This package contains implementations of the algorithm from our paper
Tate Resolutions on Products of Projective Spaces. It allows to compute the direct image complexes of a coherent sheaf along the projection onto a product of any of the factors.
In the moment the function tateExtension is not completed, a version sloppyTateExtension works however nicely in examples. The documentation and comments in the code are in a preliminary shape. Some function have to be removed, other wait for their implementation
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
From graded modules to Tate resolutions
- setupRings -- setup the Cox ring of a product of t projective space, and its exterior dual
- 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
Numerical Information
- cohomologyTable -- compute the the cohomology groups of a (part) of a Tate resolution or sheaf on products of projective spaces
- tallyDegrees -- collect the degrees of the generators of the terms in a free complex
Subcomplexes
formal ChainComplex manipulations
Beilinson monads
- beilinsonWindow -- extract the subquotient complex which contributes to the Beilinson window
- sloppyTateExtension -- extend the terms in the Beilinson window to a part of a corner complex of the corresponding Tate resolution
- tateExtension (missing documentation)
- pushAboveWindow -- push a projective resolution of the Beilinson complex out of the window
Examples from the papers
Missing pieces
- BGG functor R for complexes of S-modules
- projective resolutions of complexes
- Beilinson complex of sheaves/S-modules from a Beilinson window
- BGG functor L for complexes of E-modules
- various composition of functions
- cornerCohomologyTablesOfUa -- cohomology tables of Ua and related complexes - Example 3.6
- Example section: examples from the paper, jumping lines, $R pi_* sO_X$ for a resolution of singularities, $Rf_*sF$ for a coherent sheaf $sF$ on $X subset P^{n_1}$ and a morphism $f:X -> P^{n_2}$.