Hochschild (Co)Homology of Differential Operator Rings

Hochschild and Ordinary Cohomology Rings of Small Categories

Let C be a small category and k a field. There are two interesting mathematical subjects: the category algebra kC and the classifying space |C| = BC. We study the ring homomorphism HH∗(kC) → H∗(|C|, k) and prove it is split surjective, using the factorization category of Quillen [16] and certain techniques from functor cohomology theory. This generalizes the well-known results for finite groups...

THE TANGENT COMPLEX AND HOCHSCHILD COHOMOLOGY OF En-RINGS

In this work, we study the deformation theory of En-rings and the En analogue of the tangent complex, or topological André-Quillen cohomology. We prove a generalization of a conjecture of Kontsevich, that there is a fiber sequence A[n − 1] → TA → HHEn (A)[n], relating the En-tangent complex and En-Hochschild cohomology of an En-ring A. We give two proofs: The first is direct, reducing the probl...

Second Hochschild Cohomology of Convolution Algebras

Higher order Hochschild cohomology

Following ideas of Pirashvili, we define higher order Hochschild cohomology over spheres S defined for any commutative algebra A and module M . When M = A, we prove that this cohomology is equipped with graded commutative algebra and degree d Lie algebra structures as well as with Adams operations. All operations are compatible in a suitable sense. These structures are related to Brane topology...

Alternated Hochschild Cohomology

In this paper we construct a graded Lie algebra on the space of cochains on a Z2-graded vector space that are skew-symmetric in the odd variables. The Lie bracket is obtained from the classical Gerstenhaber bracket by (partial) skew-symmetrization; the coboundary operator is a skew-symmetrized version of the Hochschild differential. We show that an order-one element m satisfying the zero-square...