Depth (ring theory)¶

A homological invariant measuring the length of a maximal regular sequence acting on a module, equivalently the first degree of nonvanishing Ext under standard local Noetherian hypotheses.

Version: v1 · 2026-09-08 · History
Domain-specific #: 4113
Origin domain: commutative algebra
Subdomain: commutative algebra

Core Idea¶

For a finitely generated module over a commutative Noetherian local ring, depth records how many successive non-zero-divisors can be chosen from the maximal ideal before unavoidable torsion appears. Regular elements successively reduce the module without annihilating nonzero classes; Ext, local cohomology and resolution theorems identify the maximum possible length with equivalent vanishing thresholds. 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¶

Depth (ring theory) belongs to commutative algebra and is useful where the analyst can specify the typed commutative algebra carrier, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, then evaluate the ring, ideal or local maximal ideal, module and finiteness hypotheses, regular-sequence convention and equivalent Ext or local-cohomology degree are explicit. The scope is broad within that domain but bounded by the need for the ring, ideal or local maximal ideal, module and finiteness hypotheses, regular-sequence convention and equivalent Ext or local-cohomology degree are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the ring, ideal or local maximal ideal, module and finiteness hypotheses, regular-sequence convention and equivalent Ext or local-cohomology degree 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. A bare label is insufficient because the name Depth (ring theory) can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated.

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 Depth (ring theory). Depth (ring theory) 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, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the ring, ideal or local maximal ideal, module and finiteness hypotheses, regular-sequence convention and equivalent Ext or local-cohomology degree 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, defining objects and relations, parameters, conventions, evidence, boundary cases, and comparison targets, Regular elements successively reduce the module without annihilating nonzero classes; Ext, local cohomology and resolution theorems identify the maximum possible length with equivalent vanishing thresholds., and type the carrier, state every parameter and convention in the definition, test that the ring, ideal or local maximal ideal, module and finiteness hypotheses, regular-sequence convention and equivalent Ext or local-cohomology degree are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Depth (ring theory) Domain-specific

Parents (1) — more general patterns this builds on

Depth (ring theory) is a kind of Dimension Prime

The proposed strict upward parent is prime:dimension.

Hierarchy path (1) — routes to 1 parentless root

Depth (ring theory) → Dimension

Neighborhood in Abstraction Space¶

Depth (ring theory) sits in a crowded region of the domain-specific corpus (5^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Homological Ring & Scheme Invariants (13 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08