local cohomology and Matlis duality. Habilitationsschrift. Leipzig (2006)] gave the following criterion for a subvariety \(V\subset\mathbb{P}^n\) of codimension \(r\), with homogeneous ideal \(I\) in \(A\), the coordinate ring of \(\mathbb{P}^n\) to be a set-theoretic complete intersection: \(V\) is a set-theoretic complete intersection in \(\mathbb{P}^n\) if and only if the local cohomology modules \(H_I^i (A)\) are zero for all \(i\neq r\), and the Matlis dual \(D(H^r_I (A))\) has depth \(r\). In the paper under review, the authors develop the theory of coregular sequences and codepths and give a new version of Hellus's theorem: ``A variety \(V\) of codimension \(r\) in \(\mathbb{P}^n\) is a set-theoretic complete intersection if and only if \(H^i_I (A) = 0\) for all \(i > r\) and \(H^r_I (A)\) has codepth \(r\)'' (To avoid dealing with Matlis duals of large modules, they state their theorem in term of codepth). They also define quasi-cyclic modules as increasing unions of cyclic modules, and show that modules of codepth at least two are quasi-cyclic. Then they focus on curves in \(\mathbb{P}^3\) and give a number of necessary conditions for a curve to be a set-theoretic complete intersection. Thus an example of a curve for which any of these necessary conditions does not hold would provide a negative answer to the still open problem, whether every connected curve in \(\mathbb{P}^3\) is a set-theoretic complete intersection.
Reviewer: Mohammad-Reza Doustimehr (Tabriz)Local cohomology and the multigraded regularity of \(\mathcal{FI}^m\)-moduleshttps://www.zbmath.org/1475.130292022-01-14T13:23:02.489162Z"Li, Liping"https://www.zbmath.org/authors/?q=ai:li.liping"Ramos, Eric"https://www.zbmath.org/authors/?q=ai:ramos.ericIn this paper, the authors consider regularity properties of \(\mathcal{FI}^m\)-modules, working over a commutative Noetherian ring for the main results. (Here \(\mathcal{FI}^m\) is the \(m\)-fold product of the category \(\mathcal{FI}\) of finite sets and injections.)
Important players are \(\mathcal{FI}^m\)-module homology and the semi-induced \(\mathcal{FI}^m\)-modules (those admitting a finite filtration with associated graded that is induced). The latter play a fundamental rôle in the theory; for instance, a finitely-generated \(\mathcal{FI}^m\)-module is acyclic for homology if and only if it is semi-induced [\textit{L. Li} and \textit{N. Yu}, J. Pure Appl. Algebra 223, No. 8, 3436--3460 (2019; Zbl 1411.13011)].
The authors work with the notion of \(B\)-torsion. This is intimately related to the behaviour of shift functors. (For any object \(\mathbf{a}\), the shift functor \(\Sigma _{\mathbf{a}}\) is induced by precomposition with \(-\amalg \mathbf{a}\) and there is a natural transformation \(\mathrm{Id} \rightarrow \Sigma _{\mathbf{a}}\).) A \(\mathcal{FI}^m\)-module \(V\) is \(B\)-torsion-free if and only if \(V \rightarrow \Sigma_{1, \ldots , 1} V\) is injective. In general there is a natural short exact sequence \[ 0 \rightarrow V_T \rightarrow V \rightarrow V_F \rightarrow 0 \] in which \(V_T\) is \(B\)-torsion and \(V_F\) is \(B\)-torsion-free.
Generalizing ideas of Nagpal (cf. [\textit{R. Nagpal}, Algebra Number Theory 13, No. 9, 2151--2189 (2019; Zbl 1461.20004)]), Li and Yu established a profound link between torsion and semi-induced modules. For example, for \(V\) a finitely-generated \(\mathcal{FI}^m\)-module and \(N \gg 0\), the natural transformation \(\mathrm{Id} \rightarrow \Sigma_{N, \ldots, N}\) induces an exact sequence \[ 0 \rightarrow V_T \rightarrow V \rightarrow \Sigma_{N, \ldots , N} V \] with \(\Sigma_{N, \ldots, N} V\) semi-induced. This allows the construction of coresolutions (up to \(B\)-torsion) using semi-induced modules.
The authors define \(H^0_B (V):= V_T\) and the local cohomology \(H^i_B(-)\) as the right derived functors. Then, using the above techniques, they show that the vanishing of \(H^{>0}_ B(-)\) for finitely-generated \(\mathcal{FI}^m\)-modules is equivalent to \(V_F\) being semi-induced.
They then introduce a notion of Castelnuovo-Mumford regularity for finitely-generated \(\mathcal{FI}^m\)-modules, \(\mathrm{CMreg}(V)\), related to the multi-graded regularity of [\textit{D. Maclagan} and \textit{G. G. Smith}, J. Reine Angew. Math. 571, 179--212 (2004; Zbl 1062.13004)] and prove an analogue of the main result of that paper. Namely, for \(V\) a finitely-generated \(\mathcal{FI}^m\)-module, \(\mathbf{r} \in \mathrm{CMreg}(V)\) and \(\mathbf{c}\) with positive entries, there exists a complex \[ \ldots \rightarrow F_i \rightarrow F_{i-1} \rightarrow \ldots \rightarrow F_0 \rightarrow V \rightarrow 0 \] that is exact up to \(B\)-torsion and such that \(F_i\) is an induced module generated in degree \(\mathbf{r} + i \mathbf{c}\). The proof exploits the Koszul complex that calculates \(\mathcal{FI}^m\)-module homology.
Reviewer: Geoffrey Powell (Angers)On cohomologically complete intersection moduleshttps://www.zbmath.org/1475.130302022-01-14T13:23:02.489162Z"Mahmood, Waqas"https://www.zbmath.org/authors/?q=ai:mahmood.waqasLet \(I\) denote an ideal of a local ring \((R,\mathfrak{m})\). A finitely generated \(R\)-module \(M\) is called cohomologically complete intersection with respect to \(I\) if \(H^i_I(M)\) vanishes for all \(i \not= \operatorname{grade} (I,M)\), where \(H^i_I(M)\) denotes the \(i\)-th local cohomology module of \(M\) with respect to \(I\). This is the extension of the notion of cohomologically complete intersections as introduced by \textit{M. Hellus} et al. [Semigroup Forum 97, No. 1, 64--74 (2018; Zbl 1471.20040)] for \(M= R\). This generalization to modules is defined as relative Cohen-Macaulay module with respect to \(I\) by \textit{M. R. Zargar} [J. Algebra Appl. 14, No. 3, Article ID 1550042, 7 p. (2015; Zbl 1307.13026)]. In the paper under review the author generalizes some the main results of the paper by Hellus et al. [loc. cit.], i.e. the characterization of cohomologically complete intersections in terms of some localizations of natural homomorphisms of certain Ext- resp. Tor-modules. The main technical tool is the so-called truncation complex as introduced by the reviewer [J. Algebra 275, No. 2, 751--770 (2004; Zbl 1103.13014)]. The present work generalzes the authors results in [\textit{W. Mahmood}, Math. Rep., Buchar. 20(70), No. 1, 39--50 (2018; Zbl 1399.13022)].
Reviewer: Peter Schenzel (Halle)Colocalization of formal local cohomology moduleshttps://www.zbmath.org/1475.130312022-01-14T13:23:02.489162Z"Rezaei, Shahram"https://www.zbmath.org/authors/?q=ai:rezaei.shahramLet \(\mathfrak{a}\) denote an ideal of a local ring \((R,\mathfrak{m})\). Let \(H^i_{\mathfrak{m}}(\cdot)\) denote the \(i\)-th local cohomology functor with respect to \(\mathfrak{m}\). For a finitely generated \(R\)-module \(M\) the reviewer introduced \(\mathfrak{F}^i_{\mathfrak{a}}(M) := \varprojlim H^i_{\mathfrak{m}}(M/\mathfrak{a}^nM)\) as the \(i\)-th formal local cohomology of \(M\) with respect to \(\mathfrak{a}\) [\textit{P. Schenzel}, J. Algebra 315, No. 2, 894--923 (2007; Zbl 1131.13018)]. Here the author contributes with structural results about the formal local cohomology modules: (1) The following conditions are equivalent: (i) \(\mathfrak{F}^i_{\mathfrak{a}}(M)\) is an Artinian \(R\)-module for all \(i < n\). (ii) \(\mathfrak{F}^i_{\mathfrak{a}}(M)\) is a representable \(R\)-module for all \(i < n\). (iii) The co-localization \({}_{\mathfrak{p}}(\mathfrak{F}^i_{\mathfrak{a}}(M))\) (in the sense of \textit{L. Melkersson} and \textit{P. Schenzel} [Proc. Edinb. Math. Soc., II. Ser. 38, No. 1, 121--131 (1995; Zbl 0824.13011)]) is a representable \(R_{\mathfrak{p}}\)-module for all \(i <n\) and all \(\mathfrak{p} \in \operatorname{Spec} R\). -- (2) \(\mathfrak{F}^i_{\mathfrak{a}}(M)\) is a minimax \(R\)-module for all \(i < n\) if and only if \(\operatorname{Cos}_R (\mathfrak{F}^i_{\mathfrak{a}}(M)) \subseteq V(\mathfrak{a}) \) for all \(i < n\). Here \(\operatorname{Cos}_R\) is the co-support defined by Melkersson and the reviewer (see [loc. cit.).
Reviewer: Peter Schenzel (Halle)Equivalent generating pairs of an ideal of a commutative ringhttps://www.zbmath.org/1475.130322022-01-14T13:23:02.489162Z"Guyot, Luc"https://www.zbmath.org/authors/?q=ai:guyot.lucThe author investigates two generated ideals in a commutative ring with unity. Let \(\mathrm{SL}_2(R)\) be the group of \(2\times 2\) matrices over \(R\) with determinant \(1\). They study the action of \(\mathrm{SL}_2(R)\) by matrix right-multiplication on \(V_2(I)\), the set of generating pairs of \(I\). \(\mathrm{Fitt}_1(I)\) is the second Fitting ideal of \(I\). The main result states that \(V_2(I)/\mathrm{SL}_2(R)\) identifies with a group of units of \(R/\text{Fitt}_1(I)\) via a natural generalization of the determinant if \(I\) can be generated by two regular (not zero-divisors) elements. The author provides several corollaries and other interesting propositions as well as showing the results in Bass rings. As an application, they derive a formula for the number of cusps of a modular group over a quadratic order. The article contains a nice background and history of the area to provide context to these generalizations as well as an extensive bibliography for the interested reader.
