Hilbert’s syzygy theorem
E43323
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Hilbert’s syzygy theorem canonical | 2 |
| Hilbert’s theorem on projective dimension | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T326975 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Hilbert’s syzygy theorem Context triple: [David Hilbert, notableWork, Hilbert’s syzygy theorem]
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A.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
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B.
Hilbert basis theorem
The Hilbert basis theorem is a fundamental result in commutative algebra stating that if a ring is Noetherian then any polynomial ring over it is also Noetherian, ensuring that ideals in such rings are finitely generated.
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C.
Noether normalization lemma
The Noether normalization lemma is a fundamental result in commutative algebra and algebraic geometry that shows any finitely generated algebra over a field can be made integral over a polynomial subring, providing a way to relate complicated varieties to affine space.
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D.
Noether's isomorphism theorems
Noether's isomorphism theorems are fundamental results in abstract algebra that relate quotient structures and substructures of groups, rings, and modules, providing a unifying framework for understanding homomorphic images and factor structures.
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E.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Hilbert’s syzygy theorem Target entity description: 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.
-
A.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
-
B.
Hilbert basis theorem
The Hilbert basis theorem is a fundamental result in commutative algebra stating that if a ring is Noetherian then any polynomial ring over it is also Noetherian, ensuring that ideals in such rings are finitely generated.
-
C.
Noether normalization lemma
The Noether normalization lemma is a fundamental result in commutative algebra and algebraic geometry that shows any finitely generated algebra over a field can be made integral over a polynomial subring, providing a way to relate complicated varieties to affine space.
-
D.
Noether's isomorphism theorems
Noether's isomorphism theorems are fundamental results in abstract algebra that relate quotient structures and substructures of groups, rings, and modules, providing a unifying framework for understanding homomorphic images and factor structures.
-
E.
Gauss’s lemma in number theory
Gauss’s lemma in number theory is a result that relates the Legendre symbol to the number of sign changes in a certain sequence of multiples, providing a practical criterion for determining quadratic residues modulo an odd prime.
- F. None of above. chosen
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’s basis theorem ⓘ
linked to:
Hilbert basis theorem
Hilbert’s theorem on projective dimension ⓘ
linked to:
Hilbert’s syzygy theorem
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 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Hilbert’s syzygy theorem Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Hilbert’s syzygy theorem