Multiplicative subsets and the localisation $S^{-1}R$ as equivalence classes of fractions

Depends on

• Commutative ring
• Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$
• The units of a ring are the invertible elements of its multiplicative monoid, and $R^{\times}$ is a group under multiplication; $0 \in R^{\times}$ only in the zero ring

Used by

• Rings of integers are Dedekind domains Corollary
• The quasi-finite locus of a finite-type algebra is open Corollary
• A nonradical ideal need not enlarge every tangent space Counterexample
• A nonseparated affine cover can have nonaffine intersection Counterexample
• Outside a domain, the nonzero elements need not be multiplicative: 2·3=0 in ℤ/6 Counterexample
• Quasi-finite does not imply finite Counterexample
• Affine-local quasi-coherent algebras before general sheaf theory Definition
• Localisation at a prime ideal: Rₚ=(R setminusp)⁻¹R Definition
• Localisation of a module at a multiplicative subset Definition
• Principal localisation R_f={1,f,f²,…}⁻¹R Definition
• Sheaf total quotient rings Definition
• The field of fractions Frac(D)=(D∖{0})⁻¹D of an integral domain Definition
• The HOMFLYPT coefficient ring Definition
• The Laurent polynomial ring as the principal localisation of Z[t] at t Definition
• The localization presheaf on distinguished opens Definition
• The Temperley-Lieb quotient and the Jones specialization Definition
• Total quotient ring and normalisation of a reduced plane curve germ Definition
• total ring of fractions Definition
• Localization is flat and has vanishing positive Tor Example
• S-units of Q Example
• The punctured affine line as an open finite factorization Example
• A meromorphic unit has locally finite nonzero order support Lemma
• A prime containing an ideal and avoiding a multiplicative set Lemma
• A quasi-compact image stable under specialization is closed Lemma
• A quasi-finite one-generator quotient is locally its integral closure Lemma
• Affine blowup algebras: normal form, nonzerodivisors, reducedness, domains Lemma
• Base change of standard smooth presentations Lemma
• Every ideal of a localisation is generated by the images of any generating set of its contraction Lemma
• Field extension preserves the graded pieces and the total length of a zero-dimensional projective quotient Lemma
• Finite normalization commutes with principal localization Lemma
• Finite-free local criterion for cohomology and base change Lemma
• Fraction-field coefficients of an integral LKB class are Laurent polynomials Lemma
• Kähler differentials commute with localization Lemma
• Localisation at a homogeneous element is graded, with graded kernels and dehomogenised degree-zero parts Lemma
• Primes of a localization avoid the denominator set Lemma
• Quasi-finite local fibres transfer through quotients and intermediate rings Lemma
• Radicals commute with localization Lemma
• Standard smooth algebras are finitely presented and flat Lemma
• Substituting y = 1/f and clearing denominators yields a power of f in I Lemma
• The fork-noodle pairing is well defined and equivariant Lemma

…and 12 more results.

Dependency tree · two levels

13 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 10.9: Localization (standard reference, not scraped)