Reviewer: Christopher P. Mooney (Fulton)\(\omega \)-primality in arithmetic Leamer monoidshttps://www.zbmath.org/1475.130332022-01-14T13:23:02.489162Z"Chapman, Scott T."https://www.zbmath.org/authors/?q=ai:chapman.scott-thomas"Tripp, Zack"https://www.zbmath.org/authors/?q=ai:tripp.zackThe paper under review studies \(\omega\)-primality for Leamer monoids.
For an additive monoid \(S\) and some \(x \in S\), not a unit, one defines \(\omega(x)\) as the smallest \(m\) such that whenever \(x\) divides \(x_1 + \dots +x_k\) with \(x_i\in S\) there is a subset \(I\) of \(\{1, \dots, k\}\) of at most \(m\) elements such that \(x\) divides \(\sum_{i\in I}x_i\). If \(x\) is a prime element, then \(\omega(x)=1\). In a way the \(\omega\)-function allows to quantify how far an element is from being prime.
A numerical semigroup \(\Gamma\) is a submonoid of \((\mathbb{N}, +)\) with finite complement. For \(s \in \mathbb{N} \setminus \Gamma\) one defines \(S_{\Gamma}^s = \{(0,0)\} \cup \{(x,n) \colon \{x, x+s, x+2s, \dots, x+ns\} \subset \Gamma\} \subset \mathbb{N}^2\). This is a monoid called a Leamer monoid. The authors study the \(\omega\)-function for Leamer monoids, and more specifically Leamer monoids for \(\Gamma\) being generated by an arithmetical progression. In this case exact results are obtained.
Reviewer: Wolfgang A. Schmid (Paris)An elementary proof of the two-generator property for the ring of integer-valued polynomialshttps://www.zbmath.org/1475.130342022-01-14T13:23:02.489162Z"Boulanger, Jacques"https://www.zbmath.org/authors/?q=ai:boulanger.jacques"Chabert, Jean-Luc"https://www.zbmath.org/authors/?q=ai:chabert.jean-lucLet \(\text{Int}(\mathbb{Z})\) be the ring of integer-valued polynomials, that is, \[\text{Int}(\mathbb{Z})=\{f(X)\in \mathbb{Q}[X]~|~f(\mathbb{Z})\subseteq \mathbb{Z}\}.\] In [\textit{R. Gilmer} and \textit{W. W. Smith}, J. Algebra 81, No. 1, 150--164 (1983, Zbl 0515.13016)], it is shown, by constructive methods, that \(\text{Int}(\mathbb{Z})\) has the two-generator property, that is, every finitely generated ideal of \(\text{Int}(\mathbb{Z})\) is generated by two elements. In this paper, the auothers present an elementary proof of this known result.
Reviewer: Qiao Lei (Chengdu)On resurgence via asymptotic resurgencehttps://www.zbmath.org/1475.130352022-01-14T13:23:02.489162Z"DiPasquale, Michael"https://www.zbmath.org/authors/?q=ai:dipasquale.michael-r"Drabkin, Ben"https://www.zbmath.org/authors/?q=ai:drabkin.benLet \(I\) denote an ideal of the polynomial ring \(R\) over a field \(K\). For an integer \(s \geq 1\) let \(I^s\) resp \(I^{(s)}\) the \(s\)-th ordinary resp. the \(s\)-th symbolic power of \(I\). Clearly \(I^s \subseteq I^{(s)}\) for all \(s \geq 1\). In the paper [Math. Nachr. 129, 123--148 (1986; Zbl 0606.13001)] by the reviewer it is shown that for each \(r\) there is an integer \(s \geq r\) such that \(I^{(s)} \subseteq I^r\). The present author posed the problem to describe the smallest value \(f(r) := \min \{s \mid I^{(s)} \subseteq I^r\}\). Among others \textit{I. Swanson} [Math. Z. 234, No. 4, 755--775 (2000; Zbl 1010.13015)] proved that \(f(r)\) is bounded above by a linear function on \(r\). Under the above assumptions on \(R\) by the work of \textit{L. Ein} et al. [Invent. Math. 144, No. 2, 241--252 (2001; Zbl 1076.13501)] it follows that \(f(r) \leq hr\), where \(h\) denotes the big height of \(I\). For more general rings further results are obtained by \textit{M. Hochster} and \textit{C. Huneke} [Invent. Math. 147, No. 2, 349--369 (2002; Zbl 1061.13005)] and by \textit{L. Ma} and \textit{K. Schwede} [Invent. Math. 214, No. 2, 913--955 (2018; Zbl 1436.13009)]. A subtle point of view was initiated in the work of \textit{B. Harbourne} and \textit{C. Huneke} [J. Ramanujan Math. Soc. 28A, 247--266 (2013; Zbl 1296.13018)] with several research articles in recent times. This leads to two ``statistical quantities'', namely the resurgence \(\rho(I) = \sup \{s/r \mid I^{(s)} \not\subseteq I^r\}\) and the asymptotic resurgence \(\hat{\rho}(I) = s/r \mid I^{(st)} \not\subseteq I^{rt} \mbox{ for all } t \gg 0\}\). It follows that \(\hat{\rho}(I) \leq \rho(I) \leq h\). A problem is the rationality of the resurgence, answered in the affirmative by the authors provided the symbolic Rees algebra is finitely generated as an algebra. This is a consequence of the following: If \( \hat{\rho}(I) < \rho(I)\), then \( \rho(I)\) is a maximum instead of a supremum. This follows by the authors result about two bounds on asymptotic resurgence given a single known containment between a symbolic and ordinary power. With that they deduce subsequent criteria for expected resurgence, (that is, \( \rho(I) < h\)). As an application it is shown that squarefree monomial ideals have expected resurgence.
Reviewer: Peter Schenzel (Halle)Polarization of neural ringshttps://www.zbmath.org/1475.130362022-01-14T13:23:02.489162Z"Güntürkün, Sema"https://www.zbmath.org/authors/?q=ai:gunturkun.sema"Jeffries, Jack"https://www.zbmath.org/authors/?q=ai:jeffries.jack"Sun, Jeffrey"https://www.zbmath.org/authors/?q=ai:sun.jeffrey\(\mathbb{A}_{\mathrm{inf}}\) is infinite dimensionalhttps://www.zbmath.org/1475.130372022-01-14T13:23:02.489162Z"Lang, Jaclyn"https://www.zbmath.org/authors/?q=ai:lang.jaclyn"Ludwig, Judith"https://www.zbmath.org/authors/?q=ai:ludwig.judithSummary: Given a perfect valuation ring \(R\) of characteristic \(p\) that is complete with respect to a rank-\(1\) nondiscrete valuation, we show that the ring \(\mathbb{A}_\mathrm{{\inf}}\) of Witt vectors of \(R\) has infinite Krull dimension.Doset Hibi rings with an application to invariant theoryhttps://www.zbmath.org/1475.130382022-01-14T13:23:02.489162Z"Miyazaki, Mitsuhiro"https://www.zbmath.org/authors/?q=ai:miyazaki.mitsuhiroAuthor's abstract: We define the concept of a doset Hibi ring and a generalized doset Hibi ring which are subrings of a Hibi ring and are normal affine semigroup rings. We apply the theory of (generalized) doset Hibi rings to analyze the rings of absolute orthogonal invariants and absolute special orthogonal invariants and show that these rings are normal and Cohen-Macaulay and has rational singularities if the characteristic of the base field is zero and is \(F\)-rational otherwise. We also state criteria of Gorenstein property of these rings.
Reviewer: Kriti Goel (Mumbai)Infinite families of equivariantly formal toric orbifoldshttps://www.zbmath.org/1475.130392022-01-14T13:23:02.489162Z"Bahri, Anthony"https://www.zbmath.org/authors/?q=ai:bahri.anthony-p"Sarkar, Soumen"https://www.zbmath.org/authors/?q=ai:sarkar.soumen"Song, Jongbaek"https://www.zbmath.org/authors/?q=ai:song.jongbaekIn this paper, the analysis of the simplicial wedge construction on simplicial complexes and simple polytopes has been extended to the case of toric orbifolds. The authors present a class of infinite families of toric orbifolds which is derived from a given one by simplicial wedge construction. They show in their main result (Theorem 5.5) that its integral cohomology is free of torsion and is concentrated in even degrees.
Reviewer: Irem Portakal (Magdeburg)Cohen-Macaulay edge-weighted edge ideals of very well-covered graphshttps://www.zbmath.org/1475.130402022-01-14T13:23:02.489162Z"Seyed Fakhari, Seyed Amin"https://www.zbmath.org/authors/?q=ai:seyed-fakhari.seyed-amin"Shibata, Kosuke"https://www.zbmath.org/authors/?q=ai:shibata.kosuke"Terai, Naoki"https://www.zbmath.org/authors/?q=ai:terai.naoki"Yassemi, Siamak"https://www.zbmath.org/authors/?q=ai:yassemi.siamakSummary: We characterize unmixed and Cohen-Macaulay edge-weighted edge ideals of very well-covered graphs. We also provide examples of oriented graphs that have unmixed and non-Cohen-Macaulay vertex-weighted edge ideals, while the edge ideal of their underlying graph is Cohen-Macaulay. This disproves a conjecture posed by \textit{Y. Pitones} et al. [Electron. J. Comb. 26, No. 3, Research Paper P3.44, 18 p. (2019; Zbl 1419.05099)].Freiman cover ideals of unmixed bipartite graphshttps://www.zbmath.org/1475.130412022-01-14T13:23:02.489162Z"Zhu, Guangjun"https://www.zbmath.org/authors/?q=ai:zhu.guangjun"Zhao, Yakun"https://www.zbmath.org/authors/?q=ai:zhao.yakun"Cui, Yijun"https://www.zbmath.org/authors/?q=ai:cui.yijunSummary: An equigenerated monomial ideal \(I\) in the polynomial ring \(R=k[z_1,\ldots, z_n]\) is a Freiman ideal if \(\mu(I^2)=\ell(I)\mu(I)-\binom{\ell(I)} {2}\), where \(\ell(I)\) is the analytic spread of \(I\) and \(\mu(I)\) is the number of the minimal generators of \(I\). In this paper we classify all simple connected unmixed bipartite graphs whose cover ideals are Freiman ideals.Binomial edge ideals and bounds for their regularityhttps://www.zbmath.org/1475.130422022-01-14T13:23:02.489162Z"Kumar, Arvind"https://www.zbmath.org/authors/?q=ai:kumar.arvindThe study of algebraic invariants of (binomial) edge ideals in terms of combinatorial properties of the associated graph has a long tradition in Combinatorial Commutative Algebra.
