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In [[mathematics]], '''Hilbert's syzygy theorem''' is a result of [[commutative algebra]], first proved by [[David Hilbert]] (1890) in connection with the [[Syzygy (mathematics)|syzygy]] (relation) problem of [[invariant theory]]. Roughly speaking, starting with relations between [[invariant polynomial|polynomial invariant]]s, then relations between the relations, and so on, it explains ''how far'' one has to go to reach a clarified situation. It is now considered to be an early result of [[homological algebra]], and through the [[depth (algebra)|depth]] concept, to be a measure of the [[non-singularity]] of [[affine space]]. | |||
== Formal statement == | |||
In modern language, the theorem may be stated as follows. Let ''k'' be a [[field (mathematics)|field]] and ''M'' a finitely generated [[module (mathematics)|module]] over the [[polynomial ring]] | |||
:<math>k[x_1,\ldots,x_n].</math> | |||
Hilbert's syzygy theorem then states that there exists a [[free resolution]] of ''M'' of length at most ''n''. | |||
== See also == | |||
* [[Quillen–Suslin theorem]] | |||
* [[Hilbert polynomial]] | |||
== References == | |||
* [[David Eisenbud]], ''Commutative algebra. With a view toward algebraic geometry''. Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995. xvi+785 pp. ISBN 0-387-94268-8; ISBN 0-387-94269-6 {{MathSciNet|id=1322960}} | |||
* {{springer|title=Hilbert theorem|id=p/h047410}} | |||
[[Category:Commutative algebra]] | |||
[[Category:Homological algebra]] | |||
[[Category:Invariant theory]] | |||
[[Category:Theorems in abstract algebra]] | |||
Revision as of 18:28, 7 December 2013
In mathematics, Hilbert's syzygy theorem is a result of commutative algebra, first proved by David Hilbert (1890) in connection with the syzygy (relation) problem of invariant theory. Roughly speaking, starting with relations between polynomial invariants, then relations between the relations, and so on, it explains how far one has to go to reach a clarified situation. It is now considered to be an early result of homological algebra, and through the depth concept, to be a measure of the non-singularity of affine space.
Formal statement
In modern language, the theorem may be stated as follows. Let k be a field and M a finitely generated module over the polynomial ring
![{\displaystyle k[x_{1},\ldots ,x_{n}].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c109f59bd548c669a7645880f875a11422061bb)
Hilbert's syzygy theorem then states that there exists a free resolution of M of length at most n.
See also
References
- David Eisenbud, Commutative algebra. With a view toward algebraic geometry. Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995. xvi+785 pp. ISBN 0-387-94268-8; ISBN 0-387-94269-6 Template:MathSciNet
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