nLab modules over a ring are equivalent to quasicoherent sheaves over its spectrum







A basic result in the study of coherent and quasicoherent sheaves (of modules) over affine schemes says that for a commutative ring RR the quasicoherent modules over the Zariski spectrum Spec(R)Spec(R) are equivalent to RR-modules [Serre 1955].

Statement in the formalism of locally ringed spaces

Recall that given a commutative unital ring RR, there is an adjunction

()˜Γ:𝒪(SpecR)-ModR-Mod, \widetilde{(-)} \,\dashv\, \Gamma \;\colon\; \mathcal{O}(Spec\; R)\text{-}Mod \leftrightarrows R\text{-}Mod \,,

between the category of modules over its affine spectrum, i.e., the local ring object 𝒪(SpecR)\mathcal{O}(Spec\; R) in the category of sheaves over the topological space Spec(R)Spec(R), and the category of ordinary modules over RR.

In the right adjoint direction, for an 𝒪(SpecR)\mathcal{O}(Spec\; R)-module \mathcal{F}, we have the presheaf value (SpecR)\mathcal{F}(Spec\; R), the module of global sections, usually denoted Γ()\Gamma(\mathcal{F}).

In the left adjoint direction, for a (left) RR-module MM, we have a sheaf M˜\tilde{M} over Spec(R)Spec(R), whose value M˜(D f)\tilde{M}(D_f) at a typical open D fD_f of Spec(R)Spec(R) is given by the localization R[f 1] RMR[f^{-1}]\otimes_R M, and where the restriction maps are given by canonical maps between these localizations. This gives a sheaf of modules over the sheaf of rings 𝒪(SpecR)\mathcal{O}(Spec\; R).

The unit of this adjunction, with components MΓ(M˜)M \to \Gamma(\tilde{M}), is the canonical isomorphism MR RMM \cong R \otimes_R M. The counit, with components Γ()˜\widetilde{\Gamma(\mathcal{F})} \to \mathcal{F}, is the presheaf map whose value at a typical open D fD_f is the canonical map

R[f 1] R(SpecR)(D f) R[f^{-1}] \otimes_R \mathcal{F}(Spec\; R) \longrightarrow \mathcal{F}(D_f)

induced by the restriction map

(SpecR)(D f) \mathcal{F}(Spec\; R) \longrightarrow \mathcal{F}(D_f)

of RR-modules (with RR acting on the codomain by restriction of scalars along the ring map RR[f 1]R \to R[f^{-1}]), noting that R[f 1] R()R[f^{-1}] \otimes_R (-) is left adjoint to the functor that restricts scalars along RR[f 1]R \to R[f^{-1}].

The theorem of Serre (1955) is that this adjunction restricts to an adjoint equivalence of categories, which we denote as

()˜Γ:Qcoh(SpecR)R-Mod \widetilde{(-)} \dashv \Gamma \;\colon\; Qcoh(Spec R) \leftrightarrows R\text{-}Mod

between quasicoherent modules over 𝒪 SpecR\mathcal{O}_{Spec R}, and ordinary RR-modules. In particular, an 𝒪 SpecR\mathcal{O}_{Spec\; R}-module \mathcal{F} is quasicoherent if and only if the counit map

ε :Γ()˜ \varepsilon_\mathcal{F} \;\colon\; \widetilde{\Gamma(\mathcal{F})} \longrightarrow \mathcal{F}

is an isomorphism.

Furthermore, if RR is Noetherian, the adjoint equivalence restricts further to an equivalence between coherent modules over 𝒪 SpecR\mathcal{O}_{Spec R} and finitely generated modules over RR.

Statement in the formalism of functor of points

In the formalism of functor of points, the equivalence turns into a definition: affine schemes are defined as the opposite category of the category of commutative rings (with the functor SpecSpec now being tautologically defined as the identity functor), and the category of quasicoherent modules over SpecRSpec R is now defined as the category of RR-modules. This assignment defines a stack of categories over the site of affine schemes with the Zariski topology.

The functor of points approach carries over to quasicoherent modules over non-affine schemes: given such a scheme XX, a quasicoherent module over XX is a morphism of stacks from XX to the stack of quasicoherent modules defined above. In concrete terms, this boils down to picking an open cover of XX and defining a quasicoherent module using cocycle data?.


The result is originally due to:

It appears in many texts (often without a name, but elsewhere in the nLab it is referred to as the “affine Serre theorem”), for example:

Last revised on July 31, 2023 at 09:57:24. See the history of this page for a list of all contributions to it.