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I'm interested in working with bicomplexes of modules over polynomial rings, specifically tensoring them together, and the operation of taking cohomology in one direction, and then the other. Is there any computer algebra package which will do this efficiently?

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You can use the Homalg package for GAP4 (check their manual, see chapter 13)

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There are two relevant chapters in Computations in algebraic geometry with Macaulay 2: "Resolutions and cohonology over complete intersections", and "Sheaf algorithms using exterior algebra".

The computations in Farkas G.; Syzygies of curves and the effective cone of Mg. They where done with Macaulay 2, and involve bi-complexes and tensors.

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I am not an expert in this field, but maybe Kenzo is what you need? It can do tensors and homologies and so on.

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Have you tried GAP?

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Check magma also

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