Geometrically Regular Ring¶
In algebraic geometry, a geometrically regular ring is a Noetherian ring over a field that remains a regular ring after any finite extension of the base field.
Core Idea¶
Geometrically Regular Ring is treated here as the recurring mathematics, logic, and statistics identity summarized by this source-grounded definition: In algebraic geometry, a geometrically regular ring is a Noetherian ring over a field that remains a regular ring after any finite extension of the base field. In algebraic geometry, a geometrically regular ring is a Noetherian ring over a field that remains a regular ring after any finite extension of the base field. Geometrically regular schemes are defined in a similar way.
Scope of Application¶
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Examples. gave the following two examples of local rings that are regular but not geometrically regular.
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Examples. Suppose that k is a field of characteristic p > 0 and a is an element of k that is not a pth power.
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Examples. Then every point of the curve x p + y p = a is regular.
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Examples. However over the field k[a 1/p ], every point of the curve is singular.
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Examples. So the points of this curve are regular but not geometrically regular.
Clarity¶
A clear use of Geometrically Regular Ring names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In algebraic geometry, a geometrically regular ring is a Noetherian ring over a field that remains a regular ring after any finite extension of the base field.
Manages Complexity¶
Geometrically Regular Ring compresses multiple mathematics, logic, and statistics details into a stable diagnostic relation. The source shows both the central mechanism—suppose that k is a field of characteristic p > 0 and a is an element of k that is not a pth power.—and the practical consequence—in the previous example, the equation defining the curve becomes reducible over a finite extension of the base field.
Abstract Reasoning¶
- Type the carrier. Identify the mathematics, logic, and statistics entities to which the claim applies.
- State the relation. Use the source-grounded identity: In algebraic geometry, a geometrically regular ring is a Noetherian ring over a field that remains a regular ring after any finite extension of the base field.
- Check operation and conditions. Then every point of the curve x p + y p = a is regular.
- Demand recognition evidence. However over the field k[a 1/p ], every point of the curve is singular.
Knowledge Transfer¶
Within the home domain. Knowledge about Geometrically Regular Ring transfers literally when a new case preserves the same carrier type, relation, and recognition test. gave the following two examples of local rings that are regular but not geometrically regular. Suppose that k is a field of characteristic p > 0 and a is an element of k that is not a pth power. Beyond the home domain. No canonical parent is asserted for Geometrically Regular Ring.
Relationships to Other Abstractions¶
Current abstraction Geometrically Regular Ring Domain-specific
Parents (1) — more general patterns this builds on
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Geometrically Regular Ring is a kind of Ring Domain-specific
A geometrically regular ring is a ring retaining regularity after finite base-field extensions.
Hierarchy paths (5) — routes to 5 parentless roots
- Geometrically Regular Ring → Ring → Group → Monoid → Semigroup → Set and Membership
- Geometrically Regular Ring → Ring → Group → Monoid → Identity Element
- Geometrically Regular Ring → Ring → Group → Monoid → Semigroup → Closure
- Geometrically Regular Ring → Ring → Group → Monoid → Semigroup → Associativity → Invariance
- Geometrically Regular Ring → Ring → Group → Monoid → Semigroup → Associativity → Symmetry
Neighborhood in Abstraction Space¶
Geometrically Regular Ring sits in a sparse region of the domain-specific corpus (67th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Unclustered & Miscellaneous (2551 abstractions)
Nearest neighbors
- Projective variety — 0.86
- Regular ideal — 0.84
- Terminal singularity — 0.84
- Filling radius — 0.84
- Unibranch local ring — 0.84
Computed from structural-signature embeddings · 2026-10-08