Localisation of a module at a multiplicative subset

Depends on

• Unital left and right modules over a ring; unqualified module means left module
• Multiplicative subsets and the localisation $S^{-1}R$ as equivalence classes of fractions

Used by

• A Weil divisor that is not Cartier at the vertex of the quadric cone Counterexample
• Finite type need not be locally free Counterexample
• Quasi-separatedness in pushforward cannot be omitted Counterexample
• The intersection product needs Cartier or complementary-dimension hypotheses Counterexample
• Finite type and finitely presented module sheaves Definition
• Module sheaf on an affine scheme Definition
• A coherent closed-point skyscraper Example
• Localisation need not commute with infinite products Example
• Localising Z/12Z kills exactly the torsion seen by the denominator set Example
• Restricting an associated sheaf to a localization Example
• S-units of Q Example
• A dominant map has a surjective differential on a dense source open Lemma
• A localised module fraction is zero exactly when one denominator kills its numerator Lemma
• Addition of localised module fractions is independent of representatives Lemma
• Affine charts recover the algebraic module of differentials Lemma
• Associated primes lie in the support Lemma
• Closed immersions are affine quotients and survive base change Lemma
• Descent of modules on a finite principal cover Lemma
• Exact principal-open Cech resolution Lemma
• Finite is affine and local on its target Lemma
• Kähler differentials commute with localization Lemma
• Koszul Complex Localises Termwise Lemma
• Rationalization is exact and commutes with singular homology Lemma
• Schematic closure and agreement on a dense open Lemma
• Sections of the associated sheaf on basic opens Lemma
• Surjective module maps remain surjective after localisation Lemma
• The localised Hom map is an isomorphism for finite free sources Lemma
• The localised scalar action is independent of representatives Lemma
• The module-fraction relation is an equivalence relation Lemma
• The stalk of an associated sheaf is the localisation Lemma
• Closed immersions into affine schemes are quotient spectra Theorem
• Coherent sheaves on a locally Noetherian scheme Theorem
• Downward-closed intersections of primary components are intrinsic Theorem
• Quasi-coherence of pushforward for qcqs morphisms Theorem
• The associated module sheaf exists Theorem
• Universal property of localisation for modules Theorem

Dependency tree · two levels

7 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., Section 12 (standard reference, not scraped)
• The Stacks Project, Section 10.9: Localization (standard reference, not scraped)