Related papers: Reflexivity and Hochschild Cohomology
For a finite dimensional algebra $A$, we prove that the bounded homotopy category of projective $A$-modules and the bounded derived category of $A$-modules are dual to each other via certain categories of locally-finite cohomological…
Hochschild (co)homology and Pirashvili's higher order Hochschild (co)homology are useful tools for a variety of applications including deformations of algebras. When working with higher order Hochschild (co)homology, we can consider the…
In our paper "On D-module of categories I", we provide two different methods of constructing D-module structures on the complex computing periodic cyclic homology associated to a family of stable infinity categories. One is based on a…
Let $k$ be a field, let $R$ be a commutative ring, and assume the exponential characteristic of $k$ is invertible in $R$. In this note, we prove that isomorphisms in Voevodsky's triangulated category of motives $\mathcal{DM}(k;R)$ are…
We construct the cohomology groups with compact support of stacks of shtukas with $\mathbb Z_{\ell}$-coefficients. We construct the cuspidal cohomology groups and prove that they are $\mathbb Z_{\ell}$-modules of finite type. We prove that…
Smooth and proper dg-algebras have an Euler class valued in the Hochschild homology of the algebra. This Euler class is worthy of this name since it satisfies many familiar properties including compatibility with the familiar pairing on the…
Rigid meromorphic cocycles are defined in the setting of orthogonal groups of arbitrary real signature and constructed in some instances via a $p$-adic analogue of Borcherds' singular theta lift. The values of rigid meromorphic cocycles at…
Exact categories are a natural generalisation of abelian categories and provide a fertile ground to develop relative homological algebra. In this paper, starting from a class of relative Gorenstein projective objects in an exact category…
We present an alternative construction of Soergel's category of bimodules associated to a reflection faithful representation of a Coxeter system. We show that its objects can be viewed as sheaves on the associated moment graph. We introduce…
Behrend, Liao, and Xu showed that differential graded (DG) manifolds of positive amplitude forms a category of fibrant objects. In particular, this ensures that notion of derived intersection -- more generally, homotopy fibre product -- is…
We study interactions between the categories of $\D$-modules on smooth and singular varieties. For a large class of singular varieties $Y$, we use an extension of the Grothendieck--Sato formula to show that $\D_Y$-modules are equivalent to…
We prove basic facts about reflexivity in derived categories over noetherian schemes; and about related notions such as semidualizing complexes, invertible complexes, and Gorenstein-perfect maps. Also, we study a notion of rigidity with…
We show that indecomposable exact module categories over the category Rep H of representations of a finite-dimensional Hopf algebra H are classified by left comodule algebras, H-simple from the right and with trivial coinvariants, up to…
We show that the classical Hochschild homology and (periodic and negative) cyclic homology groups are representable in the category of motives with modulus. We do this by constructing Hochschild homology and (periodic and negative) cyclic…
The notion of Hochschild cochains induces an assignment from $Aff$, affine DG schemes, to monoidal DG categories. We show that this assignment extends, under some appropriate finiteness conditions, to a functor $\mathbb H: Aff \to…
In a compactly generated triangulated category, we introduce a class of tilting objects satisfying certain purity condition. We call these the decent tilting objects and show that the tilting heart induced by any such object is equivalent…
In this paper we construct a symmetric monoidal closed model category of coherently commutative Picard groupoids. We construct another model category structure on the category of (small) permutative categories whose fibrant objects are…
We transport Steenrod's cup-i products from the singular cochains on the free loop space Maps(S^1, BG) to Hochschild's original cochain complex Hom (k[G]^*, k[G]) defining Hochschild cohomology. Here G is a discrete group, k an arbitrary…
We investigate Hochschild cohomology and homology of admissible subcategories of derived categories of coherent sheaves on smooth projective varieties. We show that the Hochschild cohomology of an admissible subcategory is isomorphic to the…
We study monoidal categories that enjoy a certain weakening of the rigidity property, namely, the existence of a dualizing object in the sense of Grothendieck and Verdier. We call them Grothendieck-Verdier categories. Notable examples…