Recent zbMATH articles in MSC 14Ahttps://www.zbmath.org/atom/cc/14A2021-04-16T16:22:00+00:00WerkzeugDegree bounds for local cohomology.https://www.zbmath.org/1456.140092021-04-16T16:22:00+00:00"Kustin, Andrew R."https://www.zbmath.org/authors/?q=ai:kustin.andrew-r"Polini, Claudia"https://www.zbmath.org/authors/?q=ai:polini.claudia"Ulrich, Bernd"https://www.zbmath.org/authors/?q=ai:ulrich.berndSummary: It has long been known how to read information about the socle degrees of the local cohomology \(\mathrm{H}_{\mathfrak{m}}^0 (M)\) of a graded module over a polynomial ring \(R\) from the twists in position \(d=\dim R\), in a resolution of \(M\) by free \(R\)-modules. It has also long been known how to use local cohomology to read valuable information from complexes which approximate resolutions in the sense that they have positive homology of small Krull dimension. The present paper reads information about the maximal generator degree (rather than the socle degree) of \(\mathrm{H}_{\mathfrak{m}}^0 ( M )\) from the twists in position \(d-1\) (rather than position \(d)\) in an approximate resolution of \(M\).
We apply the local cohomology results to draw conclusions about the maximal generator degree of the second symbolic power of the prime ideal defining a monomial curve and the second symbolic power of the ideal defining a finite set of points in projective space. There is also an application to hyperplane sections of subschemes of projective space and to partial Castelnuovo-Mumford regularity. Perhaps, the most important application is to the study of blow-up algebras and their defining equations. The techniques of the present paper are the main tool used in [the authors, Algebra Number Theory 11, No. 7, 1489--1525 (2017; Zbl 1386.13014)] to bound the degrees of these equations and thus to identify them in some cases.Linear Batalin-Vilkovisky quantization as a functor of \(\infty \)-categories.https://www.zbmath.org/1456.180182021-04-16T16:22:00+00:00"Gwilliam, Owen"https://www.zbmath.org/authors/?q=ai:gwilliam.owen"Haugseng, Rune"https://www.zbmath.org/authors/?q=ai:haugseng.runeThe authors consider a categorical construction of linear Batalin-Vilkovisky quantization in a derived setting.
The basic example that is the starting point for this article is the Weyl quantization, sending a symplectic vector space \(\mathbb R^{2n}\) to the Weyl algebra on \(2n\) generators. One can factor this construction as taking a vector space with a skew-symmetric form first to its Heisenberg Lie algebra and then to its universal envelopping algebra. The specalization at \(\hbar = 0\) of this universal envelopping algebra is a Poisson algebra and the specializiation at \(\hbar = 1\) is its quantizaiton.
The authors consider a special case of the shifted derived versions of this problem: Their starting point are chain complexes equipped with a 1-shifted symmetric pairing. Following the article we will call them quadratic modules for short.
They then construct \(\infty\)-categorical versions of both the Heisenberg Lie algebra (which is actually a shifted \(L_\infty\)-algebra) of a quadratic module, and the universal enveloping \(BD\)-algebra of a shifted Lie algebra. Both of these appear to be of independent interest.
The universal enveloping \(BD\)-algebra is a so-called Beilinson-Drinfeld algebra, a \(k[\hbar]\)-algebra over a certain operad that specialises to a shifted Poisson algebra at \(\hbar = 0\) and to an \(E_0\)-algebra at \(\hbar = 1\). (An \(E_0\)-algebra is just a pointed chain complex, but this is the correct edge case of the notion of \(E_n\)-algebras. The classical, unshifted case involves an unshifted Poisson algebra and an \(E_1\)-algebra (i.e.\ an associative algebra) as specializiations.)
Thus the authors are able to construct linear BV quantization as a symmetric monoidal \(\infty\)-functor from quadratic algebras to \(BD\)-algebras.
The proofs involve a mixture of categorical techniques (model, simplicial and \(\infty\)).
One upside of the \(\infty\)-categorical approach is that by using Lurie's descent theorem the author can consider linear BV quantization for sheaves of quadratic modules on derived stacks. Thus they are able to show that the graded vector bundle \(V \oplus V^\vee[1]\) with its obvious quadratic form quantizes to a line bundle. This is an explicit example of the BV formalism ``behaving like a determinant'', an idea the authors credit to K. Costello. The paper also provides an example that the behaviour for more general 1-shifted symplectic modules is more complicated and the quantization need only be invertible in the formal neighbourhood of a point.
