A COUNTEREXAMPLE TO A CONJECTURE OF BARR

ABSTRACT. We discuss two versions of a conjecture attributed to M. Barr. The Harrison cohomology of a commutative algebra is known to coincide with the André/ Quillen cohomology over a field of characteristic zero but not in prime characteristics. The conjecture is that a modified version of Harrison cohomology, taking into account torsion, always agrees with André/Quillen cohomology. We give a counterexample. 1. Definitions and Notation Let A be a commutative algebra over a commutative ring k and let M be an A-bimodule. We denote ⊗k simply by ⊗. Consider the Hochschild cochain complex (Homk(A n;M); ), where is the Hochschild coboundary, f(a1 ⊗ · · · ⊗ an) = a1f(a2 ⊗ · · · ⊗ an) n1∑ i=1 (−1)if(a1 ⊗ · · · ⊗ aiai+1 ⊗ · · · ⊗ an) + (−1)nf(a1 ⊗ · · · ⊗ an1)an: The symmetric group Σn acts on the left on A n by (a1 ⊗... ⊗ an) = (a11 ⊗... ⊗ a1n); and we extend this linearly to an action of the group algebra kΣn. Let T c(A) denote the cotensor algebra over A. That is, (T c(A))n = A n, with graded commutative multiplication the signed shuffle, : A p ⊗ A q → A p+q, ((a1 ⊗ · · · ⊗ ap) ⊗ (ap+1 ⊗ · · · ⊗ ap+q)) = (sgn)(a1 ⊗ · · · ⊗ ap+q); where the sum is over (p; q)-shuffles in Σp+q. Harrison cohomology, Harr(A;M), is the homology of the subcomplex of the Hoch-schild cochain complex consisting of those cochains which vanish on decomposable ele-ments for the shuffle product. This commutative algebra cohomology theory is known to coincide with the André/Quillen theory when A is projective over k and k contains Q [4]. An example of André, described in [1], shows that the theories do not coincide in prime characteristics. I would like to thank M. Barr for helpful comments