Recent zbMATH articles in MSC 19https://www.zbmath.org/atom/cc/192021-02-27T13:50:00+00:00WerkzeugCorrigendum to: ``On differential characteristic classes''.https://www.zbmath.org/1453.530312021-02-27T13:50:00+00:00"Ho, Man-Ho"https://www.zbmath.org/authors/?q=ai:ho.man-hoSummary: In this erratum we correct a mistake in [the author, J. Aust. Math. Soc. 99, No. 1, 30--47 (2015; Zbl 1323.53021)].The Bass-Quillen conjecture and Swan's question.https://www.zbmath.org/1453.130262021-02-27T13:50:00+00:00"Popescu, Dorin"https://www.zbmath.org/authors/?q=ai:popescu.dorinSummary: We present a question which implies a complete positive answer for the Bass-Quillen Conjecture.
For the entire collection see [Zbl 07248454].Congruences for critical values of higher derivatives of twisted Hasse-Weil \(L\)-functions.https://www.zbmath.org/1453.111422021-02-27T13:50:00+00:00"Bley, Werner"https://www.zbmath.org/authors/?q=ai:bley.werner"Macias Castillo, Daniel"https://www.zbmath.org/authors/?q=ai:macias-castillo.danielSummary: Let \(A\) be an abelian variety over a number field \(k\) and let \(F\) be a finite cyclic extension of \(k\) of \(p\)-power degree for an odd prime \(p\). Under certain technical hypotheses, we obtain a reinterpretation of the equivariant Tamagawa number conjecture (`eTNC') for \(A\), \(F/k\) and \(p\) as an explicit family of \(p\)-adic congruences involving values of derivatives of the Hasse-Weil \(L\)-functions of twists of \(A\), normalised by completely explicit twisted regulators. This reinterpretation makes the eTNC amenable to numerical verification and furthermore leads to explicit predictions which refine well-known conjectures of \textit{B. Mazur} and \textit{J. Tate} [Duke Math. J. 54, 711--750 (1987; Zbl 0636.14004)].Idempotent characters and equivariantly multiplicative splittings of \(K\)-theory.https://www.zbmath.org/1453.190082021-02-27T13:50:00+00:00"Böhme, Benjamin"https://www.zbmath.org/authors/?q=ai:bohme.benjaminLet \(G\) be a finite group, \(\mathbb S\) -- the \(G\)-equivarint sphere spectrum, and \(\pi_0^G(\mathbb S)\cong A(G)\) -- the Segal's identification between the \(G\)-equivariant homotopy group of \(\mathbb S\) and the Burnside ring \(A(G)\) of \(G\). In this paper, the author shows that for the \(G\)-equivariant topological \(K\)-theory \(KU_G\) and \(KO_G\), the homotopy groups of complex or orthogonal \(K\)-theories are isomorphic to the corresponding representation rings \(\pi_0^G(KU_G) \cong RU(G)\) and \(\pi_0^G(KO_G) \cong RO(G)\).
Let \(P\) be a set of prime in \(\mathbb Z\), \(\mathbb Z_{(P)}= \mathbb Z[p^{-1}|p\notin P]\) - the localization of \(\mathbb Z\) at \(P\), and \(A_{(P)} = A(G) \otimes \mathbb Z_{(P)}\) - the \(P\)-local Burnside ring. To each conjugacy class of \(P\)-perfect subgroup \(L \subseteq G\) assign the primitive Dress' idempotent \(e_L\) with the mark \(\phi^H(e_L)= \begin{cases} 1 & \mbox{ if } O^H(G) \mbox{ and } L \mbox{ are conjugate, }\\ 0 & \mbox{ otherwise. } \end{cases}\). The `linearization' map \(lin : A(G)_{(P)} \to RU(G)_{(P)}:= RU(G) \otimes Z_{(P)}\) is given by sending each finite \(G\)-set to its associated permutation representation.
