Quillen Stratification for Hochschild Cohomology of Blocks

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...

Second Hochschild Cohomology of Convolution Algebras

Spaces of Algebra Structures and Cohomology of Operads

The aim of this paper is two-fold. First, we compare two notions of a "space" of algebra structures over an operad A: 1. the classification space, which is the nerve of the category of weak equivalences of A-algebras, and 2. the moduli space A{X}, which is the space of maps from A to the endomorphism operad of an object X. We show that under certain hypotheses the moduli space of A-algebra stru...

On Commutative Algebra Homology in Prime Characteristics

We give decompositions of the Hochschild and cyclic homology of a com-mutative algebra in characteristic p into p?1 parts, and show these decom-positions are compatible with the shuue product structures. We also give a counterexample to a conjecture attributed to Barr, which asserts that a modiied version of Harrison cohomology coincides with Andr e/Quillen cohomology.

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...