Singular

D.15.21 tateProdCplxNegGrad_lib

Library:

tateProdCplxNegGrad.lib

Purpose:

for computing sheaf cohomology on product of projective spaces

Author:

Clara Petroll ([email protected])

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.15.21.1 productOfProjectiveSpaces		creates rings S,E corresponding to the product
D.15.21.2 truncateM		truncates module M at c
D.15.21.3 truncateCoker		truncates the cokernel at c
D.15.21.4 symExt		computes first differential of R(M)
D.15.21.5 sufficientlyPositiveMultidegree		computes a sufficiently positive multidegree for M
D.15.21.6 tateResolution		computes subquotient complex of Tate resolution T(F)
D.15.21.7 cohomologyMatrix		computes cohomologymatrix of corresponding sheaf
D.15.21.8 cohomologyMatrixFromResolution		computes dimensions of sheaf cohomology groups contained in T
D.15.21.9 eulerPolynomialTable		computes table of Euler polynomials
D.15.21.10 cohomologyHashTable		computes cohomology hash table
D.15.21.11 twist		twists module M by c
D.15.21.12 beilinsonWindow		computes Beilinson window of T
D.15.21.13 regionComplex		computes region complex
D.15.21.14 strand		computes strand
D.15.21.15 firstQuadrantComplex		computes first quadrant complex
D.15.21.16 lastQuadrantComplex		computes last quadrant complex proc shift(multigradedcomplex A, int i) shifts the multigraded complex by i