The main result of the paper is Theorem 1.2 establishing a bijection between the conjugacy classes of cyclic subgroups \(C \subseteq G\) of order not divisible by any prime in \(P\) and the primitive idempotents \(lin(e_C)\) in the \(P\)-localized representation ring \(RU(G)_{(P)}\). This results extend the corresponding classical results in the sense without adjoining roots of unity or inverting the order of the group \(G\).
Reviewer: Do Ngoc Diep (Hanoi)The generalized slices of Hermitian \(K\)-theory.https://www.zbmath.org/1453.140652021-02-27T13:50:00+00:00"Bachmann, Tom"https://www.zbmath.org/authors/?q=ai:bachmann.tomSummary: We compute the generalized slices (as defined by Spitzweck-Østvær) of the motivic spectrum \(KO\) (representing Hermitian \(K\)-theory) in terms of motivic cohomology and (a version of) generalized motivic cohomology, obtaining good agreement with the situation in classical topology and the results predicted by Markett-Schlichting. As an application, we compute the homotopy sheaves of (this version of) generalized motivic cohomology, which establishes a version of a conjecture of Morel.The fundamental group and extensions of motives of Jacobians of curves.https://www.zbmath.org/1453.190062021-02-27T13:50:00+00:00"Sarkar, Subham"https://www.zbmath.org/authors/?q=ai:sarkar.subham"Sreekantan, Ramesh"https://www.zbmath.org/authors/?q=ai:sreekantan.rameshSummary: In this paper, we construct extensions of mixed Hodge structure coming from the mixed Hodge structure on the graded quotients of the group ring of the fundamental group of a smooth projective pointed curve which correspond to the regulators of certain motivic cohomology cycles on the Jacobian of the curve essentially constructed by Bloch and Beilinson. This leads to a new iterated integral expression for the regulator. This is a generalisation of a theorem of [\textit{E. Colombo}, J. Algebr. Geom. 11, No. 4, 761--790 (2002; Zbl 1059.14032)] where she constructed the extension corresponding to Collino's cycles in the Jacobian of a hyperelliptic curve.Cobordism-framed correspondences and the Milnor \(K\)-theory.https://www.zbmath.org/1453.140702021-02-27T13:50:00+00:00"Tsybyshev, A."https://www.zbmath.org/authors/?q=ai:tsybyshev.a-eSummary: The 0th cohomology group is computed for a complex of groups of cobordism-framed correspondences. In the case of ordinary framed correspondences, an analogous computation was completed by A. Neshitov in his paper ``Framed correspondences and the Milnor-Witt \(K\)-theory''. Neshitov's result is, at the same time, a computation of the homotopy groups \(\pi_{i,i}(S^0)(\operatorname{Spec}(k))\), and the present work might be used subsequently as a basis for computing the homotopy groups \(\pi_{i,i}(MGL_{\bullet })(\operatorname{Spec}(k))\) of the spectrum \(MGL_{\bullet } \).A remark on the stable real forms of complex vector bundles over manifolds.https://www.zbmath.org/1453.190092021-02-27T13:50:00+00:00"Yang, Huijun"https://www.zbmath.org/authors/?q=ai:yang.huijunSummary: Let \(M\) be an \(n\)-dimensional closed oriented smooth manifold with \(n\equiv 4 \mod 8\), and \(\eta\) be a complex vector bundle over \(M\). We determine the final obstruction for \(\eta\) to admit a stable real form in terms of the characteristic classes of \(M\) and \(\eta\). As an application, we obtain the criteria to determine which complex vector bundles over a simply connected four-dimensional manifold admit a stable real form.Infinitesimal Bloch regulator.https://www.zbmath.org/1453.190052021-02-27T13:50:00+00:00"Ünver, Sinan"https://www.zbmath.org/authors/?q=ai:unver.sinanSummary: The aim of the paper is to define an infinitesimal analog of the Bloch regulator, which attaches to a pair of meromorphic functions on a Riemann surface, a line bundle with connection on the punctured surface. In the infinitesimal context, we consider a pair \((X, \underline{X})\) of schemes over a field of characteristic 0, such that the regular scheme \(\underline{X}\) is defined in \(X\) by a square-zero sheaf of ideals which is locally free on \(\underline{X}\). We propose a definition of the weight two motivic cohomology of \(X\) based on the Bloch group, which is defined in terms of the functional equation of the dilogarithm. The analog of the Bloch regulator is a map from a subspace of the infinitesimal part of \(\operatorname{H}_{\mathcal{M}}^2(X, \mathbb{Q}(2))\) to the first cohomology group of the Zariski sheaf associated to an André-Quillen homology group. Using Goodwillie's theorem, we deduce that this map is an isomorphism, which is an infinitesimal analog of the injectivity conjecture for the Bloch regulator.