Hilbert’s syzygy theorem

E43323

mathematical theorem theorem in commutative algebra

Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.

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Observed surface forms (1)

Surface form	Occurrences
Hilbert’s theorem on projective dimension	1

Statements (49)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in commutative algebra ⓘ
appliesTo	finitely generated modules over polynomial rings ⓘ polynomial rings over a field ⓘ standard graded polynomial rings ⓘ
assumes	finitely generated module ⓘ polynomial ring in finitely many variables ⓘ
characterizes	global dimension of polynomial rings ⓘ
concerns	homological dimension ⓘ minimal free resolutions ⓘ projective dimension of modules ⓘ syzygies of modules ⓘ
describes	finite length of minimal free resolutions over polynomial rings ⓘ structure of free resolutions of modules over polynomial rings ⓘ
field	algebraic geometry ⓘ commutative algebra ⓘ homological algebra ⓘ
formalizes	finite generation of higher syzygies over polynomial rings ⓘ
hasConsequence	bounds length of minimal free resolutions by number of variables ⓘ controls number of steps needed to resolve a module by free modules ⓘ gives homological characterization of polynomial rings ⓘ
historicalPeriod	late 19th century mathematics ⓘ
holdsFor	finitely generated graded modules over a standard graded polynomial ring ⓘ polynomial rings in finitely many variables over a field ⓘ
implies	global dimension of a polynomial ring in n variables over a field is n ⓘ polynomial rings over a field are regular rings ⓘ polynomial rings over a field have finite global dimension ⓘ
influenced	development of homological algebra ⓘ modern commutative algebra ⓘ
isPartOf	Hilbert’s work on invariant theory ⓘ
namedAfter	David Hilbert ⓘ
relatedTo	Auslander–Buchsbaum formula ⓘ Betti numbers of graded modules ⓘ Castelnuovo–Mumford regularity ⓘ Hilbert basis theorem ⓘ surface form: Hilbert’s basis theorem Hilbert’s syzygy theorem self-linksurface differs ⓘ surface form: Hilbert’s theorem on projective dimension minimal graded free resolution ⓘ
statesThat	every finitely generated module over a polynomial ring in n variables over a field has a free resolution of length at most n ⓘ projective dimension of a finitely generated module over a polynomial ring in n variables over a field is at most n ⓘ
usedIn	algebraic geometry via coordinate rings of varieties ⓘ computational commutative algebra ⓘ homological characterization of regular local rings ⓘ study of graded modules over polynomial rings ⓘ theory of minimal free resolutions of ideals ⓘ
usesConcept	Noetherian ring ⓘ free resolution ⓘ graded module ⓘ polynomial ring ⓘ syzygy ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

David Hilbert → notableWork → Hilbert’s syzygy theorem ⓘ

Hilbert’s syzygy theorem → relatedTo → Hilbert’s syzygy theorem self-linksurface differs ⓘ

this entity surface form: Hilbert’s theorem on projective dimension