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:

  • xy - (-1)^
    xy
    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)^

xy
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.

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