Abstract
This introductory Chapter aims at establishing, together with the basic notation and terminology, some elementary results about Z2-graded algebra that we shall constantly use in the sequel. The topics covered include Z2-graded rings and modules, Z2-graded tensor algebra, Lie superalgebras, and matrices with entries in a Z2-graded commutative ring.
(Number has two species, odd and even, whilst the third is the even-odd, which is a mixture of both. Many forms there are of both species, and every thing on its own reveals them.) Philolaos
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In accordance with Bourbaki’s terminology, any ring has an identity.
We only deal with tensor products of finite families of graded modules; a more general treatment can be found in [Ma2].
The characteristic of a graded ring R can be defined as follows. Let φ: Z → R0 be the unique ring morphism such that 1 ↦ 1. The kernel of φ is an ideal of Z, and therefore is the set of multiples of an integer p, which is by definition the characteristic of R.
In this discussion, ‘homogeneous’ refers, as usual, to the Z2-gradation.
The numerical factors appearing in the following equation, as well as in other equations in this subsection, are determined by the choice of the projection TPM* → Alt(MP; R). Here we follow the conventions of [KN].
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© 1991 Springer Science+Business Media Dordrecht
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Bartocci, C., Bruzzo, U., Hernández-Ruipérez, D. (1991). Elements of graded algebra. In: The Geometry of Supermanifolds. Mathematics and Its Applications, vol 71. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-3504-7_1
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DOI: https://doi.org/10.1007/978-94-011-3504-7_1
Publisher Name: Springer, Dordrecht
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