Auslander–Buchsbaum formula
E223665
The Auslander–Buchsbaum formula is a fundamental result in commutative algebra that relates the projective dimension of a finitely generated module over a Noetherian local ring to the depth of the module and the depth of the ring.
All labels observed (3)
How this entity was disambiguated
This entity first appeared as the object of triple T1994340 — 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: Auslander–Buchsbaum formula Context triple: [Hilbert’s syzygy theorem, relatedTo, Auslander–Buchsbaum formula]
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A.
Hilbert’s syzygy theorem
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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B.
Krull dimension
Krull dimension is a fundamental invariant in commutative algebra that measures the "size" of a ring by the maximum length of chains of its prime ideals.
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C.
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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D.
Deuring reduction theorem
The Deuring reduction theorem is a result in number theory that relates the reduction of elliptic curves with complex multiplication modulo primes to the theory of quaternion algebras and endomorphism rings.
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E.
Hilbert’s fourteenth problem
Hilbert’s fourteenth problem is one of David Hilbert’s famous list of 23 problems, concerning the finite generation of certain algebras of invariants in algebraic geometry and invariant theory.
- 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: Auslander–Buchsbaum formula Target entity description: The Auslander–Buchsbaum formula is a fundamental result in commutative algebra that relates the projective dimension of a finitely generated module over a Noetherian local ring to the depth of the module and the depth of the ring.
-
A.
Hilbert’s syzygy theorem
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.
-
B.
Krull dimension
Krull dimension is a fundamental invariant in commutative algebra that measures the "size" of a ring by the maximum length of chains of its prime ideals.
-
C.
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.
-
D.
Deuring reduction theorem
The Deuring reduction theorem is a result in number theory that relates the reduction of elliptic curves with complex multiplication modulo primes to the theory of quaternion algebras and endomorphism rings.
-
E.
Hilbert’s fourteenth problem
Hilbert’s fourteenth problem is one of David Hilbert’s famous list of 23 problems, concerning the finite generation of certain algebras of invariants in algebraic geometry and invariant theory.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical theorem
ⓘ
theorem in commutative algebra ⓘ |
| appliesTo |
Noetherian local ring
ⓘ
finitely generated module ⓘ |
| area | algebra ⓘ |
| assumption |
M has finite projective dimension
ⓘ
M is a finitely generated R-module ⓘ R is a Noetherian local ring ⓘ |
| category | homological algebra results ⓘ |
| conclusion | projective dimension of M equals depth(R) minus depth_R(M) ⓘ |
| context |
homological algebra
ⓘ
local algebra ⓘ |
| field | commutative algebra ⓘ |
| historicalPeriod | 20th century mathematics ⓘ |
| holdsWhen | the projective dimension of the module is finite ⓘ |
| implies | modules of finite projective dimension over a regular local ring are free ⓘ |
| involvesIdeal | maximal ideal m of R ⓘ |
| involvesInvariant |
depth(R)
ⓘ
depth_R(M) ⓘ pd_R(M) ⓘ |
| involvesRing | local ring (R, m) ⓘ |
| isFundamentalResultIn | commutative algebra ⓘ |
| languageOfFormulation | ring theory ⓘ |
| namedAfter |
David Buchsbaum
ⓘ
Maurice Auslander ⓘ |
| relatedConcept |
Cohen–Macaulay ring
ⓘ
Ext functor ⓘ Tor functor ⓘ global dimension ⓘ regular local ring ⓘ |
| relatedTo |
Auslander–Buchsbaum formula
self-linksurface differs
ⓘ
surface form:
Auslander–Buchsbaum theorem
Serre’s characterization of regular local rings ⓘ |
| relatesConcept |
depth of a module
ⓘ
depth of a ring ⓘ projective dimension ⓘ |
| requires |
Noetherian hypothesis on the ring
ⓘ
local hypothesis on the ring ⓘ |
| statement |
Auslander–Buchsbaum formula
self-linksurface differs
ⓘ
surface form:
For a Noetherian local ring (R, m) and a finitely generated R-module M of finite projective dimension, pd_R(M) + depth_R(M) = depth(R).
|
| symbolicForm | pd_R(M) + depth_R(M) = depth(R) ⓘ |
| usedFor |
characterizing regular local rings via projective dimension
ⓘ
relating homological and depth invariants of modules ⓘ |
| usedIn |
classification of modules over regular local rings
ⓘ
computations of projective dimension ⓘ study of depth and Cohen–Macaulay modules ⓘ |
| usesConcept |
depth
ⓘ
homological dimension ⓘ projective resolution ⓘ regular sequence ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Auslander–Buchsbaum formula Description of subject: The Auslander–Buchsbaum formula is a fundamental result in commutative algebra that relates the projective dimension of a finitely generated module over a Noetherian local ring to the depth of the module and the depth of the ring.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
For a Noetherian local ring (R, m) and a finitely generated R-module M of finite projective dimension, pd_R(M) + depth_R(M) = depth(R).
this entity surface form:
Auslander–Buchsbaum theorem