Graded-symmetric algebra
{{Short description|Type of algebra over a commutative ring}}
{{one source |date=May 2024}}
In algebra, given a commutative ring R, the graded-symmetric algebra of a graded R-module M is the quotient of the tensor algebra of M by the ideal I generated by elements of the form:
- when |x| is odd
for homogeneous elements x, y in M of degree |x|, |y|. By construction, a graded-symmetric algebra is graded-commutative; i.e., and is universal for this.
In spite of the name, the notion is a common generalization of a symmetric algebra and an exterior algebra: indeed, if V is a (non-graded) R-module, then the graded-symmetric algebra of V with trivial grading is the usual symmetric algebra of V. Similarly, the graded-symmetric algebra of the graded module with V in degree one and zero elsewhere is the exterior algebra of V.
References
- David Eisenbud, Commutative Algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, vol 150, Springer-Verlag, New York, 1995. {{ISBN|0-387-94268-8}}
External links
- {{cite web|url=https://mathoverflow.net/q/7080 |title=rt.representation theory - Definition of the symmetric algebra in arbitrary characteristic for graded vector spaces |publisher=MathOverflow |date= |accessdate=2017-04-18}}
{{algebra-stub}}