For a polynomial $f = x_1^n + \dots + x_N^n$ let $G_f$ be the non--abelian maximal group of symmetries of $f$. This is a group generated by all $g \in \mathrm{GL}(N,\mathbb{C})$, rescaling and permuting the variables, so that $f(\mathbf{x}) = f(g \cdot \mathbf{x})$. For any $G \subseteq G_f$ we compute explicitly Hochschild cohomology of the category of $G$--equivarint matrix factorizations of $f$. We introduce the pairing on it showing that it is a Frobenius algebra.
Given a finite group action on a (suitably enhanced) triangulated category linear over a field, we establish a formula for the Hochschild cohomology of the category of invariants, assuming the order of the group is coprime to the characteristic of the base field. The formula shows that the cohomology splits canonically with one summand given by the invariant subspace of the Hochschild cohomology of the original category. We also prove that Serre functors act trivially on Hochschild cohomology, and combine this with our formula to give a useful mechanism for computing the Hochschild cohomology …
We discuss the Hochschild cohomology of the category of D-modules associated to an algebraic stack. In particular we describe the Hochschild cohomology of the category of torus-equivariant D-modules as the cohomology of a D-module on the loop space of the quotient stack. Finally, we give an approach for understanding the Hochschild cohomology of D-modules on general stacks via a relative compactification of the diagonal. This work is motivated by a desire to understand the support theory (in the sense of Benson-Iyengar-Krause) of D-modules on stacks.
We define the normal Hochschild cohomology of an admissible subcategory of the derived category of coherent sheaves on a smooth projective variety $X$ --- a graded vector space which controls the restriction morphism from the Hochschild cohomology of $X$ to the Hochschild cohomology of the orthogonal complement of this admissible subcategory. When the subcategory is generated by an exceptional collection, we define its new invariant (the height) and show that the orthogonal to an exceptional collection of height $h$ in the derived category of a smooth projective variety $X$ has the same Hochsc…
We develop a formalism involving Atiyah classes of sheaves on a smooth manifold, Hochschild chain and cochain complexes. As an application we prove a version of the Riemann--Roch theorem.
We study the multiplicative structure of orbifold Hochschild cohomology in an attempt to generalize the results of Kontsevich and Calaque-Van den Bergh relating the Hochschild and polyvector field cohomology rings of a smooth variety. We introduce the concept of linearized derived scheme, and we argue that when $X$ is a smooth algebraic variety and $G$ is a finite abelian group acting on $X$, the derived fixed locus $\widetilde{X^G}$ admits an HKR linearization. This allows us to define a product on the cohomology of polyvector fields of the orbifold $[X/G]$. We analyze the obstructions to ass…
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 derived endomorphisms of the kernel giving the corresponding projection functor, and the Hochschild homology is isomorphic to derived morphisms from this kernel to its convolution with the kernel of the Serre functor. We investigate some basic properties of Hochschild homology and cohomology of admissible subcategories. In particular, we check that the Hoch…
We discuss how the Hochschild cohomology of a dg category can be computed as the trace of its Serre functor. Applying this approach to the principal block of the Bernstein--Gelfand--Gelfand category $\mathcal{O}$, we obtain its Hochschild cohomology as the compactly supported cohomology of an associated space. Equivalently, writing $\mathcal{O}$ as modules over the endomorphism algebra $A$ of a minimal projective generator, this is the Hochschild cohomology of $A$. In particular our computation gives the Euler characteristic of the Hochschild cohomology of $\mathcal{O}$ in type A.
We interpret Hochschild cohomology as the Lie algebra of the derived Picard group (in the sense of Rouquier-Zimmermann and Yekutieli) and deduce that it is preserved under derived equivalences.
We show that Higher Hochschild complex associated to a connected pointed simplicial set commutes with localization of commutative algebras over a field of characteristic zero. Then, we define in two ways higher order Hochschild cohomology of schemes over a field of characteristic zero. Originally, we can take the hyperext functor of the sheaf associated to Higher Hochschild presheaf. We obtain a Hodge decomposition for higher order Hochschild cohomology of smooth algebraic varieties over a field of characteristic zero which generalizes Pirashvili's Hodge decomposition. We can also define the H…
We discuss homological mirror symmetry of Fermat polynomials in terms of derived Morita equivalence between derived categories of coherent sheaves and Fukaya-Seidel categories (a.k.a. perfect derived categories of directed Fukaya categories), and some related aspects such as stability conditions, (kinds of) modular forms, and Hochschild homologies.
