Recent zbMATH articles in MSC 16Ehttps://www.zbmath.org/atom/cc/16E2021-04-16T16:22:00+00:00WerkzeugRings in which every left zero-divisor is also a right zero-divisor and conversely.https://www.zbmath.org/1456.160392021-04-16T16:22:00+00:00"Ghashghaei, E."https://www.zbmath.org/authors/?q=ai:ghashghaei.ebrahim"Koşan, M. Tamer"https://www.zbmath.org/authors/?q=ai:kosan.muhammet-tamer"Namdari, M."https://www.zbmath.org/authors/?q=ai:namdari.mehrdad"Yildirim, T."https://www.zbmath.org/authors/?q=ai:yildirim.tulayOn the Hochschild homology of involutive algebras.https://www.zbmath.org/1456.160052021-04-16T16:22:00+00:00"Fernàndez-València, Ramsès"https://www.zbmath.org/authors/?q=ai:fernandez-valencia.ramses"Giansiracusa, Jeffrey"https://www.zbmath.org/authors/?q=ai:giansiracusa.jeffreyLet \(k\) be a field. An \textit{involutive vector space} is a \(k\)-vector space \(V\) equipped with an automorphism of order \(2\), denoted \(v\mapsto v^*\) and called its \textit{involution}. In other words, an involutive vector space is a \({\mathbb Z}/2\)-module. An \textit{involutive algebra} is an involutive vector space \(A\) together with an associative product such that
\[(ab)^*\,=\,b^*a^*.\]
Observe that commutative (associative) algebras are involutive algebras equipped with the identity as involution. Therefore one may think of the category of involutive algebras as some ``first-order-non-commutative-algebras'', lying in between commutative and arbitrary associative algebras.
\textit{C. Braun} defined in [J. Homotopy Relat. Struct. 9, No. 2, 317--337 (2014; Zbl 1321.16004)] a Hochschild cohomology theory for involutive algebras. Namely, let \(i\mathrm{Coder}(C)\) denote the space of involution-preserving coderivations for an involutive coalgebra \(C\). Then, as in Hochschild theory, the commutator \([m,-]\) with the product of an involutive algebra \(A\) defines a differential on the shifted tensor coalgebra on \(A\), i.e., on \(i\mathrm{Coder}(T\Sigma A)\). Braun defined the cohomology of (a complex isomorphic to) \(\Sigma^{-1}i\mathrm{Coder}(T\Sigma A)\) with respect to this differential as the Hochschild cohomology of the involutive algebra \(A\).
The authors of the article under review develop the necessary homological algebra to interpret Braun's Hochschild cohomology of an involutive algebra as an Ext functor in a suitable category. They start with a careful analysis of the category of involutive bimodules over an involutive algebra \(A\), showing that it is equivalent to the category of left modules over the \textit{involutive enveloping algebra} \(A^{ie}:=(A\otimes A^{\mathrm{op}})\otimes k[{\mathbb Z}/2]\). Then they study the Hom-Tensor adjunction as well as the correct analogues of the center and the abelianization construction of a bimodule in the involutive world.
Next follows the study of flat and projective involutive bimodules, in order to develop an involutive bar resolution. With these preliminaries in place, they pass to the Ext and Tor functors in the category of involutive bimodules. The main result is that Braun's definition and the derived functor definition compute the same cohomology, provided that the involutive algebra is projective as an involutive vector space.\par
Notably, the authors do not restrict to characteristic different from \(2\).
Reviewer: Friedrich Wagemann (Nantes)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)\(A_\infty \)-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 \(\mathbb{Z} \).Deformation theories controlled by Hochschild cohomologies.https://www.zbmath.org/1456.160282021-04-16T16:22:00+00:00"Carolus, Samuel"https://www.zbmath.org/authors/?q=ai:carolus.samuel"Hokamp, Samuel A."https://www.zbmath.org/authors/?q=ai:hokamp.samuel-a"Laubacher, Jacob"https://www.zbmath.org/authors/?q=ai:laubacher.jacobSummary: We explore how the higher order Hochschild cohomology controls a deformation theory when the simplicial set models the 3-sphere. Besides generalizing to the \(d\)-sphere for any \(d\ge 1\), we also investigate a deformation theory corresponding to the tertiary Hochschild cohomology, which naturally reduces to those studied for the secondary and usual Hochschild cohomologies under certain conditions.