# Question about the definition of a sheaf cohomology group for a sheaf using tensor products of sheaves

In Warner's 'Foundations of differentiable manifolds and Lie groups', in the section about axiomatic sheaf theory (page 178), when establishing the conditions necessary for the existence of a cohomology theory on a manifold $M$, Warner (although the construction is by Cartan-Eilenberg) says that a fine torsionless resolution of a constant sheaf $\mathcal{H}=M\times K$,

$0\rightarrow\mathcal{H}\rightarrow S_{0}\rightarrow S_{1}\rightarrow S_{2}\rightarrow S_{3}\rightarrow\cdots$,

(where $S^{*}$ is a cochain complex of sheaves) defines canonically a cohomology theory on $M$, he then defines a sheaf cohomology group of a sheaf $\mathcal{F}$ in this cohomology theory as

$H^{q}(M,\mathcal{F})=H^{q}(\Gamma(\mathcal{S}^{*}\otimes\mathcal{F}))$,

here $\Gamma$ is the K-module of sections on the manifold. I don't know if I'm missing something here but this kind of baffled me as I thought the sheaf cohomology group of a sheaf was the right derived functor of the global sections functor, I don't understand what tensor products of sheaves are doing here quite frankly. I'm trying to use Cartan-Eilenberg's axiomatic sheaf theory construction to define an injective resolution of a constant sheaf on a SITE (with a topology given by a Grothendieck topology) and I'm practically trying to apply this axiomatic sheaf theory construction verbatim, so I don't know if I'm on the right path here or if there's a book I should refer to or something I should know, thanks.

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I like Warner's book for certain things, but I agree that his treatment of sheaf cohomology is a bit unclear. In the manifold setting fine resolutions are convenient, but there is no reason to take a tensor product above. Anyway, if you want to work on a site then you are better off using derived functors and other references. I'm not sure of your background, but Artin has some notes on Grothendieck topologies which might be better. –  Donu Arapura Aug 5 '11 at 1:59
Will do, thanks Donu –  Mario Carrasco Aug 5 '11 at 16:54
The claim follows from the fact that for any sheaf $\mathcal{F}$, the complex $\mathcal{S} ^\bullet \otimes \mathcal{F}$ is an $\Gamma$-acyclic resolution of $\mathcal{F}$, and the fact that derived functors can be computed via acyclic resolutions.
The map $\mathcal{F} \to \mathcal{S}^\bullet \otimes \mathcal{F}$ is a quasi-isomorphism of complexes because $\mathcal{F}$ is torsion-free. Moreover, tensoring with a fine sheaf with another sheaf gives a fine sheaf, so $\mathcal{S}^\bullet \otimes \mathcal{F}$ consists of fine sheaves, which are acyclic with respect to the global section functor.
Note that one doesn't need fineness of the resolution $\mathcal{S}^\bullet$: softness is enough. The reason is that tensoring a soft, flat sheaf with another sheaf on a space of finite dimension (e.g. a manifold) yields another soft sheaf. Anyway, one reason being able to compute sheaf cohomology this way is useful if you want to show that derived $\Gamma_c$ admits a right adjoint (Verdier duality): then you can write $\mathbb{R} \Gamma_c = \Gamma_c( \cdot \otimes \mathcal{S}^\bullet)$, which is easier to work with.