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2. Local cohomology and differential operators

    1. Computing local cohomology via D-modules

          The current method for computing local cohomology requires the iterated computation of Bernstein–Sato polynomials at every step of the Cech complex.

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      Problem 2.1.

      Can computing just the necessary steps of the Cech complex speed up the computation of local cohomology? What about computing just the Budur–Mustata–Saito b-function just once?

        • Local cohomology in characteristic p

              If R is a regular ring of characteristic p, then the local cohomology modules have many nice finiteness properties; one can describe many of these properties via the theory of unit $F$-modules. Work on implementing these calculations in M2 was begun at the 2023 Minneapolis workshop.

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          Problem 2.2.

          Given an F-finite F-module, how can one efficiently compute a unit F-module? For example, given the map $\mathrm{Ext}^i(R/I,R)\to \mathrm{Ext}^i(R/I^{[p]},R)$, how can one find a root for the local cohomology module $H^i_I(R)$?

            • Applying differential operators

                  Currently, the D-module package(s) doesn’t have a built-in way to act on ring elements by differential operators.

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              Problem 2.3.

              Can we add a convenient way to let the differential operators on a ring R act on elements of R? Can we have them act on elements of the fraction field?

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