Question

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.

Answer 1

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.