## Abstract

An analog in characteristic 2 for the Grassmann algebra G was essential in a counterexample to the long standing Specht conjecture. We define a generalization G of the Grassmann algebra, which is well-behaved over arbitrary commutative rings C, even when 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 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 nth co-module is a free C-module of rank 2^{n−1}.

Original language | English |
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Pages (from-to) | 227-253 |

Number of pages | 27 |

Journal | Transactions of the American Mathematical Society Series B |

Volume | 7 |

DOIs | |

State | Published - 2020 |

### Bibliographical note

Publisher Copyright:© 2020 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0).

### Funding

Received by the editors July 9, 2015, and, in revised form, February 27, 2018, and August 7, 2019. 2010 Mathematics Subject Classification. Primary 16R10; Secondary 17A70, 16R30, 16R50. Key words and phrases. Superalgebra, generalized Grassmann algebra, generalized sign, polynomial identities, trace identities. This work was supported by BSF grant 2010/149, ISF grants 1207/12 and 1994/20, and RSF grant 17-11-01377.

Funders | Funder number |
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United States-Israel Binational Science Foundation | 2010/149 |

Israel Science Foundation | 1207/12, 1994/20 |

Russian Science Foundation | 17-11-01377 |

## Keywords

- Generalized Grassmann algebra
- Generalized sign
- Polynomial identities
- Superalgebra
- Trace identities