The Grassmann algebra in arbitrary characteristic and generalized sign

TL;DR

This paper introduces a generalized Grassmann algebra applicable over any commutative ring, including those where 2 is not invertible, enabling a consistent theory of superalgebras in arbitrary characteristic.

Contribution

It defines a new algebraic structure $rak{G}$ that extends Grassmann algebra to arbitrary rings, providing bases for identities and explicit co-module descriptions.

Findings

Provides a basis for multilinear identities of the free superalgebra with supertrace.

Shows all identities of $rak{G}$ derive from the Grassmann identity.

Explicitly describes co-modules as free modules of rank $2^{n-1}$.

Abstract

We define a generalization $\mathfrak{G}$ of the Grassmann algebra $G$ which is well-behaved over arbitrary commutative rings $C$, even when $2$ is not invertible. In particular, this enables us to define a notion of superalgebras that does not become degenerate in such a setting. Using this construction we are able 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 $\mathfrak{G}$ 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 co-module is a free $C$-module of rank $2^{n-1}$.