Étale motives.https://www.zbmath.org/1453.140592021-02-27T13:50:00+00:00"Cisinski, Denis-Charles"https://www.zbmath.org/authors/?q=ai:cisinski.denis-charles"Déglise, Frédéric"https://www.zbmath.org/authors/?q=ai:deglise.fredericSummary: We define a theory of étale motives over a noetherian scheme. This provides a system of categories of complexes of motivic sheaves with integral coefficients which is closed under the six operations of Grothendieck. The rational part of these categories coincides with the triangulated categories of Beilinson motives (and is thus strongly related to algebraic \(K\)-theory). We extend the rigidity theorem of Suslin and Voevodsky over a general base scheme. This can be reformulated by saying that torsion étale motives essentially coincide with the usual complexes of torsion étale sheaves (at least if we restrict ourselves to torsion prime to the residue characteristics). As a consequence, we obtain the expected results of absolute purity, of finiteness, and of Grothendieck duality for étale motives with integral coefficients, by putting together their counterparts for Beilinson motives and for torsion étale sheaves. Following Thomason's insights, this also provides a conceptual and convenient construction of the \(\ell\)-adic realization of motives, as the homotopy \(\ell\)-completion functor.Quasi-elliptic cohomology and its power operations.https://www.zbmath.org/1453.550062021-02-27T13:50:00+00:00"Huan, Zhen"https://www.zbmath.org/authors/?q=ai:huan.zhenSummary: Quasi-elliptic cohomology is a variant of Tate K-theory. It is the orbifold K-theory of a space of constant loops. For global quotient orbifolds, it can be expressed in terms of equivariant K-theories. In this paper we show how this theory is equipped with power operations. We also prove that the Tate K-theory of symmetric groups modulo a certain transfer ideal classify the finite subgroups of the Tate curve.Descent in algebraic \(K\)-theory and a conjecture of Ausoni-Rognes.https://www.zbmath.org/1453.180112021-02-27T13:50:00+00:00"Clausen, Dustin"https://www.zbmath.org/authors/?q=ai:clausen.dustin"Mathew, Akhil"https://www.zbmath.org/authors/?q=ai:mathew.akhil"Naumann, Niko"https://www.zbmath.org/authors/?q=ai:naumann.niko"Noel, Justin"https://www.zbmath.org/authors/?q=ai:noel.justinSummary: Let \(A \to B\) be a \(G\)-Galois extension of rings, or more generally of \(\mathbb{E}_\infty \)-ring spectra in the sense of \textit{J. Rognes} [in: Proceedings of the International Congress of Mathematicians (ICM 2014), Seoul, Korea, August 13--21, 2014. Vol. II: Invited lectures. Seoul: KM Kyung Moon Sa. 1259--1283 (2014; Zbl 1373.19002)]. A basic question in algebraic \(K\)-theory asks how close the map \(K(A) \to K(B)^{hG}\) is to being an equivalence, i.e., how close algebraic \(K\)-theory is to satisfying Galois descent. An elementary argument with the transfer shows that this equivalence is true rationally in most cases of interest. Motivated by the classical descent theorem of \textit{R. W. Thomason} [Ann. Sci. Éc. Norm. Supér. (4) 18, 437--552 (1985; Zbl 0596.14012)], one also expects such a result after periodic localization. We formulate and prove a general result which enables one to promote rational descent statements as above into descent statements after periodic localization. This reduces the localized descent problem to establishing an elementary condition on \(K_0(-)\otimes \mathbb{Q} \). As applications, we prove various descent results in the periodically localized \(K\)-theory, \(TC, THH\), etc. of structured ring spectra, and verify several cases of a conjecture of \textit{C. Ausoni} and \textit{J. Rognes} [Geom. Topol. 