具有任意特征和广义符号的Grassmann代数
The Grassmann algebra in arbitrary characteristic and generalized sign
摘要：
我们定义了Grassmann代数$G$的一个推广$\mathfrak{G}$，它在任意交换环$C$上是良好的，即使当$2$不可逆时也是如此.特别是，这使我们能够定义一个概念的超代数，不成为退化在这样的设置。利用这种构造，我们能够提供一个基的非分次多线性恒等式的自由超代数的超迹，有效的任何环。我们还证明了$\mathfrak{G}$的所有恒等式都是从Grassmann恒等式导出的，并明确给出了它的余模，这些余模是符号表示的推广.特别地，我们证明了余模是秩为$2^{n-1}$的自由$C$ -模。
英文摘要：
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 $2^{n-1}$.
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