The Stacks project

Lemma 17.10.5. Let $(X, \mathcal{O}_ X)$ be ringed space. Let $\alpha : R \to \Gamma (X, \mathcal{O}_ X)$ be a ring homomorphism from a ring $R$ into the ring of global sections on $X$. Let $M$ be an $R$-module. The following three constructions give canonically isomorphic sheaves of $\mathcal{O}_ X$-modules:

  1. Let $\pi : (X, \mathcal{O}_ X) \longrightarrow (\{ *\} , R)$ be the morphism of ringed spaces with $\pi : X \to \{ *\} $ the unique map and with $\pi $-map $\pi ^\sharp $ the given map $\alpha : R \to \Gamma (X, \mathcal{O}_ X)$. Set $\mathcal{F}_1 = \pi ^*M$.

  2. Choose a presentation $\bigoplus _{j \in J} R \to \bigoplus _{i \in I} R \to M \to 0$. Set

    \[ \mathcal{F}_2 = \mathop{\mathrm{Coker}}\left( \bigoplus \nolimits _{j \in J} \mathcal{O}_ X \to \bigoplus \nolimits _{i \in I} \mathcal{O}_ X \right). \]

    Here the map on the component $\mathcal{O}_ X$ corresponding to $j \in J$ given by the section $\sum _ i \alpha (r_{ij})$ where the $r_{ij}$ are the matrix coefficients of the map in the presentation of $M$.

  3. Set $\mathcal{F}_3$ equal to the sheaf associated to the presheaf $U \mapsto \mathcal{O}_ X(U) \otimes _ R M$, where the map $R \to \mathcal{O}_ X(U)$ is the composition of $\alpha $ and the restriction map $\mathcal{O}_ X(X) \to \mathcal{O}_ X(U)$.

This construction has the following properties:

  1. The resulting sheaf of $\mathcal{O}_ X$-modules $\mathcal{F}_ M = \mathcal{F}_1 = \mathcal{F}_2 = \mathcal{F}_3$ is quasi-coherent.

  2. The construction gives a functor from the category of $R$-modules to the category of quasi-coherent sheaves on $X$ which commutes with arbitrary colimits.

  3. For any $x \in X$ we have $\mathcal{F}_{M, x} = \mathcal{O}_{X, x} \otimes _ R M$ functorial in $M$.

  4. Given any $\mathcal{O}_ X$-module $\mathcal{G}$ we have

    \[ \mathop{\mathrm{Mor}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}_ M, \mathcal{G}) = \mathop{\mathrm{Hom}}\nolimits _ R(M, \Gamma (X, \mathcal{G})) \]

    where the $R$-module structure on $\Gamma (X, \mathcal{G})$ comes from the $\Gamma (X, \mathcal{O}_ X)$-module structure via $\alpha $.

Proof. The isomorphism between $\mathcal{F}_1$ and $\mathcal{F}_3$ comes from the fact that $\pi ^*$ is defined as the sheafification of the presheaf in (3), see Sheaves, Section 6.26. The isomorphism between the constructions in (2) and (1) comes from the fact that the functor $\pi ^*$ is right exact, so $\pi ^*(\bigoplus _{j \in J} R) \to \pi ^*(\bigoplus _{i \in I} R) \to \pi ^*M \to 0$ is exact, $\pi ^*$ commutes with arbitrary direct sums, see Lemma 17.3.3, and finally the fact that $\pi ^*(R) = \mathcal{O}_ X$.

Assertion (1) is clear from construction (2). Assertion (2) is clear since $\pi ^*$ has these properties. Assertion (3) follows from the description of stalks of pullback sheaves, see Sheaves, Lemma 6.26.4. Assertion (4) follows from adjointness of $\pi _*$ and $\pi ^*$. $\square$

Comments (2)

Comment #24 by Pieter Belmans on

The first sentence of the proof should be about the isomorphism between and .

Comment #25 by Johan on

Fixed. Thanks!

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