Let \(S=K[x_1,\dots,x_n]\) be the standard polynomial ring, with \(K\) a field. Let \(G\) be a finite simple graph on vertex set \([n]:=\{1,2,\dots,n\}\), \(J_G\) the binomial edge ideal associated to \(G\) and \(c(G)\) be the number od maximal cliques in \(G\). In this paper, the author focuses its attention to the following conjecture due to Madani and Kiani:\\
\((*)\): Let \(G\) be a simple graph on \([n]\). Then, \(\text{reg}(S/J_G)\le c(G)\).
In particular, it proves conjecture \((*)\) for quasi-block graphs, semi-block graphs and gives a new proof for the class of chordal graphs. Finally, the regularity of Jahangir graphs is computed.
It would be nice to settle conjecture \((*)\) in full generality.
Reviewer: Antonino Ficarra (Messina)Some generalizations of strongly prime idealshttps://www.zbmath.org/1475.130432022-01-14T13:23:02.489162Z"Ansari-Toroghy, Habibollah"https://www.zbmath.org/authors/?q=ai:ansari-toroghy.habibollah"Farshadifar, Faranak"https://www.zbmath.org/authors/?q=ai:farshadifar.faranak"Maleki-Roudposhti, Sepideh"https://www.zbmath.org/authors/?q=ai:maleki-roudposhti.sepidehSummary: In this paper, we introduce the concepts of strongly 2-absorbing primary ideals (resp., submodules) and strongly 2-absorbing ideals (resp., submodules) as generalizations of strongly prime ideals. Furthermore, we investigate some basic properties of these classes of ideals.Almost canonical ideals and GAS numerical semigroupshttps://www.zbmath.org/1475.130442022-01-14T13:23:02.489162Z"D'Anna, Marco"https://www.zbmath.org/authors/?q=ai:danna.marco"Strazzanti, Francesco"https://www.zbmath.org/authors/?q=ai:strazzanti.francescoThis manuscript collects and relates several approaches given in the literature to the generalization of the concept of symmetry in numerical semigroups. In all these approaches the underlying algebraic geometric motivation is also described.
The authors realized that the families of symmetric, almost symmetric and 2-AGL semigroups can be gathered in a single family, which they call generalized almost symmetric numerical semigroups.
Let \(S\) is a numerical semigroup, that is, a submonoid of the set of nonnegative integers under addition for which \(\operatorname{F}(S)=\max(\mathbb{Z}\setminus S)\) exists (called the Frobenius number of \(S\)). Let \(M=S\setminus\{0\}\) and \(K=\{ x\in \mathbb{N} \mid \operatorname{F}(S)-x\not\in S\}\) be its maximal and canonical ideal, respectively. The authors recall that the symmetry, almost symmetry and the 2-AGL property of \(S\) can be characterized in terms of \(K\) and \(M\). More precisely, (1) \(S\) is symmetric if and only if \(S=K\) (equivalently \(2K=K\)), (2) \(S\) is almost symmetric if and only if \(S-M=K\cup \{ \operatorname{F}(S)\}\) (equivalently \(2K\setminus K\subseteq \{\operatorname{F}(S)\}\)), and (3) \(S\) is 2-AGL if and only if \(2K=3K\) and \(|2K\setminus K|=2\). The authors prove that \(S\) is 2-AGL if and only if \(2K\setminus K=\{\operatorname{F}(S)-x, \operatorname{F}(S)\}\) for some minimal generator \(x\) of \(S\). By looking at these characterizations of symmetric, almost symmetric and 2-AGL numerical semgigroups, the authors propose the following generalization: \(S\) is said to be a generalized almost symmetric numerical semigroup if either \(2K=K\) of \(2K\setminus K=\{\operatorname{F}(S)-x_1,\dots, \operatorname{F}(S)-x_r,\operatorname{F}(S)\}\) for some \(x_1,\dots,x_r \in M\setminus 2M\) (minimal generators) and \(x_i-x_j\not\in (S-M)\setminus S\) (not pseudo-Frobenius numbers). The authors provide several examples and the study of particular cases such as being of maximal embedding dimension.
Again, inspired by the existing results in the literature describing the relationship between \(M-M\) and \(M-e\) and the concepts of symmetry and almost symmetry, the authors show that \(S\) is a generalized almost symmetric numerical semigroup if and only if \(M-e\) is an almost canonical ideal of \(M-M\). An ideal \(I\) of a semigroup \(S\) is almost canonical if \(I-M=K-M\) (just paraphrasing one of the many characterizations of almost canonical ideals given in the second section of the manuscript).
The last section is devoted to show how the property of being a generalized almost symmetric numerical semigroup behaves under gluings, numerical duplication, and dilatations.
Reviewer: Pedro A. García Sánchez (Granada)On Ratliff-Rush closure of moduleshttps://www.zbmath.org/1475.130452022-01-14T13:23:02.489162Z"Endo, Naoki"https://www.zbmath.org/authors/?q=ai:endo.naokiFor an ideal \(I\) in a Noetherian ring \(A\), the Ratliff-Rush closure \(\widetilde{I}\) of \(I\) given by
\[
\widetilde{I} := \bigcup_{\ell>0} [I^{\ell+1}:_A I^\ell]
\]
was introduced by \textit{L. J. Ratliff jun.} and \textit{D. E. Rush} [Indiana Univ. Math. J. 27, 929--934 (1978; Zbl 0368.13003)]. It is known that \(\widetilde{I}\) is the largest ideal in \(A\) such that \(I^n = (\widetilde{I})^n\) for \(n \gg 0.\) The present paper deals with the Ratliff-Rush closure of modules. \vskip 2mm Let \(M\) be a finitely generated \(A\)-module which is contained in a free module \(F\) of finite rank \(r>0.\) Let Sym\(_A(M)\) and Sym\(_A(F)\) denote the symmetric algebras of \(M\) and \(F\), respectively. Recall that the Rees algebra \(\mathcal{R}(M)\) of \(M\) is the image of Sym\((i):\) Sym\(_A(M) \to \) Sym\(_A(F)\) induced by the embedding \(i: M \hookrightarrow F.\) Let \(M^n\) be the \(n\)-the homogeneous component of the Rees algebra \(\mathcal{R}(M)\). In the case where \(A\) is a Noetherian domain, \textit{J.-C. Liu} [J. Algebra 201, No. 2, 584--603 (1998; Zbl 0918.13002)] defined the Ratliff-Rush closure \(\widetilde{M}\) of \(M\) to be the largest \(A\)-submodule \(N\) of \(F\), which satisfies \(M \subseteq N \subseteq F\) and \(M^n=N^n\) for \(n \gg 0.\) In the paper under review the author defines the Ratliff-Rush closure \(\widetilde{M}\) of \(M\) for arbitrary Noetherian ring as
\[
\widetilde{M} = \bigcup_{\ell>0} [M^{\ell+1}:_F M^\ell].
\]
The author shows that this definition coincides with the definition of Liu when \(A\) is a domain. \textit{S. Goto} and \textit{N. Matsuoka} [``The Rees algebras of ideals in two-dimensional regular local rings'', Proceedings of the 27th Symposium on Commutative Algebra. 81--89 (2006)] have investigated when does the Ratliff-Rush closure \(\widetilde{I}\) of \(I\) coincides with the integral closure \(\overline{I} \) of \(I\). They proved that in a two-dimensional regular local ring \(A,\) \(\widetilde{I}=\overline{I}\) if and only if the Rees algebra \(\mathcal{R}(I)\) of \(I\) is locally normal on Spec \(\mathcal{R}(I) \setminus \{\mathcal{M}\}\) where \(\mathcal{M}\) denotes the graded maximal ideal in \(\mathcal{R}(I).\) The author's aim in this paper is to investigate when does the Ratliff-Rush closure of \(M\) coincides with the integral closure \(\overline{M}\) of \(M\). In a two-dimensional regular local ring he proves, among other equivalent conditions, that \(\widetilde{M}=\overline{M}\) if and only if for every \(P \in \) Spec \(\mathcal{R}(M) \setminus \{ \mathcal M\}\), \(\mathcal{R}(M)_P\) is normal. As a consequence he provides a criterion for \(\mathcal{R}(M)\) to be Buchsbaum for a module \(M\) with \(\widetilde{M}=\overline{M}.\) He also produces several examples of Buchsbaum Rees algebras, including an example of Buchsbaum Rees algebra \(\mathcal{R}(M) \) where \(M\) is an indecomposable module.
