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D.15.18 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.15.18.1 productOfProjectiveSpaces  creates rings S,E corresponding to the product
D.15.18.2 truncateM  truncates module M at c
D.15.18.3 truncateCoker  truncates the cokernel at c
D.15.18.4 symExt  computes first differential of R(M)
D.15.18.5 sufficientlyPositiveMultidegree  computes a sufficiently positive multidegree for M
D.15.18.6 tateResolution  computes subquotient complex of Tate resolution T(F)
D.15.18.7 cohomologyMatrix  computes cohomologymatrix of corresponding sheaf
D.15.18.8 cohomologyMatrixFromResolution  computes dimensions of sheaf cohomology groups contained in T
D.15.18.9 eulerPolynomialTable  computes table of Euler polynomials
D.15.18.10 cohomologyHashTable  computes cohomology hash table
D.15.18.11 twist  twists module M by c
D.15.18.12 beilinsonWindow  computes Beilinson window of T
D.15.18.13 regionComplex  computes region complex
D.15.18.14 strand  computes strand
D.15.18.15 firstQuadrantComplex  computes first quadrant complex
D.15.18.16 lastQuadrantComplex  computes last quadrant complex proc shift(multigradedcomplex A, int i) shifts the multigraded complex by i