In
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, given a
commutative ring
In mathematics, a commutative ring is a Ring (mathematics), ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring prope ...
''R'', the graded-symmetric algebra of a
graded ''R''-module ''M'' is the
quotient
In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...
of the
tensor algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra over a field, algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', ...
of ''M'' by the
ideal ''I'' generated by elements of the form:
*
*
when , ''x'', is
odd
for
homogeneous element
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that . The index set is usually the set of nonnegative integers or the set of integers, but ...
s ''x'', ''y'' in ''M'' of degree , ''x'', , , ''y'', . By construction, a graded-symmetric algebra is
graded-commutative In Abstract algebra, 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
:xy = (-1)^ yx,
where , ''x'', and , ''y'', ...
; i.e.,
and is universal for this.
In spite of the name, the notion is a common generalization of a
symmetric algebra
In mathematics, the symmetric algebra (also denoted on a vector space over a field is a commutative algebra over that contains , and is, in some sense, minimal for this property. Here, "minimal" means that satisfies the following universal ...
and an
exterior algebra
In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector ...
: 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
David Eisenbud (born 8 April 1947 in New York City) is an American mathematician. He is a professor of mathematics at the University of California, Berkeley and former director of the then Mathematical Sciences Research Institute (MSRI), now k ...
, ''Commutative Algebra. With a view toward algebraic geometry'',
Graduate Texts in Mathematics
Graduate Texts in Mathematics (GTM) () is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with va ...
, vol 150,
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 in ...
, New York, 1995.
External links
*
Ring theory
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