affine Serre's theorem

Given a commutative unital ring RR there is an equivalence of categories

RModQcoh(SpecR){}_R Mod\to Qcoh(Spec R)

between the category of RR-modules and the category of quasicoherent sheaves of 𝒪 SpecR\mathcal{O}_{Spec R}-modules given on objects by MM˜M\mapsto \tilde{M} where M˜\tilde{M} is the unique sheaf such that the restriction on the principal Zariski open subsets is given by the localization M˜(D f)=R[f 1] RM\tilde{M}(D_f) = R[f^{-1}]\otimes_R M where D fD_f is the principal Zariski open set underlying SpecR[f 1]SpecRSpec R[f^{-1}]\subset Spec R, and the restrictions are given by the canonical maps among the localizations. The action of 𝒪 SpecR\mathcal{O}_{Spec R} is defined using a similar description of 𝒪 SpecR=R˜\mathcal{O}_{Spec R} = \tilde{R}. Its right adjoint (quasi)inverse functor is given by the global sections functor (SpecR)\mathcal{F}\mapsto\mathcal{F}(Spec R).

Created on June 1, 2012 at 15:36:05. See the history of this page for a list of all contributions to it.