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

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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?
[1] : R.D. Swan. Cup merchandise in sheaf cohomology, absolute injectives, and an alternative to projective resolutions.

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