Principal localisation $R_f=\{1,f,f^2,\dots {\}}^{-1}R$

Depends on

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

Used by

• Primes of a principal localization Corollary
• Quasi-finite algebras are source locally localizations of finite algebras Corollary
• The quasi-finite locus of a finite-type algebra is open Corollary
• A nonseparated affine cover can have nonaffine intersection Counterexample
• Global sections do not determine a sheaf on P1 Counterexample
• Quasi-finite does not imply finite Counterexample
• Quasi-separatedness in pushforward cannot be omitted Counterexample
• The conductor of a normalization Definition
• The Laurent polynomial ring as the principal localisation of Z[t] at t Definition
• A coherent closed-point skyscraper Example
• A finite algebra is its own Zariski Main factor Example
• Dual numbers compute the tangent spaces of the general and special linear groups Example
• Finite power map of the affine line Example
• Open immersions are etale Example
• The punctured affine line as an open finite factorization Example
• The square-root standard etale chart Example
• ℤ[1/6] consists exactly of rationals a/6ⁿ and inverts precisely the primes 2 and 3 Example
• An affine scheme of finite type over a field has a finitely generated coordinate ring Lemma
• Coprime polynomial factorisations lift after an etale localisation Lemma
• Exact principal-open Cech resolution Lemma
• Finite is affine and local on its target Lemma
• Finite normalization commutes with principal localization Lemma
• Finite-type field extensions with zero Ω Lemma
• Generic freeness over a Noetherian domain Lemma
• Localisation at a homogeneous element is graded, with graded kernels and dehomogenised degree-zero parts Lemma
• One-variable integral correction after leading-coefficient localization Lemma
• Prime correspondence on a Proj chart Lemma
• Rational maps from a smooth curve to a proper scheme are morphisms Lemma
• The general linear group scheme and its coordinate ring Lemma
• The Laurent polynomial ring is Noetherian and a unique factorisation domain Lemma
• The spectrum of a finite product ring is the disjoint union of the factor spectra Lemma
• Localising twice is localising once at the multiplicative set generated by both denominator sets Proposition
• A quasi-finite algebra factors openly through a finite algebra Theorem
• Algebraic Zariski Main localization at a quasi-finite prime Theorem
• Cartier's theorem: affine group schemes in characteristic zero are smooth Theorem
• Global regular functions on a classical affine variety are its coordinate ring Theorem
• Proj carries a scheme structure Theorem
• Quasi-coherence of pushforward for qcqs morphisms Theorem
• Regular functions on a principal open are the principal localization Theorem
• Regular functions on a principal open are the principal localization of the coordinate ring Theorem

…and 1 more result.

Dependency tree · two levels

4 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)