%0 Journal Article
%T The Grassmann algebra in arbitrary characteristic and generalized sign
%A Gal Dor
%A Alexei Kanel-Belov
%A Uzi Vishne
%J Mathematics
%D 2015
%I arXiv
%X 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 $2^{n-1}$.
%U http://arxiv.org/abs/1501.02464v1