Given a graded-commutative ring acting centrally on a triangulated category, the main result of this talk shows that if the cohomology of a pair of objects of the triangulated category is finitely generated over the ring acting centrally, then the asymptotic vanishing of the cohomology is well-behaved. In particular, enough consecutive asymptotic vanishing of cohomology implies all eventual vanishing. Several key applications are also given.
A few years ago, Bhatt-Morrow-Scholze introduced an invariant of p-adic formal schemes called syntomic cohomology, which has a close relationship to (étale-localized) algebraic K-theory. In a recent paper, Antieau-Mathew-Morrow-Nikolaus showed that, after inverting p, syntomic cohomology admits a concrete description in terms of more familiar invariants, such as de Rham and crystalline cohomology. In this talk, I'll explain an alternative perspective on their result, which avoids the use of K-theoretic methods.
In this talk, I will first describe how classical Dieudonne module of finite flat group schemes and p-divisible groups can be recovered from crystalline cohomology of classifying stacks. Then, I will explain how in mixed characteristics, using classifying stacks, one can define Dieudonné module of a finite locally free group scheme as a prismatic F-gauge (prismatic F-gauges have been recently introduced by Drinfeld and Bhatt-Lurie), which gives a fully faithful functor from finite locally free group schemes over a quasi-syntomic algebra to the category of prismatic F-gauges. This can be seen as a generalization of the work of Anschütz-Le Bras on "prismatic Dieudonne theory" to torsion situations.
Stable equivalences occur frequently in the representation theory of finite-dimensional algebras; however, these equivalences are poorly understood. An interesting class of stable equivalences is obtained by ‘gluing’ two idempotents. More precisely, let A be a finite-dimensional algebra with a simple projective module and a simple injective module. Assume that B is a subalgebra of A having the same Jacobson radical. Then B is constructed by identifying the two idempotents belonging to the simple projective module and to the simple injective module, respectively. In this talk we will compare the first Hochschild cohomology groups of finite-dimensional monomial algebras under gluing two arbitrary idempotents (hence not necessarily inducing a stable equivalence). As a corollary, we will show that stable equivalences obtained by gluing two idempotents provide 'some functoriality' to the first Hochschild cohomology, that is, HH1(A) is isomorphic to a quotient of HH1(B).
The whole structure given by the Hochschild cohomology and homology of an associative algebra A together with the cup and cap products, the Gerstenhaber bracket and the Connes differential is called the Tamarkin-Tsygan calculus. It is invariant under derived equivalence and if we can compute all these invariants provides a lot of information. The calculation of the whole Tamarkin-Tsygan calculus is very difficult and generally not even possible for particular algebras. However, there exist some calculations for individual algebras. The problem is, in general, that the minimal projective bimodule resolutions are difficult to find and even if one is able to compute such a resolution, it might be so complicated that the computation of the Tamarkin-Tsygan calculus is not within reach. For monomial algebras the minimal projective bimodule resolution is known and in the case of quadratic monomial algebras it is simple enough, to embark on the extensive calculations of the Tamarkin Tsygan calculus. Yet even for quadratic monomial algebras, the combinatorial level of the calculations is such that it is too complicated to calculate the whole calculus. On the other hand for gentle algebras, the additional constraints on their structure are such that the calculations become possible. We will focus on the concrete aspects of these calculations.
In this talk, I will discuss some recent advances in the theory of motives in the context of rigid analytic geometry. Building on work of Ayoub, Bondarko, we provide an equivalence between the category of “unipotent” rigid analytic motives over a non-archimedean field and the category of “monodromy maps” M → M (−1) of algebraic motives over the residue field. This allows us to build a unified framework for the study of monodromy operators and weight filtrations of cohomology theories for varieties over a local field. As an application, we give a streamlined definition of Hyodo–Kato cohomology without recourse to log-geometry, as predicted by Fontaine, and we produce an induced Clemens–Schmid chain complex.
The theory of condensed sets, developed by Dustin Clausen and Peter Scholze, provides a framework well-suited to study algebraic objects that carry a topology. In my talk, I will discuss the basic properties of the cohomology of condensed groups and its relation to continuous group cohomology. Johannes Anschütz and Arthur-César le Bras showed that for locally profinite groups and solid (e.g. discrete) coefficients, condensed group cohomology agrees with continuous group cohomology. On the other hand, if G is a locally compact and locally contractible topological group (e.g., a Lie group), and M is a discrete group with trivial G-action, then the condensed group cohomology of G, M (the sheaves of continuous functions into G and M) is isomorphic to the singular cohomology of the classifying space of G with coefficients in M, whereas the continuous group cohomology of G with coefficients in M is isomorphic to the singular cohomology of the classifying space of π0(G) with coefficients in M.
Generalizing results of Johannes Anschütz and Arthur-César le Bras on locally profinite groups, I will explain that continuous group cohomology with solid coefficients can be described as a cohomological δ-functor in the condensed setting for a large class of topological groups.
We show that the celebrated Friedlander-Suslin theorem - on finite generation of cohomology of a finite group scheme G over a field - remains valid for a finite flat group scheme G over a commutative noetherian ring. In view of earlier work it suffices to put a uniform bound, depending on G only, on torsion in cohomology of G-modules.
Murray Gerstenhaber constructed a graded Lie structure (Gerstenhaber bracket) on Hochschild cohomology, which makes Hochschild cohomology a Lie algebra. However, it is not easy to calculate bracket structure with the original definition. There is an alternative technique to compute Gerstenhaber bracket on Hochschild cohomology, introduced by Chris Negron and Sarah Witherspoon. It is also known that Hopf algebra cohomology has a bracket and the bracket is trivial when a Hopf algebra is quasi-triangular. We use a similar technique to the technique given by Negron and Witherspoon to calculate the Lie structure on Hochschild cohomology of the Taft algebra Tp for any integer p>2 which is a nonquasi-triangular Hopf algebra. Then, we find the corresponding bracket on Hopf algebra cohomology of Tp. We show that the bracket is indeed zero on Hopf algebra cohomology of Tp, as in all known quasi-triangular Hopf algebras. This example is the first known bracket computation for a nonquasi triangular algebra.
Murray Gerstenhaber constructed Lie structure (Gerstenhaber bracket) on Hochschild cohomology, which makes Hochschild cohomology a graded Lie algebra. Later, it is shown that Hopf algebra cohomology also has a Lie structure. We will introduce a general formula for the bracket on Hopf algebra cohomology of any Hopf algebra with bijective antipode on the bar resolution that is reminiscent of Gerstenhaber’s original formula for Hochschild cohomology.
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