Graded-symmetric algebra

Short description: Type of algebra over a commutative ring

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 odd

for 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. ISBN:0-387-94268-8

External links

• "rt.representation theory - Definition of the symmetric algebra in arbitrary characteristic for graded vector spaces". MathOverflow. https://mathoverflow.net/q/7080. Retrieved 2017-04-18.