Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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Equicharacteristic local rings and coefficient fields
Definition
Let be a local ring with residue field .
The local ring is equicharacteristic when and are equal.
A coefficient field of is a subfield such that the residue map restricts to an isomorphism
Thus a coefficient field is not merely an embedded field: it is an embedded copy of the residue field itself.
Depends on
Used by
- Complete equicharacteristic local rings have coefficient fields Corollary
- A coefficient field maps isomorphically to the residue field Lemma
- Completeness resolves the purely inseparable prime-field case Lemma
- Maximal residue-injective subfields exist Lemma
- The prime field lifts in the equicharacteristic case Lemma
- Transcendental residue elements adjoin across a maximal subfield Lemma
Dependency tree · two levels
6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- The Stacks Project, Section 10.160: The Cohen structure theorem (standard reference, not scraped)
- Allen B. Altman and Steven L. Kleiman, A Term of Commutative Algebra, 13th ed., Chapter 22 (standard reference, not scraped)