On Kontsevich’s Hochschild cohomology conjecture

Quantizing Poisson Manifolds

This paper extends Kontsevich’s ideas on quantizing Poisson manifolds. A new differential is added to the Hodge decomposition of the Hochschild complex, so that it becomes a bicomplex, even more similar to the classical Hodge theory for complex manifolds. These notes grew out of the author’s attempt to understand Kontsevich’s ideas [Kon95a] on quantizing Poisson manifolds. We introduce a new di...

Second Hochschild Cohomology of Convolution Algebras

A Counterexample to a Conjecture of Barr

We discuss two versions of a conjecture attributed to M Barr The Harrison cohomology of a commutative algebra is known to coincide with the Andr e Quillen cohomology over a eld of characteristic zero but not in prime characteristics The conjecture is that a modi ed version of Harrison cohomology taking into account torsion always agrees with Andr e Quillen cohomology We give a counterexample De...

A Solution of Deligne's Hochschild Cohomology Conjecture

Deligne asked in 1993 whether the Hochschild cochain complex of an associative ring has a natural action by the singular chains of the little 2-cubes operad. In this paper we give an affirmative answer to this question. We also show that the topological Hochschild cohomology spectrum of an associative ring spectrum has an action by an operad that is equivalent to the little 2-cubes operad.

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...