The classical HKR-theorem gives an isomorphism of the n-th Hochschild cohomology of a smooth algebra and the n-th exterior power of its module of Kähler differentials. Here we generalize it for simplicial, graded and anticommutative objects in ``good pairs of categories''. We apply this generalization to complex spaces and noetherian schemes and deduce two decomposition theorems for their (relative) Hochschild cohomology (special cases of those were recently shown by Buchweitz-Flenner and Yekutieli). The first one shows that Hochschild cohomology contains tangent cohomology: $\HH^n(X/Y,\sM)=\c…
We determine the product structure on Hochschild cohomology of commutative algebras in low degrees, obtaining the answer in all degrees for complete intersection algebras. As applications, we consider cyclic extension algebras as well as Hochschild and ordinary cohomology of finite abelian groups.
For an affine toric variety $\mathrm{Spec}(A)$, we give a convex geometric description of the Hodge decomposition of its Hochschild cohomology. Under certain assumptions we compute the dimensions of the Hodge summands $T^1_{(i)}(A)$, generalizing the existing results about the Andre-Quillen cohomology group $T^1_{(1)}(A)$. We prove that every Poisson structure on a possibly singular affine toric variety can be quantized in the sense of deformation quantization.
We present a detailed computation of the cyclic and the Hochschild homology and cohomology of generic and 3-Calabi-Yau homogeneous down-up algebras. This family was defined by Benkart and Roby in their study of differential posets. Our calculations are completely explicit, by making use of the Koszul bimodule resolution and arguments similar to those used by the second and third authors for Yang-Mills algebras.
For an odd prime $p$ satisfying Vandiver's conjecture, we give explicit formulae for the action of the absolute Galois group $G_{\mathbb{Q}(ζ_p)}$ on the homology of the degree $p$ Fermat curve, building on work of Anderson. Further, we study the invariants and the first Galois cohomology group which are associated with obstructions to rational points on the Fermat curve.
We give a description of the Hochschild cohomology for noncommutative planes (resp. quadrics) using the automorphism groups of the elliptic triples (resp. quadruples) that classify the Artin-Schelter regular $\mathbb{Z}$-algebras used to define noncommutative planes and quadrics. For elliptic triples the description of the automorphism groups is due to Bondal-Polishchuk, for elliptic quadruples it is new.
For a flat commutative $k$-algebra $A$ such that the enveloping algebra $A\otimes_k A$ is noetherian, given a finitely generated bimodule $M$, we show that the adic completion of the Hochschild cohomology module $HH^n(A/k,M)$ is naturally isomorphic to $HH^n(\widehat{A}/k,\widehat{M})$. To show this, we (1) make a detailed study of derived completion as a functor $D(A) \to D(\widehat{A})$ over a non-noetherian ring $A$; (2) prove a flat base change result for weakly proregular ideals; and (3) Prove that Hochschild cohomology and analytic Hochschild cohomology of complete noetherian local rings…
We show that for a noetherian algebra $A$ whose bounded dg derived category is smooth, the singular Hochschild cohomology (=Tate--Hochschild cohomology) is isomorphic, as a graded algebra, to the Hochschild cohomology of the dg singularity category of $A$. The existence of such an isomorphism is suggested by recent work of Zhengfang Wang.
We perform Hochschild homology calculations in the algebro-geometric setting of motives. The motivic Hochschild homology coefficient ring contains torsion classes which arise from the mod-$p$ motivic Steenrod algebra and from generating functions on the natural numbers with finite non-empty support. Under the Betti realization, we recover Bökstedt's calculation of the topological Hochschild homology of finite prime fields.
Dimer models provide a method of constructing noncommutative crepant resolutions of affine toric Gorenstein threefolds. In homological mirror symmetry, they can also be used to describe noncommutative Landau--Ginzburg models dual to punctured Riemann surfaces. For a zigzag consistent dimer embedded in a torus, we explicitly compute the Hochschild cohomology of its Jacobi algebra in terms of dimer combinatorics. This includes a full characterization of the Batalin--Vilkovisky structure induced by the Calabi--Yau structure of the Jacobi algebra. We then compute the compactly supported Hochschild…
