Tor as a module over an exterior algebra

Source abstract

Let S be a regular local ring with residue field k and let M be a finitely generated S -module. Suppose that f_1,\dots ,f_c \in 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,\dots,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 non-minimal) resolution of M over S starting from a resolution of M over R and its higher CI operators.