16, No. 4, 2037--2065 (2012; Zbl 1260.19004)].About Bredon motivic cohomology of a field.https://www.zbmath.org/1453.140712021-02-27T13:50:00+00:00"Voineagu, Mircea"https://www.zbmath.org/authors/?q=ai:voineagu.mirceaSummary: We prove that, over a perfect field, Bredon motivic cohomology can be computed by Suslin-Friedlander complexes of equivariant equidimensional cycles. Partly based on this result we completely identify Bredon motivic cohomology of a quadratically closed field and of a euclidian field in weights 1 and \(\sigma\). We also prove that Bredon motivic cohomology of an arbitrary field in weight 0 with integer coefficients coincides (as abstract groups) with Bredon cohomology of a point.cdh descent in equivariant homotopy \(K\)-theory.https://www.zbmath.org/1453.140682021-02-27T13:50:00+00:00"Hoyois, Marc"https://www.zbmath.org/authors/?q=ai:hoyois.marcSummary: We construct geometric models for classifying spaces of linear algebraic groups in \(G\)-equivariant motivic homotopy theory, where \(G\) is a tame group scheme. As a consequence, we show that the equivariant motivic spectrum representing the homotopy \(K\)-theory of \(G\)-schemes (which we construct as an \(E_\infty \)-ring) is stable under arbitrary base change, and we deduce that the homotopy \(K\)-theory of \(G\)-schemes satisfies cdh descent.Cokernels of homomorphisms from Burnside rings to inverse limits.https://www.zbmath.org/1453.190012021-02-27T13:50:00+00:00"Morimoto, Masaharu"https://www.zbmath.org/authors/?q=ai:morimoto.masaharuSummary: Let \(G\) be a finite group and let \(A(G)\) denote the Burnside ring of \(G\). Then an inverse limit \(L(G)\) of the groups \(A(H)\) for proper subgroups \(H\) of \(G\) and a homomorphism \({\operatorname{res}}\) from \(A(G)\) to \(L(G)\) are obtained in a natural way. Let \(Q(G)\) denote the cokernel of \({\operatorname{res}}\). For a prime \(p\), let \(N(p)\) be the minimal normal subgroup of \(G\) such that the order of \(G/N(p)\) is a power of \(p\), possibly 1. In this paper we prove that \(Q(G)\) is isomorphic to the cartesian product of the groups \(Q(G/N(p))\), where \(p\) ranges over the primes dividing the order of \(G\).
For part II, see [\textit{M. Morimoto} and \textit{M. Sugimura}, Kyushu J. Math. 72, No. 1, 95--105 (2018; Zbl 1400.19001)].On the projectivity of finitely generated flat modules.https://www.zbmath.org/1453.130272021-02-27T13:50:00+00:00"Tarizadeh, A."https://www.zbmath.org/authors/?q=ai:tarizadeh.abolfazlAuthor's abstract: In this paper, the projectivity of a finitely generated flat module of a commutative ring is studied through its exterior powers and invariant factors and then various new results are obtained. Specially, the related results of Endo, Vasconcelos, Wiegand, Cox-Rush and Puninski-Rothmaler on the projectivity of finitely generated flat modules are generalized.
Reviewer: François Couchot (Caen)Cotorsion pairs and a \(K\)-theory localization theorem.https://www.zbmath.org/1453.190042021-02-27T13:50:00+00:00"Sarazola, Maru"https://www.zbmath.org/authors/?q=ai:sarazola.maruSummary: We show that a complete hereditary cotorsion pair \((\mathcal{C}, \mathcal{C}^\bot)\) in an exact category \(\mathcal{E} \), together with a subcategory \(\mathcal{Z} \subseteq \mathcal{E}\) containing \(\mathcal{C}^\bot\), determines a Waldhausen category structure on the exact category \(\mathcal{C}\), in which \(\mathcal{Z}\) is the class of acyclic objects. This allows us to prove a new version of Quillen's Localization Theorem, relating the \(K\)-theory of exact categories \(\mathcal{A} \subseteq \mathcal{B}\) to that of a cofiber. The novel idea in our approach is that, instead of looking for an exact quotient category that serves as the cofiber, we produce a Waldhausen category, constructed through a cotorsion pair. Notably, we do not require \(\mathcal{A}\) to be a Serre subcategory, which produces new examples.
