Tor as a module over an exterior algebra

Let S be a regular local ring with residue field k and let M be a finitely generated S-module. Suppose that f(1), . . . , f(c) is an element of S is a regular sequence that annihilates M, and let E be an exterior algebra over k generated by c elements. The homotopies for the f(i) on a free resolution of M induce a natural structure of graded E-module on Tor(S) (M, k). In the case where M is a high syzygy over the complete intersection R:= S/(f(1), ..., f(c) ) we describe this E-module structure in detail, including its minimal free resolution over E. Turning to Ext(R) (M, k) we show that, when M is a high syzygy over R, the minimal free resolution of Ext(R) (M, k) as a module over the ring of CI operators is the Bernstein-Gel'fand- Gel'fand dual of the E-module Tor(S) (M, k). For the proof we introduce higher CI operators, and give a construction of a (generally nonminimal) resolution of M over S starting from a resolution of M over R and its higher CI operators.