Regular scheme¶
A locally Noetherian scheme whose every local ring is regular, so local dimension equals the minimal number of generators of its maximal ideal.
Core Idea¶
A regular scheme is nonsingular in the intrinsic local-ring sense at every point. Local algebra compares tangent-space dimension with Krull dimension; equality at all points rules out excess infinitesimal directions while remaining sensitive to base-field issues. 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.
The load-bearing residual is not the broad topic of algebraic geometry. It is A locally Noetherian scheme whose every local ring is regular, so local dimension equals the minimal number of generators of its maximal ideal.
Scope of Application¶
Regular scheme belongs to algebraic geometry and is useful where the analyst can specify a locally Noetherian scheme, points, local rings, maximal ideals, Krull dimensions and embedding dimensions, then evaluate every local ring is Noetherian and regular under the declared dimension criterion. The scope is broad within that domain but bounded by the need for every local ring is Noetherian and regular under the declared dimension criterion. 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 every local ring is Noetherian and regular under the declared dimension criterion 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 Regular scheme 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 Regular scheme. Regular scheme 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: a locally Noetherian scheme, points, local rings, maximal ideals, Krull dimensions and embedding dimensions. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express every local ring is Noetherian and regular under the declared dimension criterion independently of one notation or implementation. This step prevents the canonical example from becoming the definition.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of algebraic geometry because they reuse a locally Noetherian scheme, points, local rings, maximal ideals, Krull dimensions and embedding dimensions, Local algebra compares tangent-space dimension with Krull dimension; equality at all points rules out excess infinitesimal directions while remaining sensitive to base-field issues., and type the carrier, state every parameter and convention in the definition, test that every local ring is Noetherian and regular under the declared dimension criterion, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Regular scheme Domain-specific
Parents (1) — more general patterns this builds on
-
Regular scheme is a kind of Classification Prime
The proposed strict upward parent is
prime:classification.
Hierarchy path (1) — routes to 1 parentless root
- Regular scheme → Classification
Neighborhood in Abstraction Space¶
Regular scheme sits in a crowded region of the domain-specific corpus (26th 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
- Formal scheme — 0.92
- Sheaf of algebras — 0.91
- Depth (ring theory) — 0.91
- Morphism of schemes — 0.91
- Dimension of an algebraic variety — 0.90
Computed from structural-signature embeddings · 2026-09-08