Reviewer: Shreedevi K. Masuti (Dharwad)Ulrich ideals and 2-AGL ringshttps://www.zbmath.org/1475.130462022-01-14T13:23:02.489162Z"Goto, Shiro"https://www.zbmath.org/authors/?q=ai:goto.shiro"Isobe, Ryotaro"https://www.zbmath.org/authors/?q=ai:isobe.ryotaro"Taniguchi, Naoki"https://www.zbmath.org/authors/?q=ai:endo.naokiLet \((R, \mathfrak m)\) be a Cohen-Macaulay local ring of dimension \(1\) and let \(K_R\) be its canonical module. Let \(I\) be a canonical ideal of \(R\) which means \(I\simeq K_R.\) Let \(Q=(a)\) be a minimal reduction of \(I\). Let \(\mathcal R=\mathcal R(I)\) and \(\mathcal T=\mathcal R(Q)\) be the Rees algebras of \(I\) and \(Q\) respectively. Let \(S_Q(I)=I{\mathcal R}/I {\mathcal T}\) be the Sally module of \(I\) with respect to \(Q.\) For all large \(n,\) the length \(\ell(R/I^{n+1})\) is given by a polynomial \(P_I(n)\) written as \(P_I(n)=e_0(I)\binom{n+1}{1}-e_1(I).\) Here \(e_0(I), e_1(I)\) are called the Hilbert coefficients of \(R\) with respect to \(I.\) The rank of \(S_Q(I)\) is defined to be \(\ell_{T_{\mathfrak p}}[S_Q(I)]_{\mathfrak p}\) where \(\mathfrak p=\mathfrak m {\mathcal T}.\) We say that \(R\) is a 2-AGL ring if \(\text{rank } S_Q(I)=2.\) This is equivalent to \(e_1(I)=e_0(I)-\ell(R/I)+2.\) \textit{T. D. M. Chau} et al. [J. Algebra 521, 299--330 (2019; Zbl 1454.13043)] obtained several characterisations of 2-AGL rings in terms of reduction number of \(I\) with respect to \(Q\) and co-length of the canonical fractional ideal \(K\) and the conductor ideal \(\mathcal C\) of \(R[K]\) into \(R.\) The authors consider the minimal presentations of \(K\) to characterise 2-AGL rings. It is proved that \(R\) is a 2-AGL ring if and only if the finer product \(R\times_{R/\mathcal C} R\) is so.
Ulrich ideals are studied in 2-AGL rings. Let \(R\) be a 2-AGL ring with minimal multiplicity and let \(K\) be its canonical fractional ideal. If \(K/R\) is \(R/{\mathcal C}\)-free then \({\mathcal C}\) and \(\mathfrak m\) are the only Ulrich ideals of \(R.\) In case \(K/R\) is not \(R/{\mathcal C}\)-free then the only Ulrich ideal is \(\mathfrak m.\)
Reviewer: Jugal K. Verma (Mumbai)Nearly Gorenstein rings arising from finite graphshttps://www.zbmath.org/1475.130472022-01-14T13:23:02.489162Z"Hibi, Takayuki"https://www.zbmath.org/authors/?q=ai:hibi.takayuki"Stamate, Dumitru I."https://www.zbmath.org/authors/?q=ai:stamate.dumitru-ioanLet \(R\) be a Cohen-Macaulay graded \(k\)-algebra over a field \(k\) with a canonical module \(\omega_R\). The trace of \(\omega_R\) is the ideal generated by the images of \(\omega_R\) through all the homomorphisms of \(R\)-modules into \(R\), and it does not depend on the choice of \(\omega_R\). Following [\textit{J. Herzog} et al., Isr. J. Math. 233, No. 1, 133--165 (2019; Zbl 1428.13037)], a ring is said to be nearly Gorenstein if the trace of \(\omega_R\) contains the maximal graded ideal of \(R\). This notion generalizes the notion of Gorenstein rings, which are exactly the rings that are equal to the traces of their canonical modules.
In this paper the authors classify the complete multipartite graphs whose edge rings are nearly Gorenstein and the perfect graphs whose stable set rings are nearly Gorenstein.
Reviewer: Francesco Strazzanti (Torino)Type and conductor of simplicial affine semigroupshttps://www.zbmath.org/1475.130482022-01-14T13:23:02.489162Z"Jafari, Raheleh"https://www.zbmath.org/authors/?q=ai:jafari.raheleh"Yaghmaei, Marjan"https://www.zbmath.org/authors/?q=ai:yaghmaei.marjanLet \(S\), \(\mathbb{K}\) and \(\mathbb{K}[S]\) be a simplicial affine semigroup, a field and the affine semigroup ring, respectively. In this paper, the auhtors characterize the property of being Cohen-Macaulay and Buchsbaum using the Apery set of \(S\) for the ring \(\mathbb{K}[S]\).
They also prove that if \(\mathbb{K}[S]\) is Cohen-Macaulay then the type of \(S\) is bounded and they give this boundary. Morever, it is shown that if this ring is not Cohen-Macaulay, then the type of \(S\) is unbounded.
If we define the normalization of \(S\) as \(\bar{S}=\{a\in\mbox{group}(S):na\in S\mbox{ for some }n\in\mathbb{N}\}\), we say that \(S\) is normal if \(S=\bar{S}\). Characterization of normal simplicial affine semigroups can be found in this paper using its Apery set.
Reviewer: Daniel Marín Aragon (Cádiz)Multiplicity of the saturated special fiber ring of height three Gorenstein idealshttps://www.zbmath.org/1475.130492022-01-14T13:23:02.489162Z"Cid-Ruiz, Yairon"https://www.zbmath.org/authors/?q=ai:cid-ruiz.yairon"Mukundan, Vivek"https://www.zbmath.org/authors/?q=ai:mukundan.vivekSummary: Let \(R\) be a polynomial ring over a field and let \(I \subset R\) be a Gorenstein ideal of height three that is minimally generated by homogeneous polynomials of the same degree. We compute the multiplicity of the \textit{saturated special fiber ring} of \(I\). The obtained formula depends only on the number of variables of \(R\), the minimal number of generators of \(I\), and the degree of the syzygies of \(I\). Applying results from \textit{L. Busé} et al. [Proc. Lond. Math. Soc. (3) 121, No. 4, 743--787 (2020; Zbl 1454.13017)] we get a formula for the \(j\)-multiplicity of \(I\) and an effective method to study a rational map determined by a minimal set of generators of \(I\).Toward the best algorithm for approximate GCD of univariate polynomialshttps://www.zbmath.org/1475.130502022-01-14T13:23:02.489162Z"Nagasaka, Kosaku"https://www.zbmath.org/authors/?q=ai:nagasaka.kosakuSummary: Approximate polynomial GCD (greatest common divisor) of polynomials with a priori errors on their coefficients, is one of interesting problems in Symbolic-Numeric Computations. In fact, there are many known algorithms: QRGCD, UVGCD, STLN based methods, Fastgcd and so on. The fundamental question of this paper is ``which is the best?'' from the practical point of view, and subsequently ``is there any better way?'' by any small extension, any effect by pivoting, and any combination of sub-routines along the algorithms. In this paper, we consider a framework that covers those algorithms and their sub-routines, and makes their sub-routines being interchangeable between the algorithms (i.e. disassembling the algorithms and reassembling their parts). By this framework along with/without small new extensions and a newly adapted refinement sub-routine, we have done many performance tests and found the current answer. In summary, 1) UVGCD is the best way to get smaller tolerance, 2) modified Fastgcd is better for GCD that has one or more clusters of zeros with large multiplicity, and 3) modified ExQRGCD is better for GCD that has no cluster of zeros.Multilinear polynomial systems: root isolation and bit complexityhttps://www.zbmath.org/1475.130512022-01-14T13:23:02.489162Z"Emiris, Ioannis Z."https://www.zbmath.org/authors/?q=ai:emiris.ioannis-z"Mantzaflaris, Angelos"https://www.zbmath.org/authors/?q=ai:mantzaflaris.angelos"Tsigaridas, Elias P."https://www.zbmath.org/authors/?q=ai:tsigaridas.elias-pSummary: We exploit structure in polynomial system solving by considering polynomials that are linear in subsets of the variables. We focus on algorithms and their Boolean complexity for computing isolating hyperboxes for all the isolated complex roots of well-constrained, unmixed systems of multilinear polynomials based on resultant methods. We enumerate all expressions of the multihomogeneous (or multigraded) resultant of such systems as a determinant of Sylvester-like matrices, aka \textit{generalized Sylvester matrices}. We construct these matrices by means of Weyman homological complexes, which generalize the Cayley-Koszul complex. The computation of the determinant of the resultant matrix is the bottleneck for the overall complexity. We exploit the quasi-Toeplitz structure to reduce the problem to efficient matrix-vector multiplication, which corresponds to multivariate polynomial multiplication, by extending the seminal work on Macaulay matrices of [\textit{J. F. Canny} et al., ``Solving systems of nonlinear polynomial equations faster'', in: Proceedings of the ACM-SIGSAM 1989 international symposium on Symbolic and algebraic computation, ISSAC '89. New York, NY: Association for Computing Machinery (ACM). 121--128 (1989; \url{doi:10.1145/74540.74556})] to the multihomogeneous case. We compute a rational univariate representation of the roots, based on the primitive element method. In the case of 0-dimensional systems we present a Monte Carlo algorithm with probability of success \(1 - 1 / 2^\varrho \), for a given \(\varrho \geq 1\), and bit complexity \(\widetilde{\mathcal{O}}_B( n^2 D^{4 + \epsilon}( n^{N + 1} + \tau) + n D^{2 + \epsilon} \varrho(D + \varrho))\) for any \(\epsilon > 0\), where \(n\) is the number of variables, \(D\) equals the multilinear Bézout bound, \(N\) is the number of variable subsets, and \(\tau\) is the maximum coefficient bitsize. We present an algorithmic variant to compute the isolated roots of overdetermined and positive-dimensional systems. Thus our algorithms and complexity analysis apply in general with no assumptions on the input.Homological characterization of bounded \(\mathbb{F}_2\)-regularityhttps://www.zbmath.org/1475.130522022-01-14T13:23:02.489162Z"Hodges, Timothy J."https://www.zbmath.org/authors/?q=ai:hodges.timothy-j"Molina, Sergio D."https://www.zbmath.org/authors/?q=ai:molina.sergio-dSummary: Semi-regular sequences over \(\mathbb{F}_2\) are sequences of homogeneous elements of the algebra \(B^{(n)}=\mathbb{F}_2[X_1,\dots,X_n]/(X_12,\dots,X_n^2)\), which have as few relations between them as possible. It is believed that most such systems are \(\mathbb{F}_2\)-semi-regular and this property has important consequences for understanding the complexity of Gröbner basis algorithms such as \textbf{F4} and \textbf{F5} for solving such systems. In fact even in one of the simplest and most important cases, that of quadratic sequences of length \(n\) in \(n\) variables, the question of the existence of semi-regular sequences for all \(n\) remains open. In