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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 trying 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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# 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 trying practically 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.