This package contains implementations of the algorithm from our paper
Tate Resolutions on Products of Projective Spaces. It allows computing the direct image complexes of a coherent sheaf along the projection onto a product of any of the factors.
The main differences from the paper are:
- the exterior algebra E is positively graded
- we use E instead of omega_E
- all complexes are chain complexes instead of cochain complexes
Beilinson monads
- beilinsonWindow -- extract the subquotient complex which contributes to the Beilinson window
- tateResolution -- compute the Tate resolution
- tateExtension -- extend the terms in the Beilinson window to a part of a corner complex of the corresponding Tate resolution
- beilinson -- apply the beilinson functor
- bgg -- make a linear free complex from a module over an exterior algebra or a symmetric algebra
- directImageComplex -- compute the direct image complex
- actionOnDirectImage -- recover the module structure via a Noether normalization
- composedFunctions -- composed functions
Numerical Information
- cohomologyMatrix -- cohomology groups of a sheaf on P^{n_1}xP^{n_2}, or of (part) of a Tate resolution
- eulerPolynomialTable -- cohomology groups of a sheaf on a product of projective spaces, or of (part) of a Tate resolution
- cohomologyHashTable -- cohomology groups of a sheaf on a product of projective spaces, or of (part) of a Tate resolution
- tallyDegrees -- collect the degrees of the generators of the terms in a free complex
From graded modules to Tate resolutions
- productOfProjectiveSpaces -- Cox ring of a product of projective spaces and it Koszul dual exterior algebra
- symExt -- from linear presentation matrices over S to linear presentation matrices over E and conversely
- lowerCorner -- compute the lower corner
- upperCorner -- compute the upper corner
Subcomplexes
Acknowledgement: The work of Yeongrak Kim and Frank-Olaf Schreyer was supported by Project I.6 of the SFB-TRR 195 ''Symbolic Tools in Mathematics and their Application'' of the German Research Foundation (DFG).