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.