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

• The quasi-finite locus of a finite-type algebra is open Corollary
• Outside a domain, the nonzero elements need not be multiplicative: 2·3=0 in ℤ/6 Counterexample
• Quasi-finite does not imply finite Counterexample
• 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
• The field of fractions Frac(D)=(D∖{0})⁻¹D of an integral domain Definition
• The localization presheaf on distinguished opens Definition
• total ring of fractions Definition
• Localization is flat and has vanishing positive Tor Example
• The punctured affine line as an open finite factorization Example
• A prime containing an ideal and avoiding a multiplicative set Lemma
• A quasi-finite one-generator quotient is locally its integral closure 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
• 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 localised scalar action is independent of representatives Lemma
• Algebraic Zariski Main localization at a quasi-finite prime Theorem
• Base change and composition of standard smooth presentations Theorem
• Every quotient and every localisation of a Noetherian ring is Noetherian Theorem
• Integrality and integral closure commute with localisation Theorem
• The localisation relation is an equivalence relation and fraction arithmetic is well defined Theorem
• Two coprime projective plane forms meet in total length equal to their degree product Theorem

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)