Graded-symmetric algebra explained

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:

xy - ( -1 ) ^{|x||y|} yx x ^{2} when |x | is oddfor homogeneous elements x, y in M of degree |x |, |y |. By construction, a graded-symmetric algebra is graded-commutative; i.e., xy = ( -1 ) ^{|x||y|} yx 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.

External links

• Web site: rt.representation theory - Definition of the symmetric algebra in arbitrary characteristic for graded vector spaces . MathOverflow . 2017-04-18.