Assuming the Axiom of Choice, Nakayama's lemma

Depends on

• The Axiom of Choice
• Assuming the Axiom of Choice, an element lies in the Jacobson radical exactly when one minus any multiple is a unit
• Determinant trick for Nakayama
• The submodule $IM$ generated by products of elements of an ideal $I$ with elements of a module $M$

Used by

• Assuming the Axiom of Choice, generators modulo an ideal in the Jacobson radical lift to generators Corollary
• Depth as the first nonzero Ext degree Corollary
• Euler characteristic in a proper flat family is locally constant Corollary
• Polynomial extension preserves Cohen--Macaulayness Corollary
• Upper semicontinuity of fibre cohomology dimensions Corollary
• Ramification index of a morphism of curves Definition
• Cohen–Macaulayness over a finite regular local base Example
• fields as regular local rings Example
• First-order deformations of a plane conic and of a quadric surface Example
• Ramification of the double cover y²=f(x) Example
• A birational morphism of regular surfaces factors through the blowup of a point where its inverse is undefined Lemma
• A finite module with free fibre and flat base is free over a Noetherian target Lemma
• A nonsingular formal arc on a normal Noetherian surface becomes regular Lemma
• A square-conic blowup has cubic-controlled singular successors Lemma
• Canonical pullback is surjective after blowing up a rational singular point Lemma
• Degree-p differential trace extends across normal surface valuations Lemma
• Depth at a prime is bounded by local support dimension Lemma
• Depth is infinite when the ideal acts surjectively Lemma
• depth two excludes finite punctured extension Lemma
• finite local modules admit minimal free resolutions Lemma
• Finite twisted resolutions over a regular local base Lemma
• Finite-free local criterion for cohomology and base change Lemma
• Finite-type algebraic group monomorphisms are closed immersions Lemma
• Finite-type field extensions with zero Ω Lemma
• Flat maps with geometrically regular fibres have standard smooth local presentations Lemma
• Flat schematic closure over an arbitrary valuation ring Lemma
• Free differentials imply regularity in characteristic zero Lemma
• H One Regular Local Implies Koszul Regular Lemma
• Hilbert divisor charts and the Picard diagonal Lemma
• koszul euler characteristic first element reduction Lemma
• koszul homology finite length for an ideal of definition Lemma
• Line bundles on a principal localization of a regular local ring are trivial Lemma
• Local algebra tools for elementary etale changes of classical varieties Lemma
• Local Koszul Acyclicity Inductive Converse Lemma
• Local Koszul H One Detects First Regularity Failure Lemma
• Local support and index bound for the different of a curve map Lemma
• Maximal regular sequences have a common Ext length Lemma
• minimal free resolutions unique up to chain isomorphism Lemma
• Poincare cohomology at the identity Lemma
• Positive canonical twists on projective Cohen–Macaulay curves Lemma

…and 22 more results.

Dependency tree · two levels

12 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

• A. Altman and S. Kleiman, A Term of Commutative Algebra, 13th ed., Exercise 10.12 (standard reference, not scraped)
• J. S. Milne, A Primer of Commutative Algebra, v4.03, Lemma 3.9 (standard reference, not scraped)