# ac.commutative algebra – How to compute cup product of derived limits / presheaf cohomology

I have a finite category $$mathcal{C}$$, along with a functor $$F colon mathcal{C} to mathsf{GradedCommRings}$$. If $$F_j$$ is $$j$$-th graded piece of $$F$$, then I write $$H^i(mathcal{C},F_j)$$ for the $$i$$-th derived inverse limit of the diagram $$mathcal{C} to mathsf{Ab}$$ of abelian groups. Equivalently, it’s the $$i$$-th sheaf cohomology of the sheaf $$F_j$$, where I regard $$mathcal{C}$$ as the site with trivial Grothendieck topology.

I have computed the various $$H^i(mathcal{C},F_j)$$. Assembling them, there should be a cup product structure $$H^i(mathcal{C},F_j) otimes H^{i’}(mathcal{C},F_{j’}) to H^{i+i’}(mathcal{C},F_{j + j’})$$. I would like to compute this product structure.

The only method I’m aware of is through sheaf cohomology, involving explicit resolutions, tensor products, and total complexes (see (1)). Unfortunately, I do not have an explicit resolution of $$F$$ or $$F otimes F$$: it seems too complicated to do by hand, especially because my $$F(c)$$ are typically infinitely generated. (In my computation of $$H^i(mathcal{C},F_j)$$ I circumvented this by using spectral sequences but these obscure the product structure.)

I’m led to the following questions:

• Does anyone know of a more efficient method for computing cup products of presheaf cohomology / derived limits?
• If not, is there computer software that might be capable of taking over some of the tasks outline above?

(1) : R.D. Swan. Cup products in sheaf cohomology, pure injectives, and a substitute for projective resolutions.