On inverting the Koszul complex

Abstract

Let V be an n-dimensional vector space. We give a direct construction of an exact sequence that gives a GL(V)-equivariant "resolution" of each symmetric power S^t V in terms of direct sums of tensor products of the form \wedge^{i_1} V \otimes ... \otimes \wedge^{i_p} V. This exact sequence corresponds to inverting the relation in the representation ring of GL(V) that is described by the Koszul complex, and has appeared before in work by B. Totaro, analogously to a construction of K. Akin involving the normalized bar resolution. Our approach yields a concrete description of the differentials, and provides an alternate direct proof that Ext^t_{\wedge (V^*)}(k,k) = S^t V.