this paper we present a new framework for the concept of \(\mathbb{F}_2\)-semi-regularity which we hope will allow the use of ideas and machinery from homological algebra to be applied to this interesting and important open question. First we introduce an analog of the Koszul complex and show that \(\mathbb{F}_2\)-semi-regularity can be characterized by the exactness of this complex. We show how the well known formula for the Hilbert series of a \(\mathbb{F}_2\)-semi-regular sequence can be deduced from the Koszul complex. Finally we show that the concept of first fall degree also has a natural description in terms of the Koszul complex.Symmetric ideals, Specht polynomials and solutions to symmetric systems of equationshttps://www.zbmath.org/1475.130532022-01-14T13:23:02.489162Z"Moustrou, Philippe"https://www.zbmath.org/authors/?q=ai:moustrou.philippe"Riener, Cordian"https://www.zbmath.org/authors/?q=ai:riener.cordian"Verdure, Hugues"https://www.zbmath.org/authors/?q=ai:verdure.huguesSummary: An ideal of polynomials is symmetric if it is closed under permutations of variables. We relate general symmetric ideals to the so called Specht ideals generated by all Specht polynomials of a given shape. We show a connection between the leading monomials of polynomials in the ideal and the Specht polynomials contained in the ideal. This provides applications in several contexts. Most notably, this connection gives information about the solutions of the corresponding set of equations. From another perspective, it restricts the isotypic decomposition of the ideal viewed as a representation of the symmetric group.A Morse theoretic approach to non-isolated singularities and applications to optimizationhttps://www.zbmath.org/1475.130542022-01-14T13:23:02.489162Z"Maxim, Laurentiu G."https://www.zbmath.org/authors/?q=ai:maxim.laurentiu-g"Rodriguez, Jose Israel"https://www.zbmath.org/authors/?q=ai:rodriguez.jose-israel"Wang, Botong"https://www.zbmath.org/authors/?q=ai:wang.botongLet \(X\) be a smooth submanifold of \(\mathbb C^N\) and let \(f\vert X\) be a nonconstant polynomial function with isolated critical points \(P_1, \dots , P_l\). Given a generic linear function on \(\mathbb C^N\), the perturbation \(f_t = f-tg\) is a holomorphic Morse function for all but finitely many \(t \in \mathbb C\). In this case the limit of the critical points of \(f_t\) is equal to the total sum \(\sum n_iP_i\), where \(n_i\) is the Milnor number of \(f\vert X\) at \(P_i\). The authors generalize this easy result to the case of arbitrary complex affine varieties. First, they define a limit of the set \(\mathrm{Crit}(f_t\vert X_{\mathrm{reg}})\), \(t \rightarrow 0\), of critical points of the restriction of \(f_t\) to the regular locus of \(X \subset \mathbb C^N\) and then prove that the limit is equal to the sum \(\sum n_i\mathrm{Crit}(g\vert X_i)\), where \(X_i\) are smooth locally closed subvarieties which are components of a suitable stratification given by the Lagrangian cycles of the perverse vanishing cycle functors \(^p \Phi_{f-c}([T^\ast_X\mathbb C^N])\) locally constant along \(X_i\) for all values of \(c \in \mathbb C\) and all \(i\), while \(n_i\) are nonnegative integers equal to the multiplicities of the corresponding components \([T^\ast_{\overline{X_i}}\mathbb C^N]\) of the associated characteristic cycle [\textit{D. B. Massey}, Topology Appl. 103, No. 1, 55--93 (2000; Zbl 0952.32019)]. The authors also discuss several concrete examples, some applications in convex algebraic geometry and related topics similarly to [\textit{H.-C. Graf von Bothmer} and \textit{K. Ranestad}, Bull. Lond. Math. Soc. 41, No. 2, 193--197 (2009; Zbl 1185.14047); \textit{D. B. Massey}, Numerical control over complex analytic singularities. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1025.32010)].
Reviewer: Aleksandr G. Aleksandrov (Moskva)Structural results on harmonic rings and lessened ringshttps://www.zbmath.org/1475.140042022-01-14T13:23:02.489162Z"Tarizadeh, Abolfazl"https://www.zbmath.org/authors/?q=ai:tarizadeh.abolfazl"Aghajani, Mohsen"https://www.zbmath.org/authors/?q=ai:aghajani.mohsenSummary: In this paper, a combination of algebraic and topological methods are applied to obtain new and structural results on harmonic rings. Especially, it is shown that if a Gelfand ring \(A\) modulo its Jacobson radical is a zero dimensional ring, then \(A\) is a clean ring. It is also proved that, for a given Gelfand ring \(A\), then the retraction map \(\mathrm{Spec}(A)\rightarrow\mathrm{Max}(A)\) is flat continuous if and only if \(A\) modulo its Jacobson radical is a zero dimensional ring. Dually, it is proved that for a given mp-ring \(A\), then the retraction map \(\mathrm{Spec}(A)\rightarrow\mathrm{Min}(A)\) is Zariski continuous if and only if \(\mathrm{Min}(A)\) is Zariski compact. New criteria for zero dimensional rings, mp-rings and Gelfand rings are given. The new notion of lessened ring is introduced and studied which generalizes ``reduced ring'' notion. Especially, a technical result is obtained which states that the product of a family of rings is a lessened ring if and only if each factor is a lessened ring. As another result in this spirit, the structure of locally lessened mp-rings is also characterized. Finally, it is characterized that a given ring \(A\) is a finite product of lessened quasi-prime rings if and only if \(\mathrm{Ker}\pi_{\mathfrak{p}}\) is a finitely generated and idempotent ideal for all \(\mathfrak{p}\in\mathrm{Min}(A)\).Dedekind sums and parsing of Hilbert serieshttps://www.zbmath.org/1475.140132022-01-14T13:23:02.489162Z"Zhou, Shengtian"https://www.zbmath.org/authors/?q=ai:zhou.shengtianSummary: Given a polarized variety \(X,D\), we can associate a graded ring and a Hilbert series. Assume \(D\) is an ample \(\mathbb{Q}\) Cartier divisor, and \(X,D\) is quasi smooth and projectively Gorenstein, we give a parsing formula for the Hilbert series according to their singularities. Here we allow the variety to have singularities of dimension \(\le 1\), that is, both singularities of dimension \(1\) and singular points, extending a result of \textit{A. Buckley} et al. [Izv. Math. 77, No. 3, 461--486 (2013; Zbl 1273.14023); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 77, No. 3, 29--54 (2013)] about varieties with only isolated singularities.Rationality problem for norm one torihttps://www.zbmath.org/1475.140252022-01-14T13:23:02.489162Z"Hoshi, Akinari"https://www.zbmath.org/authors/?q=ai:hoshi.akinari"Yamasaki, Aiichi"https://www.zbmath.org/authors/?q=ai:yamasaki.aiichiSummary: We classify stably/retract rational norm one tori in dimension \(p-1\) where \(p\) is a prime number and in dimension up to ten with some minor exceptions.Quasiexcellence implies strong generationhttps://www.zbmath.org/1475.140302022-01-14T13:23:02.489162Z"Aoki, Ko"https://www.zbmath.org/authors/?q=ai:aoki.koSummary: We prove that the bounded derived category of coherent sheaves on a quasicompact separated quasiexcellent scheme of finite dimension has a strong generator in the sense of \textit{A. Bondal} and \textit{M. van den Bergh} [Mosc. Math. J. 3, No. 1, 1--36 (2003; Zbl 1135.18302)]. This simultaneously extends two results of \textit{S. B. Iyengar} and \textit{R. Takahashi} [Int. Math. Res. Not. 2016, No. 2, 499--535 (2016; Zbl 1355.13015)] and \textit{A. Neeman} [Ann. Math. (2) 193, No. 3, 689--732 (2021; Zbl 07353240)] and is new even in the affine case. The main ingredient includes Gabber's weak local uniformization theorem and the notions of boundedness and descendability of a morphism of schemes.Cubic surfaces of characteristic twohttps://www.zbmath.org/1475.140652022-01-14T13:23:02.489162Z"Kadyrsizova, Zhibek"https://www.zbmath.org/authors/?q=ai:kadyrsizova.zhibek"Kenkel, Jennifer"https://www.zbmath.org/authors/?q=ai:kenkel.jennifer"Page, Janet"https://www.zbmath.org/authors/?q=ai:page.janet"Singh, Jyoti"https://www.zbmath.org/authors/?q=ai:singh.jyoti-prakash"Smith, Karen E."https://www.zbmath.org/authors/?q=ai:smith.karen-e"Vraciu, Adela"https://www.zbmath.org/authors/?q=ai:vraciu.adela-n"Witt, Emily E."https://www.zbmath.org/authors/?q=ai:witt.emily-eSummary: Cubic surfaces in characteristic two are investigated from the point of view of prime characteristic commutative algebra. In particular, we prove that the non-Frobenius split cubic surfaces form a linear subspace of codimension four in the 19-dimensional space of all cubics, and that up to projective equivalence, there are finitely many non-Frobenius split cubic surfaces. We explicitly describe defining equations for each and characterize them as extremal in terms of configurations of lines on them. In particular, a (possibly singular) cubic surface in characteristic two fails to be Frobenius split if and only if no three lines on it form a ``triangle''.Non-Ulrich representation typehttps://www.zbmath.org/1475.140792022-01-14T13:23:02.489162Z"Faenzi, Daniele"https://www.zbmath.org/authors/?q=ai:faenzi.daniele"Malaspina, Francesco"https://www.zbmath.org/authors/?q=ai:malaspina.francesco"Sanna, Giangiacomo"https://www.zbmath.org/authors/?q=ai:sanna.giangiacomoInspired by similar ideas from commutative algebra, a possible approach to classify the complexity of a projective variety is in terms of their associated category of aCM sheaves. \textit{D. Eisenbud} and \textit{J. Herzog} [Math. Ann. 280, No. 2, 347--352 (1988; Zbl 0616.13011)] gave a complete classification of aCM varieties with support a finite number of aCM sheaves, the so-called \textit{CM varieties of finite representation type}. Indeed they fall into a very short list. In the next step of complexity, there exist the \textit{ CM-discrete varieties} supporting an infinite but discrete number of aCM sheaves (for instance, quadrics of corank one) and \textit{CM-tame varieties}, supporting only one-dimensional families of aCM sheaves (for instance, Atiyah's classical case of elliptic curves, see [\textit{M. F. Atiyah}, Proc. Lond. Math. Soc. (3) 7, 414--452 (1957; Zbl 0084.17305)]).
