Artin–Tate lemma¶
If A is commutative Noetherian, C is a finite-type A-algebra and finite as a module over an intermediate A-subalgebra B, then B is finite type over A.
Core Idea¶
Finite means module-finite while finite type means algebra-finitely-generated, commutativity and Noetherian hypotheses are essential and the inclusion chain must be fixed. Generators of C over A and module generators over B yield finitely many coefficients in B; their Noetherian subalgebra controls relations and proves that B is module-finite over a finite-type A-algebra. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
Artin–Tate lemma belongs to commutative algebra and is useful where the analyst can specify the typed commutative algebra carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the commutative rings and inclusions A to B to C, Noetherian property of A, finite-type A-algebra structure of C, finite B-module structure of C, conclusion that B is finite type over A, distinction of finite and finite type and relation to Nullstellensatz and Eakin–Nagata are explicit.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the commutative rings and inclusions A to B to C, Noetherian property of A, finite-type A-algebra structure of C, finite B-module structure of C, conclusion that B is finite type over A, distinction of finite and finite type and relation to Nullstellensatz and Eakin–Nagata are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Artin–Tate lemma. Artin–Tate lemma compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed commutative algebra carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the commutative rings and inclusions A to B to C, Noetherian property of A, finite-type A-algebra structure of C, finite B-module structure of C, conclusion that B is finite type over A, distinction of finite and finite type and relation to Nullstellensatz and Eakin–Nagata are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of commutative algebra because they reuse the typed commutative algebra carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, Generators of C over A and module generators over B yield finitely many coefficients in B; their Noetherian subalgebra controls relations and proves that B is module-finite over a finite-type A-algebra., and type the carrier, state every parameter and convention in the definition, test that the commutative rings and inclusions A to B to C, Noetherian property of A, finite-type A-algebra structure of C, finite B-module structure of C, conclusion that B is finite type over A, distinction of finite and finite type and relation to Nullstellensatz and Eakin–Nagata are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Artin–Tate lemma Domain-specific
Parents (1) — more general patterns this builds on
-
Artin–Tate lemma is a kind of Inference Prime
The proposed strict upward parent is
prime:inference.
Hierarchy path (1) — routes to 1 parentless root
- Artin–Tate lemma → Inference → Rationality → Normativity → Constraint
Neighborhood in Abstraction Space¶
Artin–Tate lemma sits in a crowded region of the domain-specific corpus (16th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Commutative Algebra & Localization (16 abstractions)
Nearest neighbors
- Manin matrix — 0.92
- Ring of mixed characteristic — 0.92
- Hall algebra — 0.91
- Associated graded ring — 0.91
- Koszul–Tate resolution — 0.91
Computed from structural-signature embeddings · 2026-09-08