Graded-symmetric algebra

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; here the ideal I is generated by elements of the form:

• ${\displaystyle xy-(-1)^{|x||y|}yx}$
• $x^{2}$ 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., $xy=(-1)^{{|x||y|}}yx$.

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

• http://mathoverflow.net/questions/7080/definition-of-the-symmetric-algebra-in-arbitrary-characteristic-for-graded-vecto