Discrete and tame aCM varieties are also classified into a very short list (see [\textit{Y. A. Drozd} and \textit{G.-M. Greuel}, J. Algebra 246, No. 1, 1--54 (2001; Zbl 1065.14041); \textit{D. Faenzi} and \textit{F. Malaspina}, Adv. Math. 310, 663--695 (2017; Zbl 1387.14117)]). It was shown in [\textit{D. Faenzi} and \textit{J. Pons-Llopis}, Épijournal de Géom. Algébr., EPIGA 5, Article 8, 37 p. (2021; Zbl 07402131)] that the rest of aCM varieties are of \textit{wild representation type}, namely they support families of arbitrarily large dimension of aCM sheaves.
Among ACM sheaves, a special role is played by \textit{Ulrich sheaves}, defined as those aCM sheaves having the maximal permitted number of global sections. In the paper under review, the authors study the contribution of Ulrich sheaves to the determination of the representation type of the underlying projective variety.
More particulary, after excluding Ulrich sheaves, they show that the two rational scrolls of degree \(4\) and the Segre product \(\mathbb{P}^1\times\mathbb{P}^2\subset \mathbb{P}^5\) become of finite CM representation type, while all other varieties keep their representation type unchanged (Theorem \(1\)). More over, they present an explicit classification of aCM sheaves on the aforementioned downgraded varieties (Theorem \(2\)).
Reviewer: Joan Pons-Llopis (Maó)Grothendieck-Lefschetz and Noether-Lefschetz for bundleshttps://www.zbmath.org/1475.140852022-01-14T13:23:02.489162Z"Ravindra, G. V."https://www.zbmath.org/authors/?q=ai:ravindra.g-v"Tripathi, Amit"https://www.zbmath.org/authors/?q=ai:tripathi.amitSummary: We prove a mild strengthening of a theorem of \textit{K. Česnavičius} [Algebr. Geom. 7, No. 4, 503--511 (2020; Zbl 1450.14003)] which gives a criterion for a vector bundle on a smooth complete intersection of dimension at least \(3\) to split into a sum of line bundles. We also prove an analogous statement for bundles on a general complete intersection surface.On facets of the Newton polytope for the discriminant of the polynomial systemhttps://www.zbmath.org/1475.140942022-01-14T13:23:02.489162Z"Antipova, Irina Avgustovna"https://www.zbmath.org/authors/?q=ai:antipova.i-a"Kleshkova, Ekaterina Andreevna"https://www.zbmath.org/authors/?q=ai:kleshkova.ekaterina-andreevnaSummary: We study normal directions to facets of the Newton polytope of the discriminant of the Laurent polynomial system via the tropical approach. We use the combinatorial construction proposed by \textit{A. Dickenstein} et al. [J. Am. Math. Soc. 20, No. 4, 1111--1133 (2007; Zbl 1166.14033)] for the tropicalization of algebraic varieties admitting a parametrization by a linear map followed by a monomial map.Toric varieties and Gröbner bases: the complete \(\mathbb{Q}\)-factorial casehttps://www.zbmath.org/1475.140972022-01-14T13:23:02.489162Z"Rossi, Michele"https://www.zbmath.org/authors/?q=ai:rossi.michele"Terracini, Lea"https://www.zbmath.org/authors/?q=ai:terracini.leaThis paper is primarily concerned with questions about complete and non-projective toric varieties. The authors note early on that many examples of complete and non-projective varieties are ad-hoc variations of the classical example of Oda, and highlight a general lack of understanding as to how non-projective varieties arise among complete toric varieties. The authors highlight the need for more examples to better understand this aspect of the theory, and given the difficulty of these constructions point to the need to adopt a computer-aided approach. To this end, two algorithms are prposed for computing complete \(\mathbb{Q}\)-factorial fans over a set of vectors.
The first algorithm proposed is inefficient but of theoretical interest, leading to a conjecture and several possible applications. The second algorithm relies on the theory of Gröbner bases, where the authors deal with toric ideals arising from complete configurations, which cannot be homogeneous. The results here lead to an efficient algorithm for computing all projective complete simplicial fans with a given 1-skeleton that is based on the computation of the Gröbner fan of the associated toric ideal.
The paper concludes by presenting several nice concrete examples and outlining many opportunities and directions for future work.
Reviewer: Jeremiah Johnson (Manchester)Locally nilpotent derivations of factorial domainshttps://www.zbmath.org/1475.141172022-01-14T13:23:02.489162Z"El Kahoui, M'Hammed"https://www.zbmath.org/authors/?q=ai:el-kahoui.mhammed"Ouali, Mustapha"https://www.zbmath.org/authors/?q=ai:ouali.mustaphaLet \(R\subset A\) be factorial domains containing \(\mathbb Q\). In the paper under review the authors give an equivalent condition in terms of locally nilpotent derivations for \(A\) to be \(R\)-isomorphic to \(R[v,w]/(cw-h(v))\), where \(0\not=c\in R\) is not a unit and \(h(v)\in R[v]\) is nonconstant modulo every prime factor of \(c\): There exists an irreducible locally nilpotent \(R\)-derivation \(\xi\) of \(A\) with ring of constants \(A^{\xi}\) equal to \(R\) and such that there exists \(c=\xi(s)\in{\mathfrak p}{\mathfrak l}=\xi(A)\cap A^{\xi}\) with the property that the ideal \(R[s]\cap cA\) of \(R[s]\) is generated by \(c\) and a polynomial \(h(s)\in R[s]\). The result implies the isomorphism of the differential rings \((A,\xi)\) and \((R[v,w]/(cw-h(v)),\delta)\) where \(\delta(v)=c\) and \(\delta(w)=\partial_vh(v)\). The authors show that an example from a paper by Daigle gives that the result does not hold if \(A\) in not factorial. In the particular case when \(R\) is a polynomial ring in one variable, the result yields a natural generalization of of a result in Masuda characterizing Danielewski hypersurfaces whose coordinate ring is factorial. Finally, the authors apply their result to the study of triangularizable locally nilpotent \(R\)-derivations of the polynomial ring in two variables over \(R\).
Reviewer: Vesselin Drensky (Sofia)On a family of Caldero-Chapoton algebras that have the Laurent phenomenonhttps://www.zbmath.org/1475.160212022-01-14T13:23:02.489162Z"Labardini-Fragoso, Daniel"https://www.zbmath.org/authors/?q=ai:labardini-fragoso.daniel"Velasco, Diego"https://www.zbmath.org/authors/?q=ai:velasco.diegoSummary: We realize a family of generalized cluster algebras as Caldero-Chapoton algebras of quivers with relations. Each member of this family arises from an unpunctured polygon with one orbifold point of order 3, and is realized as a Caldero-Chapoton algebra of a quiver with relations naturally associated to any triangulation of the alluded polygon. The realization is done by defining for every arc \(j\) on the polygon with orbifold point a representation \(M(j)\) of the referred quiver with relations, and by proving that for every triangulation \(\tau\) and every arc \(j\in\tau\), the product of the Caldero-Chapoton functions of \(M(j)\) and \(M(j^\prime)\), where \(j^\prime\) is the arc that replaces \(j\) when we flip \(j\) in \(\tau\), equals the corresponding exchange polynomial of Chekhov-Shapiro in the generalized cluster algebra. Furthermore, we show that there is a bijection between the set of generalized cluster variables and the isomorphism classes of \(E\)-rigid indecomposable decorated representations of \(\Lambda\).A correspondence between rigid modules over path algebras and simple curves on Riemann surfaceshttps://www.zbmath.org/1475.160222022-01-14T13:23:02.489162Z"Lee, Kyu-Hwan"https://www.zbmath.org/authors/?q=ai:lee.kyu-hwan"Lee, Kyungyong"https://www.zbmath.org/authors/?q=ai:lee.kyungyongSummary: We propose a conjectural correspondence between the set of rigid indecomposable modules over the path algebras of acyclic quivers and the set of certain non-self-intersecting curves on Riemann surfaces, and prove the correspondence for the two-complete rank 3 quivers.Maximal forward hom-orthogonal sequences for cluster-tilted algebras of finite typehttps://www.zbmath.org/1475.160232022-01-14T13:23:02.489162Z"Nasr-Isfahani, Alireza"https://www.zbmath.org/authors/?q=ai:nasr-isfahani.alireza-rThis article tackles the following conjecture of \textit{K. Igusa} and \textit{G. Todorov} [``Maximal green sequences for cluster-tilted algebras of finite representation type'', Preprint, \url{arXiv:1706.06503}, Conjecture~3.28]:
``Let \(\Lambda\) be a cluster-tilted algebra of finite type and let \(B\) be one of the associated tilted algebras.
\begin{enumerate}
\item[(a)] The \(B\)-modules, ordered from right to left in the Auslander-Reiten quiver of \(\Lambda\) form a maximal forward hom-orthogonal sequence of \(\Lambda\)-modules whose dimension vectors form the \(c\)-vectors of a maximal green sequence for \(\Lambda\).
\item[(b)] The longest maximal green sequence for \(\Lambda\) is given in this way.''
