A ring of cohomological operators on Ext and Tor

Abstract

For any homomorphism of commutative rings $A \to B$ and a $B$-module $M$, we construct a structure of graded $\mathrm{S}_B^{\ast}\mathrm{H}^1(A,B,B)$-module on $\mathrm{Ext}_B^{\ast}(M,M)$ and $\mathrm{Tor}_{\ast}^B(M,M)$, where $\mathrm{H}^1(A,B,B)$ is the first Andr\'e-Quillen cohomology module and $\mathrm{S}^{\ast}$ denotes the symmetric algebra. This structure generalizes the well known structures of $B[X_1, \dots, X_n]$-module constructed by Gulliksen when $B=R/I$ and $I$ is generated by a regular sequence of length $n$ (in this case, $\mathrm{S}_B^{\ast}\mathrm{H}^1(R,B,B)= \mathrm{S}_B^{\ast} \left(\widehat{I/I^2}\right)=B[X_1, \dots, X_n]$), but the main interest is that for any such $B=R/I$, Gulliksen operations factorize through the ring $\mathrm{S}_B^{\ast}\mathrm{H}^1(A,B,B)$ for any ring $A$ such that $R$ is an $A$-algebra, allowing us to think that, for some purposes, in order to define these cohomological operations, Andr\'e-Quillen cohomology is a more natural choice than the normal module $\widehat{I/I^2}$.