The Stacks project

Lemma 17.3.3. Let $f : (X, \mathcal{O}_ X) \to (Y, \mathcal{O}_ Y)$ be a morphism of ringed spaces.

  1. The functor $f_* : \textit{Mod}(\mathcal{O}_ X) \to \textit{Mod}(\mathcal{O}_ Y)$ is left exact. In fact it commutes with all limits.

  2. The functor $f^* : \textit{Mod}(\mathcal{O}_ Y) \to \textit{Mod}(\mathcal{O}_ X)$ is right exact. In fact it commutes with all colimits.

  3. Pullback $f^{-1} : \textit{Ab}(Y) \to \textit{Ab}(X)$ on abelian sheaves is exact.

Proof. Parts (1) and (2) hold because $(f^*, f_*)$ is an adjoint pair of functors, see Sheaves, Lemma 6.26.2 and Categories, Section 4.24. Part (3) holds because exactness can be checked on stalks (Lemma 17.3.1) and the description of stalks of the pullback, see Sheaves, Lemma 6.22.1. $\square$

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