The Grassmann algebra over arbitrary rings and minus sign in arbitrary characteristic

Abstract

An analog in characteristic 2 2 for the Grassmann algebra G G was essential in a counterexample to the long standing Specht conjecture. We define a generalization G \mathfrak {G} of the Grassmann algebra, which is well-behaved over arbitrary commutative rings C C , even when 2 2 is not invertible. This lays the foundation for a supertheory over arbitrary base ring, allowing one to consider general deformations of superalgebras. The construction is based on a generalized sign function. It enables us to provide a basis of the non-graded multilinear identities of the free superalgebra with supertrace, valid over any ring. We also show that all identities of G \mathfrak {G} follow from the Grassmann identity, and explicitly give its co-modules, which turn out to be generalizations of the sign representation. In particular, we show that the n n th co-module is a free C C -module of rank 2 n − 1 2^{n-1} .