\end{enumerate}
The work proves part (a) of the conjecture using cluster repetitive algebras [\textit{I. Assem} et al., J. Pure Appl. Algebra 213, No. 7, 1450--1463 (2009; Zbl 1183.16013)] and gives a counterexample to part (b). The author then provides a reformulation of this part of the conjecture [Conjecture~3.4]:
``Let \(\Lambda\) be a cluster-tilted algebra of finite type. Then there exists a tilted algebra \(B\) such that \(\Lambda \simeq B \ltimes \operatorname{Ext}^2_B (DB, B)\) and the number of isomorphism classes of indecomposable \(\Lambda\)-modules which are indecomposable \(B\)-modules is equal to the length of the longest maximal green sequence for \(\Lambda\).''
Reviewer: Alexis Langlois-Rémillard (Gent)A brief survey on semiprime and weakly compressible moduleshttps://www.zbmath.org/1475.160282022-01-14T13:23:02.489162Z"Dehghani, N."https://www.zbmath.org/authors/?q=ai:dehghani.najmeh"Vedadi, M. R."https://www.zbmath.org/authors/?q=ai:vedadi.mohammad-rezaSummary: This is a brief survey on a module generalization of the concepts of prime (semiprime) for a ring. An \(R\)-module \(M\) is called weakly compressible if \(\Hom_R(M,N)N\) is nonzero for every \(0\neq N \leq M_R\). They are semiprime (i.e., \(M\in \operatorname{Cog}(N)\) for every \(N\leq_{ess} M_R)\). We show that there exist semiprime modules which are not weakly compressible. Further, we investigate when a semiprime module is weakly compressible over any ring \(R\). For certain rings \(R\), including prime hereditary Noetherian rings, weakly compressible (semiprime) modules are characterized. In continue, we study rings \(R\) whose every semiprime module is weakly compressible (say in this case, \(R\) is a SW ring). Duo Noetherian SW rings are shown to be local. If \(R\) is Morita equivalent to a Dedekind domain, then \(R\) is SW if and only if it is either simple Artinian or \(J(R)\neq 0\).
For the entire collection see [Zbl 1455.53022].Actions of the additive group \(G_a\) on certain noncommutative deformations of the planehttps://www.zbmath.org/1475.160302022-01-14T13:23:02.489162Z"Kaygorodov, Ivan"https://www.zbmath.org/authors/?q=ai:kaigorodov.i-b"Lopes, Samuel A."https://www.zbmath.org/authors/?q=ai:lopes.samuel-a"Mashurov, Farukh"https://www.zbmath.org/authors/?q=ai:mashurov.farukh-aSummary: We connect the theorems of and on locally nilpotent derivations and automorphisms of the polynomial ring \(A_0\) and of the Weyl algebra \(A_1\), both over a field of characteristic zero, by establishing the same type of results for the family of algebras
\[
A_h = \langle x,y \, | \, yx - xy = h(x)\rangle,
\]
where \(h\) is an arbitrary polynomial in \(x\). In the second part of the paper we consider a field \(\mathbb{F}\) of prime characteristic and study \(\mathbb{F}[t]\)-comodule algebra structures on \(A_h\). We also compute the Makar-Limanov invariant of absolute constants of \(A_h\) over a field of arbitrary characteristic and show how this subalgebra determines the automorphism group of \(A_h\).New perspectives in algebra, topology and categories. Summer school, Louvain-la-Neuve, Belgium, September 12--15, 2018 and September 11--14, 2019https://www.zbmath.org/1475.180012022-01-14T13:23:02.489162ZPublisher's description: This book provides an introduction to some key subjects in algebra and topology. It consists of comprehensive texts of some hours courses on the preliminaries for several advanced theories in (categorical) algebra and topology. Often, this kind of presentations is not so easy to find in the literature, where one begins articles by assuming a lot of knowledge in the field. This volume can both help young researchers to quickly get into the subject by offering a kind of \flqq{} roadmap \frqq{} and also help master students to be aware of the basics of other research directions in these fields before deciding to specialize in one of them. Furthermore, it can be used by established researchers who need a particular result for their own research and do not want to go through several research papers in order to understand a single proof. Although the chapters can be read as \flqq{} self-contained \frqq{} chapters, the authors have tried to coordinate the texts in order to make them complementary. The seven chapters of this volume correspond to the seven courses taught in two Summer Schools that took place in Louvain-la-Neuve in the frame of the project \textit{Fonds d'Appui à l'Internationalisation} of the Université catholique de Louvain to strengthen the collaborations with the universities of Coimbra, Padova and Poitiers, within the Coimbra Group.
The articles of this volume will be reviewed individually.Recollements of derived categories. I: Construction from exact contextshttps://www.zbmath.org/1475.180172022-01-14T13:23:02.489162Z"Chen, Hongxing"https://www.zbmath.org/authors/?q=ai:chen.hongxing"Xi, Changchang"https://www.zbmath.org/authors/?q=ai:xi.changchangSummary: A new method is established to construct recollements of derived categories of rings from exact contexts and noncommutative tensor products. This method is applicable to a large variety of situations, including Milnor squares, localizations, ring extensions and ring epimorphisms.Degenerating 0 in triangulated categorieshttps://www.zbmath.org/1475.180212022-01-14T13:23:02.489162Z"Saorín, Manuel"https://www.zbmath.org/authors/?q=ai:saorin.manuel"Zimmermann, Alexander"https://www.zbmath.org/authors/?q=ai:zimmermann.alexander.1Summary: In previous work [Appl. Categ. Struct. 24, No. 4, 385--405 (2016; Zbl 1375.18065); Algebr. Represent. Theory 22, No. 4, 801--836 (2019; Zbl 1423.18045)], based on the work of \textit{G. Zwara} [Arch. Math. 71, No. 6, 437--444 (1998; Zbl 0964.16008); Compos. Math. 121, No. 2, 205--218 (2000; Zbl 0957.16007)] and \textit{Y. Yoshino} [J. Algebra 332, No. 1, 500--521 (2011; Zbl 1267.13022)], we defined and studied degenerations of objects in triangulated categories analogous to the degeneration of modules. In triangulated categories \({\mathcal{T}}\), it is surprising that the zero object may degenerate. We show that the triangulated subcategory of \({\mathcal{T}}\) generated by the objects that are degenerations of zero coincides with the triangulated subcategory of \({\mathcal{T}}\) consisting of the objects with a vanishing image in the Grothendieck group \(K_0({\mathcal{T}})\) of \({\mathcal{T}}\).An algorithm for constructing the resultant of two entire functionshttps://www.zbmath.org/1475.300722022-01-14T13:23:02.489162Z"Kuzovatov, V. I."https://www.zbmath.org/authors/?q=ai:kuzovatov.vyacheslav-igorevich"Kytmanov, A. A."https://www.zbmath.org/authors/?q=ai:kytmanov.a-a"Myshkina, E. K."https://www.zbmath.org/authors/?q=ai:myshkina.evgeniya-konstantinovnaSummary: On the basis of Newton's recurrent formulas, an algorithm for constructing the resultant of two entire functions is described. This algorithm helps eliminate unknowns from systems of nonalgebraic equations. This algorithm is implemented in the computer algebra system Maple. Examples demonstrating the operation of the algorithm are discussed.Periodicity, linearizability, and integrability in seed mutations of type \(\boldsymbol{A}_N^{(1)}\)https://www.zbmath.org/1475.370642022-01-14T13:23:02.489162Z"Nobe, Atsushi"https://www.zbmath.org/authors/?q=ai:nobe.atsushi"Matsukidaira, Junta"https://www.zbmath.org/authors/?q=ai:matsukidaira.juntaThis paper regards connections between the theories of dynamical systems and cluster algebras. The authors propose studying the dynamics of seed mutations in some suitable cluster algebras. The approach allows giving results concerning the Laurent phenomenon, the periodicity conjecture, and the discrete integrability in such algebras.
The analysis starts by defining the object of study, which are generalized Cartan matrices of type \(A^{(1)}_{N}\) associated with a unique path \(w=(t_{0}, t_{1},t_{2},\dots)\) in a \(n\)-regular tree \(\mathbb{T}_{N+1}\). The partial assignment \(w\subset\mathbb{T}_{N+1} \rightarrow \Sigma=(\Sigma_{0},\Sigma_{1},\Sigma_{2},\dots)\) is defined where for \(t\geq 0\), \(\Sigma_{t}=(\textbf{x}_{t}, \textbf{y}_{t}, B_{t}=(b^{t}_{ij}))\), is a seed with \(\textbf{x}_{t}=(x_{1;t},\dots, x_{n;t})\), \(\textbf{y}_{t}=(y_{1;t},\dots, y_{n;t})\), \(\textbf{x}_{t}\) is the cluster of the seed, \(\textbf{y}_{t}\) is the coefficient tuple and \(B_{t}\) is a suitable exchange matrix associated with a point \(t\in\mathbb{T}_{n}\).
The map \(f: w\subset \mathbb{T}_{N+1}\rightarrow \Sigma\) with \(f(t_{l(N+1)+k})=\Sigma_{l(N+1)+k}\) is said to be the subcluster pattern.
The periodicity of the quivers \(Q_{l(N+1)+k}\) associated with the exchange matrices \(B_{l(N+1)+k}\) is given by the following identity:
\[Q_{l(N+1)+k}=(\sigma_{N+1})^{k}Q_{} \] where \(Q_{0}\) has the shape
\[Q_0=\quad \begin{tikzcd}[row sep=0pt]
\bigcirc\ar[rrrrrr, bend left=25]\rar&\bigcirc\rar&\bigcirc\rar&\dots\rar&\bigcirc\rar&\bigcirc\rar&\bigcirc\\
N+1 & N & N-1 & & 3 & 2 & 1
\end{tikzcd}\tag{F}\]
and \(\sigma_{N+1}\) is the \(N+1\)-permutation \((N+1,1,2,\dots, N-1,N)\).
In the third section, the authors study the dynamics of the coefficients \(\textbf{y}_{t}\) by proving the following identity via mutations and the periodicity of the exchange matrices (\(B_{l(N+1)+k}=B_{k}\)) :
\[v^{l+1}_{j}=\sigma_{N}v^{l}_{j}=v^{l}_{j+1} \]
which means that for \(l\geq1\) and \(j=1,2,\dots, N\), \(v^{l}_{j}\) has period \(N\) on \(l\). Where,
\[\begin{aligned}
v^l_1 &=y_{N+1;l}y_{1;l(N+1)},\\
v^l_j &=y_{j;l(N+1)},\quad j=2,3,\dots, N.
