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4. Sheaf cohomology and endomorphisms

    1.     M2 currently has sheaves on projective varieties, but no morphisms between them, and no way to compute the associated maps on cohomology.

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      Problem 4.1.

      What is the best way to implement a "morphism of sheaves" data type? What’s the best way to calculate the maps on sheaf cohomology induced by such a morphism on a projective variety?

        •     There is preliminary code to decompose a module into indecomposables, but it would benefit from being much faster. Right now the bottleneck is the computation of the endomorphisms of module M.

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          Problem 4.2.

          Can we compute all the degree-0 (or degree-≤ e) homomorphisms of a graded module M without computing the entirety of End M? What are other ways to speed up the decomposition of modules into indecomposable summands? If M is the direct sum of $N_1$ and $N_2$, how can we extract $\mathrm{End}(N_1)$ and $\mathrm{End}(N_2)$ from $\mathrm{End}(M)$ without recalculating?

              Cite this as: AimPL: Macaulay2: expanded functionality and improved efficiency, available at http://aimpl.org/macaulay2efie.