https://en.formulasearchengine.com/w/api.php?action=feedcontributions&feedformat=atom&user=91.239.236.16 2026-08-03T16:49:01Z User contributions MediaWiki 1.47.0-wmf.7 https://en.formulasearchengine.com/w/index.php?title=Bond_order&diff=8100 2013-12-07T16:28:02Z

<p>91.239.236.16: Dropped unmeaningful conjunction &#039;while&#039; from second sentence.</p> <hr /> <div>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]].<br /> <br /> == Formal statement ==<br /> 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]] <br /> <br /> :&lt;math&gt;k[x_1,\ldots,x_n].&lt;/math&gt;<br /> <br /> Hilbert's syzygy theorem then states that there exists a [[free resolution]] of ''M'' of length at most ''n''.<br /> <br /> == See also ==<br /> * [[Quillen–Suslin theorem]]<br /> * [[Hilbert polynomial]]<br /> <br /> == References ==<br /> * [[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}}<br /> * {{springer|title=Hilbert theorem|id=p/h047410}}<br /> <br /> [[Category:Commutative algebra]]<br /> [[Category:Homological algebra]]<br /> [[Category:Invariant theory]]<br /> [[Category:Theorems in abstract algebra]]</div>

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