The paper under review contains some interesting discussions in the introduction: Section 1.3 considers higher BV quantizations (which should arise from more general \((1-n)\)-shifted skew-symmetric forms) and a possible application to quantization of AKSZ field theories. Section 1.4 discusses the physical perspective on linear BV quantizations, providing useful context and motivation.
Reviewer: Julian Holstein (Hamburg)\(A_\infty \)-structures associated with pairs of 1-spherical objects and noncommutative orders over curves.https://www.zbmath.org/1456.140232021-04-16T16:22:00+00:00"Polishchuk, Alexander"https://www.zbmath.org/authors/?q=ai:polishchuk.alexander-eSummary: We show that pairs \((X,Y)\) of 1-spherical objects in \(A_\infty \)-categories, such that the morphism space \(\operatorname{Hom}(X,Y)\) is concentrated in degree 0, can be described by certain noncommutative orders over (possibly stacky) curves. In fact, we establish a more precise correspondence at the level of isomorphism of moduli spaces which we show to be affine schemes of finite type over \(\mathbb{Z} \).Noncommutative fiber products and lattice models.https://www.zbmath.org/1456.580072021-04-16T16:22:00+00:00"Hartwig, Jonas T."https://www.zbmath.org/authors/?q=ai:hartwig.jonas-tThis paper studies the representation theory of certain
noncommutative singular varieties using two-dimensional
lattice models. In more detail, the first main result
of the paper describes categories of weight modules over a
noncommutative biparametric deformation \(\mathcal{A}\) of
the fiber product of two Kleinian singularities of type \(A\)
in terms of periodic higher spin vertex configurations.
The second main result provides a combinatorial classification
of simple weight \(\mathcal{A}\)-modules. Finally, the third main
result describes the center of \(\mathcal{A}\), it turns our
that in some cases the center is trivial, while in some other
cases it is isomorphic to the algebra of Laurent polynomials
in one variable.
Reviewer: Volodymyr Mazorchuk (Uppsala)Representability theorem in derived analytic geometry.https://www.zbmath.org/1456.140182021-04-16T16:22:00+00:00"Porta, Mauro"https://www.zbmath.org/authors/?q=ai:porta.mauro"Yu, Tony Yue"https://www.zbmath.org/authors/?q=ai:yu.tony-yueIn the paper under review, the authors prove a representability theorem in derived analytic geometry, analogous to
Lurie's generalization of Artin's representablility criteria to derived algebraic geometry.
This is an important, standard type result for the study of moduli problems and
a crucial step towards a solid theory of derived analytic geometry.
More specifically, the authors show that a derived stack for the étale site of derived analytic spaces is a derived analytic stack
if and only if
it is compatible with Postnikov towers, has a global analytic cotangent complex, and its truncation is an analytic stack in the classical
(underived) sense.
The result applies both to complex analytic geometry and non-archimedean analytic geometry.
Central to representability results as in the present paper is deformation theory, which the authors develop here for the derived analytic setup.
The authors define an analytic version of the cotangent complex which controls the deformation theory of the derived stack.
As in the algebraic setting, the cotangent complex represents a functor of derivations.
One key step in order to define the analytic cotangent complex is the elegant description of the \(\infty\)-category of modules over a
derived analytic space \(X\) as the \(\infty\)-category of spectrum objects of a certain \(\infty\)-category associated with \(X\).
Another important construction is the analytification functor which they establish in the derived setting.
To apply derived geometry to classical moduli problems, one may try to enrich classical moduli spaces with derived structures. The paper under review is an important tool in verifying when such enrichments are indeed the correct ones.
Reviewer: Eric Ahlqvist (Stockholm)Noncommutative fibrations.https://www.zbmath.org/1456.160062021-04-16T16:22:00+00:00"Kaygun, Atabey"https://www.zbmath.org/authors/?q=ai:kaygun.atabeyThis paper shows that flat smooth extensions of associative unital algebras are reduced flat.
First, the author recalls the relevant results on reduced flat and smooth extensions. Then the reduced flat extensions and smooth extensions are identified in terms of homological conditions on the induction and restriction functors.
Then, the author shows that for reduced flat Galois fibration, the required long exact sequences in Hochschild homology and cyclic cohomology are obtained. The results are then applied to graph extension algebras.
Reviewer: Angela Gammella-Mathieu (Metz)