In algebra, a graded-commutative ring (also called a skew-commutative ring) is a graded ring that is commutative in the graded sense; that is, homogeneous elements x, y satisfy
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x
y
=
(
−
1
)
|
x
|
|
y
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y
x
,
{\displaystyle xy=(-1)^{|x||y|}yx,}
where |x| and |y| denote the degrees of x and y.
A commutative (non-graded) ring, with trivial grading, is a basic example. For a nontrivial example, an exterior algebra is generally not a commutative ring but is a graded-commutative ring.
A cup product on cohomology satisfies the skew-commutative relation; hence, a cohomology ring is graded-commutative. In fact, many examples of graded-commutative rings come from algebraic topology and homological algebra.
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
- Beck, Kristen A.; Sather-Wagstaff, Keri Ann (2013-07-01). "A somewhat gentle introduction to differential graded commutative algebra". arXiv:1307.0369 [math.AC].
See also