Due to the algebraic nature of our Waldhausen categories, we are able to recover a version of Quillen's Resolution Theorem, now in a more homotopical setting that allows for weak equivalences.On very effective Hermitian \(K\)-theory.https://www.zbmath.org/1453.140642021-02-27T13:50:00+00:00"Ananyevskiy, Alexey"https://www.zbmath.org/authors/?q=ai:ananyevskiy.alexey"Röndigs, Oliver"https://www.zbmath.org/authors/?q=ai:rondigs.oliver"Østvær, Paul Arne"https://www.zbmath.org/authors/?q=ai:ostvaer.paul-arneSummary: We argue that the very effective cover of hermitian \(K\)-theory in the sense of motivic homotopy theory is a convenient algebro-geometric generalization of the connective real topological \(K\)-theory spectrum. This means the very effective cover acquires the correct Betti realization, its motivic cohomology has the desired structure as a module over the motivic Steenrod algebra, and that its motivic Adams and slice spectral sequences are amenable to calculations.On the Farrell-Jones conjecture for Waldhausen's \(A\)-theory.https://www.zbmath.org/1453.190022021-02-27T13:50:00+00:00"Enkelmann, Nils-Edvin"https://www.zbmath.org/authors/?q=ai:enkelmann.nils-edvin"Lück, Wolfgang"https://www.zbmath.org/authors/?q=ai:luck.wolfgang"Pieper, Malte"https://www.zbmath.org/authors/?q=ai:pieper.malte"Ullmann, Mark"https://www.zbmath.org/authors/?q=ai:ullmann.mark"Winges, Christoph"https://www.zbmath.org/authors/?q=ai:winges.christophSummary: We prove the Farrell-Jones conjecture for (nonconnective) \(A\)-theory with coefficients and finite wreath products for hyperbolic groups, \(\operatorname{CAT}(0)\)-groups, cocompact lattices in almost connected Lie groups and fundamental groups of manifolds of dimension less or equal to three. Moreover, we prove inheritance properties such as passing to subgroups, colimits of direct systems of groups, finite direct products and finite free products. These results hold also for Whitehead spectra and spectra of stable pseudoisotopies in the topological, piecewise linear and smooth categories.A new index theorem for monomial ideals by resolutions.https://www.zbmath.org/1453.190072021-02-27T13:50:00+00:00"Douglas, Ronald G."https://www.zbmath.org/authors/?q=ai:douglas.ronald-george"Jabbari, Mohammad"https://www.zbmath.org/authors/?q=ai:jabbari.mohammad"Tang, Xiang"https://www.zbmath.org/authors/?q=ai:tang.xiang"Yu, Guoliang"https://www.zbmath.org/authors/?q=ai:yu.guoliangSummary: We prove an index theorem for the quotient module of a monomial ideal. We obtain this result by resolving the monomial ideal by a sequence of essentially normal Hilbert modules, each of which is a direct sum of (weighted) Bergman spaces on balls.The A-theoretic Farrell-Jones conjecture for virtually solvable groups.https://www.zbmath.org/1453.190032021-02-27T13:50:00+00:00"Kasprowski, Daniel"https://www.zbmath.org/authors/?q=ai:kasprowski.daniel"Ullmann, Mark"https://www.zbmath.org/authors/?q=ai:ullmann.mark"Wegner, Christian"https://www.zbmath.org/authors/?q=ai:wegner.christian"Winges, Christoph"https://www.zbmath.org/authors/?q=ai:winges.christophSummary: We prove the A-theoretic Farrell-Jones conjecture for virtually solvable groups. As a corollary, we obtain that the conjecture holds for \(S\)-arithmetic groups and lattices in almost connected Lie groups.