# Formulation of cap product in group-equivariant sheaf cohomology + applications?

Note: moved to Math Overflow here.

Suppose one has a distinguished cocycle in the group-equivariant sheaf cohomology $$\Phi \in H^n(X, G, \mathcal{F})$$ for a "nice" -- at least locally compact -- scheme $$X$$, a group $$G$$, and a possibly coherent sheaf $$\mathcal{F}$$. I would like to write down a natural "homology" space $$H_r(X, G, \mathcal{F})$$ together with a pairing $$H^n(X, G, \mathcal{F}) \times H_r(X, G, \mathcal{F}) \to H_{r-n}(X, G, \mathcal{F})$$ so that I can compute $$\Phi \times \alpha$$ for suitably chosen $$\alpha \in H_r(X, G, \mathcal{F})$$. Is there a natural such space $$H_r(X, G, \mathcal{F})$$? Note that it need not be a homology theory despite the suggestive notation.

In the derived category $$D(R)$$ of $$R$$-modules, there is a natural pairing $$\operatorname{Ext}^n(A,B) \times \operatorname{Tor}_r(A,B) \to \operatorname{Tor}_{r-n}(A,B)$$ using the fact that $$\mathrm{Ext}^n(A,B) = \mathrm{Hom}_{D(R)}(A, T^n B)$$. So I thought maybe one could adopt this pairing to the sheaf cohomological context using $$\mathrm{Ext}^n(\mathcal{O}_X, \mathcal{F}) = H^n(X, \mathcal{F}),$$ where $$\mathcal{O}_X$$ denotes the structure sheaf of $$X$$. Then the desired $$H_r(X, G, \mathcal{F})$$ could be an appropriate formulation of a $$G$$-equivariant $$\mathrm{Tor}''(X, \mathcal{F})$$, but is not clear to me what that might be.

Alternatively, I read a bit about the theory of Borel-Moore homology. The existence of a cap product in the sheaf-cohomological framework here seems useful, but the derived perspective strikes me as more natural for incorporating group equivariance.

Is anyone aware of what might function as a sort of $$G$$-equivariant sheaf $$\mathrm{Tor}$$, or of an existing use/application of cap products with group equivariant sheaf cohomology?

• Okie will do, I felt unsure about what rises to that level. Commented Aug 12, 2023 at 13:11