### D.15.29 tateProdCplxNegGrad_lib

Library:
  tateProdCplxNegGrad.lib
Purpose:
  for computing sheaf cohomology on product of projective spaces
Author:
  Clara Petroll (petroll@mathematik.uni-kl.de)

Overview:
  In this library, we use Tate resolutions for computing sheaf cohomology of coherent sheaves on products of projective spaces. The algorithms can be used for arbitrary products. We work over the multigraded Cox ring and the corresponding exterior algebra. Multigraded complexes are realized as the newstruct `multigradedcomplex`.

  The main algorithm is the one for computing subquotient complexes of a Tate resolution. It allows to compute cohomologytables, respectively hash table of the dimensions of sheaf cohomology groups.

References:
  [1] Eisenbud, Erman, Schreyer: Tate Resolutions for Products of Projective Spaces, Acta Mathematica Vietnamica (2015) [2] Eisenbud, Erman, Schreyer: Tate Resolutions on Products of Projective Spaces: Cohomology and Direct Image Complexes (2019)

Procedures:

> D.15.29.1 productOfProjectiveSpaces |  | creates rings S,E corresponding to the product
> D.15.29.2 truncateM |  | truncates module M at c
> D.15.29.3 truncateCoker |  | truncates the cokernel at c
> D.15.29.4 symExt |  | computes first differential of R(M)
> D.15.29.5 sufficientlyPositiveMultidegree |  | computes a sufficiently positive multidegree for M
> D.15.29.6 tateResolution |  | computes subquotient complex of Tate resolution T(F)
> D.15.29.7 cohomologyMatrix |  | computes cohomologymatrix of corresponding sheaf
> D.15.29.8 cohomologyMatrixFromResolution |  | computes dimensions of sheaf cohomology groups contained in T
> D.15.29.9 eulerPolynomialTable |  | computes table of Euler polynomials
> D.15.29.10 cohomologyHashTable |  | computes cohomology hash table
> D.15.29.11 twist |  | twists module M by c
> D.15.29.12 beilinsonWindow |  | computes Beilinson window of T
> D.15.29.13 regionComplex |  | computes region complex
> D.15.29.14 strand |  | computes strand
> D.15.29.15 firstQuadrantComplex |  | computes first quadrant complex
> D.15.29.16 lastQuadrantComplex |  | computes last quadrant complex proc shift(multigradedcomplex A, int i) shifts the multigraded complex by i

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