\end{aligned} \]
In the general case, the authors give explicit solutions \(y^{l}=(y^{l}_{1}, y^{l}_{2},\dots, y^{l}_{N+1})\) to the dynamical system governed by the map \(y^{l+1}=\psi(y^{l})\). Thus, an explicit formula for coefficients \(y_{k, l(N+1)+k-1}\) is given based on the coefficient \(y_{N+1}\) and some suitable constants.
The fourth section of the paper is devoted to the dynamics of the cluster variables, firstly it is proved that the cluster variables assigned to the path \(w\) of type \(A^{(1)}_{N}\) are given by using the solutions to the dynamical systems:
\[ \begin{aligned} z^{t+1}_{i}&=\frac{z^{t+1}_{i-1}z^{t}_{i+1}+1}{z^{t}_{i}},\quad i=1,2,\dots, N+1,\quad t\geq 1,\\
z^{t+1}_{0}&=z^{t}_{N+1},\\
z^{t}_{N+2}&=z^{t+1}_{1}. \end{aligned} \tag{1}\]
The proof of this result is based on the exchange rules and the explicit formulas given by the dynamic system governed by \(\psi\).
The authors also prove that for any \(t\geq 1\) and \(i=1,2,\dots, n\), it holds that \(\lambda^{t+N}_{i}=\lambda^{t}_{i}\), where
\[ \begin{aligned} \lambda_{i}&=\lambda_{i}(z^{1}_{1},\dots, z^{1}_{N+1})=\frac{z^{1}_{1}+z^{1}_{i+2}}{z^{1}_{i+1}},\\
\lambda_{N}&=\lambda_{N}(z^{1}_{1},\dots, z^{1}_{N+1})=\frac{z^{1}_{1}z^{1}_{N}+z^{1}_{2}z^{1}_{N+1}+1}{z^{1}_{1}z^{1}_{N+1}}. \end{aligned} \]
The system (1) can be linearized and the general solution to the dynamical system governed by the birational map \(\varphi\) such that \(z^{t+1}=\varphi(z^{t})\) is explicitly given. This results allows concluding that the cluster variables \(x^{t}_{1}, x^{t}_{2},\dots x^{t}_{N+1} \) assigned to the path \(w\) of type \(A^{(1)}_{N}\) exhibit the Laurent phenomenon.
Some properties of the birational map \(\varphi\) are given in the appendix of the paper.
Reviewer: Agustín Moreno Cañadas (Bogotá)Reduction formulas for higher order derivations and a hypergeometric identityhttps://www.zbmath.org/1475.390012022-01-14T13:23:02.489162Z"Ebanks, Bruce"https://www.zbmath.org/authors/?q=ai:ebanks.bruce-r"Kézdy, André E."https://www.zbmath.org/authors/?q=ai:kezdy.andre-eSummary: A derivation (of order 1) satisfies the reduction formula \(f(x^k) = kx^{k-1}f(x)\) for any integer \(k\). In this article we find corresponding reduction formulas for derivations of higher order on commutative rings. More precisely, for every derivation \(f\) of order \(n\) and every positive integer \(k\) we find an explicit formula for \(f(x^k)\) as a linear combination of \(x^{k-1}f(x),x^{k-2}f(x^2), \ldots , x^{k-n}f(x^n)\). The proof hinges on the hypergeometric identity
\[
\sum_{k \ge 0} (-1)^k \begin{pmatrix} n \\ k \end{pmatrix} \begin{pmatrix} n+d+1-k \\ n-j \end{pmatrix} \begin{pmatrix} d+j-k \\ j \end{pmatrix} = \begin{pmatrix} n \\ j\end{pmatrix}
\] for any positive integer \(n\), nonnegative integer \(d\), and integer \(j\) satisfying \(0 \le j \le n\). We prove this identity via the WZ-method.On Hodge-Riemann relations for translation-invariant valuationshttps://www.zbmath.org/1475.520112022-01-14T13:23:02.489162Z"Kotrbatý, Jan"https://www.zbmath.org/authors/?q=ai:kotrbaty.janIn 2001, \textit{S. Alesker} [Geom. Funct. Anal. 11, No. 2, 244--272 (2001; Zbl 0995.52001)] proved McMullen's conjecture that the space of valuations on convex bodies is the weak completion of the linear space of mixed volumes. It is also the closure of the subspace of smooth valuations, which has much better algebraic properties. In particular, it has a product structure, due to Alesker, which makes it into a (commutative, associative, unital) algebra; this algebra has properties analogous to those given by the hard Lefschetz theorem and the Poincaré duality of algebraic topology.
These last are two-thirds of the Kähler package, a powerful set of tools, originally used in the study of Kähler manifolds, that has productively been generalized to other areas of mathematics. The third leg of this tripod are the Hodge-Riemann relations.
In this paper, the author conjectures a Hodge-Riemann analogue for smooth valuations, proves it for even valuations and for 1-homogeneous valuations, and explores some implications of this result. He also conjectures mixed versions of the hard Lefschetz theorem and the Hodge-Riemann relations, which if true would have even broader implications for convex geometry.
Reviewer: Robert Dawson (Halifax)On subrings of the form \(I+\mathbb{R}\) of \(C(X)\)https://www.zbmath.org/1475.540092022-01-14T13:23:02.489162Z"Azarpanah, F."https://www.zbmath.org/authors/?q=ai:azarpanah.f"Namdari, M."https://www.zbmath.org/authors/?q=ai:namdari.mehrdad"Olfati, A. R."https://www.zbmath.org/authors/?q=ai:olfati.ali-rezaIn this article the authors study subrings of the form \(I+ \mathbb{R}\) of \(C(X).\) It is shown that for each ideal \(I\) in \(C(X)\), the sum of two prime ideals of the ring \(I+ \mathbb{R}\) is prime or all of \(I+ \mathbb{R}\) if and only if \(X\) is an \(F\)-space; and it is shown that a \(z\)-ideal in \(I+ \mathbb{R}\) containing a prime ideal of \(I+ \mathbb{R}\) is a prime ideal of \(I+ \mathbb{R}\). Also \(I+ \mathbb{R}\) is a maximal subring of \(C(X)\) if and only if \(I= M^{p}\cap M^{q}\), for some distinct \(p, q \in \upsilon X\). A necessary and sufficient condition for \(C(X)\) being integral over \(I+ \mathbb{R}\) is given which is: \(C(X)\) is to be a countably generated \(I+ \mathbb{R}\) algebra. It is characterized when \(I+ \mathbb{R}\) is integrally closed in \(C(X)\) as well as when \(C(X)\) is a ring of quotients of \(I+ \mathbb{R}\) for an ideal \(I\) in \(C(X).\)
Reviewer: Bikram Banerjee (Kolkata)Autocovariance varieties of moving average random fieldshttps://www.zbmath.org/1475.622892022-01-14T13:23:02.489162Z"Améndola, Carlos"https://www.zbmath.org/authors/?q=ai:amendola.carlos"Pham, Viet Son"https://www.zbmath.org/authors/?q=ai:pham.viet-sonSummary: We study the autocovariance functions of moving average random fields over the integer lattice \(\mathbb{Z}^d\) from an algebraic perspective. These autocovariances are parametrized polynomially by the moving average coefficients, hence tracing out algebraic varieties. We derive dimension and degree of these varieties and we use their algebraic properties to obtain statistical consequences such as identifiability of model parameters. We connect the problem of parameter estimation to the algebraic invariants known as euclidean distance degree and maximum likelihood degree. Throughout, we illustrate the results with concrete examples. In our computations we use tools from commutative algebra and numerical algebraic geometry.On the regulator problem for linear systems over rings and algebrashttps://www.zbmath.org/1475.930312022-01-14T13:23:02.489162Z"Hermida-Alonso, José Ángel"https://www.zbmath.org/authors/?q=ai:hermida-alonso.jose-angel"Carriegos, Miguel V."https://www.zbmath.org/authors/?q=ai:carriegos.miguel-v"Sáez-Schwedt, Andrés"https://www.zbmath.org/authors/?q=ai:saez-schwedt.andres"Sánchez-Giralda, Tomás"https://www.zbmath.org/authors/?q=ai:sanchez-giralda.tomasSummary: The regulator problem is solvable for a linear dynamical system \(\Sigma\) if and only if \(\Sigma\) is both pole assignable and state estimable. In this case, \(\Sigma\) is a canonical system (i.e., reachable and observable). When the ring \(R\) is a field or a Noetherian total ring of fractions the converse is true. Commutative rings which have the property that the regulator problem is solvable for every canonical system (RP-rings) are characterized as the class of rings where every observable system is state estimable (SE-rings), and this class is shown to be equal to the class of rings where every reachable system is pole-assignable (PA-rings) and the dual of a canonical system is also canonical (DP-rings).A generalization of the remainder theorem and factor theoremhttps://www.zbmath.org/1475.970042022-01-14T13:23:02.489162Z"Laudano, F."https://www.zbmath.org/authors/?q=ai:laudano.francescoSummary: We propose a generalization of the classical Remainder Theorem for polynomials over commutative coefficient rings that allows calculating the remainder without using the long division method. As a consequence we obtain an extension of the classical Factor Theorem that provides a general divisibility criterion for polynomials. The arguments can be used in basic algebra courses and are suitable for building classroom/homework activities for college and high school students.Divisibility tests for polynomialshttps://www.zbmath.org/1475.970052022-01-14T13:23:02.489162Z"Laudano, F."https://www.zbmath.org/authors/?q=ai:laudano.francesco"Donatiello, A."https://www.zbmath.org/authors/?q=ai:donatiello.aSummary: We propose a divisibility criterion for elements of a generic Unique Factorization Domain. As a consequence, we obtain a general divisibility criterion for polynomials over Unique Factorization Domains. The arguments can be used in basic algebra courses and are suitable for building classroom/homework activities for college and high school students.