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A localised module fraction is zero exactly when one denominator kills its numerator
Statement
For and ,
Facts & Assumptions
Given: A commutative ring , a multiplicative subset , a left -module , an element , and an element .
In , the equality means that for some (Localisation of a module at a multiplicative subset).
Proof
If , then [L1] gives for some .
If for some , then , so [L1] gives in .
Steps 1.1 and 1.2 prove the equivalence.
Depends on
Used by
- Global sections do not determine a sheaf on P1 Counterexample
- Quasi-separatedness in pushforward cannot be omitted Counterexample
- Finite power map of the affine line Example
- Localising cyclic abelian groups and Q/Z at a prime Example
- Localising Z/12Z kills exactly the torsion seen by the denominator set Example
- A prime lies in the support exactly when some element has annihilator inside it Lemma
- Associated primes lie in the support Lemma
- Finite is affine and local on its target Lemma
- Finite projective complex for proper flat coherent cohomology Lemma
- Finite-type field extensions with zero Ω Lemma
- Fpqc covers are universally submersive Lemma
- Geometric Nakayama for finite-type sheaves Lemma
- Injective module maps remain injective after localisation Lemma
- Sections of a sheaf flat over the base are flat over affine opens Lemma
- The Alexander module of a link complement is finitely presented Lemma
- The localised Hom map is an isomorphism for finite free sources Lemma
- The quotient by a maximal line subbundle is locally free Lemma
- Assuming the Axiom of Choice, local criteria for zero modules and for injective, surjective, and bijective maps Theorem
- Coherent sheaves on a locally Noetherian scheme Theorem
- Finite coherent cohomology for proper schemes Theorem
- Localisation of Hom for finite and finitely presented modules Theorem
- Quasi-coherence of pushforward for qcqs morphisms Theorem
Dependency tree · two levels
3 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., (12.2) (standard reference, not scraped)