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I have two sheaves of modules $\mathcal F$ and $\mathcal G$ over some scheme $X$. What I want to do is really to compare the sheaf cohomologies of $\mathcal F, \mathcal G$ and $\mathcal F \oplus \mathcal G$. Given an injective resolution $I^*_\mathcal F$ of $\mathcal F$ and $I^*_\mathcal G$ of $\mathcal G$, $I^*_\mathcal F \oplus I^*_\mathcal G$ is a coresolution of $\mathcal F \oplus \mathcal G$, but is it injective?

I also assume that direct sums over sheaves commute with sections, so that $\Gamma(-, \mathcal F \oplus \mathcal G)$ is isomorphic to $\Gamma(-, \mathcal F) \oplus \Gamma(-, \mathcal G)$, and that $\mathcal F \oplus \mathcal G$ is actually a sheaf.

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By general abstract nonsense, the product of any number of injective objects (should it exist) is automatically an injective object. Fortunately, in an abelian category, finite products and finite coproducts coincide because they are both given by the direct sum, so the direct sum of finitely many injective modules/sheaves is again an injective module/sheaf.

Also by general abstract nonsense, there is a canonical isomorphism $\Gamma(-, A \times B \times C \times \cdots) \cong \Gamma(-, A) \times \Gamma(-, B) \times \Gamma(-, C) \times \cdots$ because $\Gamma(Y, -)$ is a representable functor for any open set $Y$... at least in the category of sheaves of sets, the category of abelian sheaves, and the category of $\mathscr{O}_X$-modules.

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  • $\begingroup$ Thank you. I assumed it was true, but it's neccessary some times to check assumptions. All the googling I did only turned up infinite products and Grothendieck topologies and I don't know what else of heavy machinery which is unnecessary in the simple case of a finite sum / product. $\endgroup$ – Arthur Oct 14 '12 at 12:28

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