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

Idea

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

Statement in the formalism of locally ringed spaces

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

$$\widetilde{(-)} \,\dashv\, \Gamma \;\colon\; \mathcal{O}(Spec\; R)\text{-}Mod \leftrightarrows R\text{-}Mod \,,$$

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

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

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

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

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

induced by the restriction map

$$\mathcal{F}(Spec\; R) \longrightarrow \mathcal{F}(D_f)$$

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

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

$$\widetilde{(-)} \dashv \Gamma \;\colon\; Qcoh(Spec R) \leftrightarrows R\text{-}Mod$$

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

$$\varepsilon_\mathcal{F} \;\colon\; \widetilde{\Gamma(\mathcal{F})} \longrightarrow \mathcal{F}$$

is an isomorphism.

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

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 $Spec$ now being tautologically defined as the identity functor), and the category of quasicoherent modules over $Spec R$ is now defined as the category of $R$-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 $X$, a quasicoherent module over $X$ is a morphism of stacks from $X$ to the stack of quasicoherent modules defined above. In concrete terms, this boils down to picking an open cover of $X$ and defining a quasicoherent module using cocycle data?.

Related concepts

• Serre-Swan theorem

• duality between algebra and geometry

References

The result is originally due to:

• Jean-Pierre Serre, §48 & §49 in: Faisceaux Algebriques Coherents, Ann. Math. 61 2 (1955) 197. [doi:10.2307/1969915]

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:

• Robin Hartshorne, Chapter II, Corollary 5.5 in Algebraic Geometry, Graduate Texts in Mathematics 52, Springer (1977) [doi.org/10.1007/978-1-4757-3849-0]

• The Stacks Project, Lemma 26.7.5.

category: algebra, algebraic geometry