A Note on Symmetrizing Forms for Fully Group-Graded Algebras
✍ Scribed by Morton E Harris
- Publisher
- Elsevier Science
- Year
- 2001
- Tongue
- English
- Weight
- 98 KB
- Volume
- 237
- Category
- Article
- ISSN
- 0021-8693
No coin nor oath required. For personal study only.
✦ Synopsis
Throughout this paper, all rings are assumed to have identities and all modules over a ring are assumed to be unitary and finitely generated over the ring. We shall also assume that O is a commutative ring and that R is an O-algebra. We shall let R × denote the multiplicative group of units of R.
Recall (cf. [2, Definition 2.1]), a linear functional t ∈ Hom O R O on R that is symmetric (i.e., t rs = t sr for all r s ∈ R) and is such that the O-linear map t R → Hom O R O such that r -→ t r * for all r ∈ R is an isomorphism is called a symmetrizing form for R. If R is projective in O -mod and has a symmetrizing form, then R is said to be a symmetric algebra.
From now on and throughout the remainder of the paper, we shall also assume that G is a finite group and that R is a fully G-graded O-algebra; that is, R = g∈G R g in O -mod where R g is an O-submodule of R for each g ∈ G and R g R h = R gh for all g h ∈ G.
If R g contains a unit of R for each g ∈ G, then R is called a G-crossed product algebra. If there is a subgroup of r R , the group of graded units of R, such that ∩ R g = 1 for all g ∈ G, then R is called a skew G-algebra. (Note that in the literature, fully G-graded and strongly G-graded are synonymous (cf. [5]).)
The classic example of a skew G-algebra is the group ring of G over O OG = g∈G Og. Moreover t OG → O such that g∈G a g g → a 1 for all a g ∈ O and g ∈ G is, as is well known, a symmetrizing form for OG. 789