Graded-symmetric algebra

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:

  • x y − ( − 1 ) | x | | y | y x {\displaystyle xy-(-1)^{|x||y|}yx}
  • x 2 {\displaystyle 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., x y = ( − 1 ) | x | | y | y x {\displaystyle 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
  • "rt.representation theory - Definition of the symmetric algebra in arbitrary characteristic for graded vector spaces". MathOverflow. Retrieved 2017-04-18.


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