affine Serre's theorem

category: algebra, algebraic geometry

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

$${}_{R}\mathrm{Mod}\to \mathrm{Qcoh}(\mathrm{Spec}R)$$

between the category of $R$-modules and the category of quasicoherent sheaves of ${\mathcal{O}}_{\mathrm{Spec}R}$-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 $\mathrm{Spec}R[{f}^{-1}]\subset \mathrm{Spec}R$, and the restrictions are given by the canonical maps among the localizations. The action of ${\mathcal{O}}_{\mathrm{Spec}R}$ is defined using a similar description of ${\mathcal{O}}_{\mathrm{Spec}R}=\tilde{R}$. Its right adjoint (quasi)inverse functor is given by the global sections functor $\mathcal{F}\mapsto \mathcal{F}(\mathrm{Spec}R)$.

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