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

I’ve a finite class $mathcal{C}$, together 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 restrict of the diagram $mathcal{C} to mathsf{Ab}$ of abelian teams. Equivalently, it is the $i$-th sheaf cohomology of the sheaf $F_j$, the place I account $mathcal{C}$ as the location with trifling Grothendieck topology.

I’ve computed the varied $H^i(mathcal{C},F_j)$. Assembling them, there ought to breathe a cup product construction $H^i(mathcal{C},F_j) otimes H^{i’}(mathcal{C},F_{j’}) to H^{i+i’}(mathcal{C},F_{j + j’})$. I might love to compute this product construction.

The solely system I’m cognizant of is thru sheaf cohomology, moving categorical resolutions, tensor merchandise, and complete complexes (behold [1]). Unfortunately, I should not have an categorical decision of $F$ or $F otimes F$: it appears too sophisticated to do by hand, particularly as a result of my $F(c)$ are usually infinitely generated. (In my computation of $H^i(mathcal{C},F_j)$ I circumvented this through the use of spectral sequences however these recondite the product construction.)

I’m led to the next questions:

- Does anybody know of a extra environment friendly system for computing cup merchandise of presheaf cohomology / derived limits?
- If not, is there laptop software program that energy breathe able to taking up a few of the duties contour above?

*Cup merchandise in sheaf cohomology, absolute injectives, and an alternative to projective resolutions.*

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