affine Serre's theorem

category: algebra, algebraic geometry

Given a commutative unital ring $R$ there is an equivalence of categories

${}_R Mod\to Qcoh(Spec R)$

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

Created on June 1, 2012 15:36:05
by Stephan Alexander Spahn
(178.195.221.252)