TateOnProducts -- Computation of parts of the Tate resolution on products

Description

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

cornerComplex -- form the corner complex
regionComplex -- region complex
strand -- take the strand
firstQuadrantComplex -- form the first quadrant complex
lastQuadrantComplex -- form the last quadrant complex

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).

Authors

Version

This documentation describes version 1.2 of TateOnProducts.

Source code

The source code from which this documentation is derived is in the file TateOnProducts.m2.

Exports

Functions and commands
- actionOnDirectImage -- recover the module structure via a Noether normalization
- beilinson -- apply the beilinson functor
- beilinsonBundle -- compute a basic Beilinson bundle
- beilinsonContraction -- compute a Beilinson contraction
- beilinsonWindow -- extract the subquotient complex which contributes to the Beilinson window
- bgg -- make a linear free complex from a module over an exterior algebra or a symmetric algebra
- coarseMultigradedRegularity -- A truncation that has linear resolution
- cohomologyHashTable -- cohomology groups of a sheaf on a product of projective spaces, or of (part) of a Tate resolution
- cohomologyMatrix -- cohomology groups of a sheaf on P^{n_1}xP^{n_2}, or of (part) of a Tate resolution
- composedFunctions -- composed functions
- contractionData -- Compute the action of monomials in the exterior algebra on the Beilinson monad
- cornerComplex -- form the corner complex
- directImageComplex -- compute the direct image complex
- eulerPolynomialTable -- cohomology groups of a sheaf on a product of projective spaces, or of (part) of a Tate resolution
- firstQuadrantComplex -- form the first quadrant complex
- isAction -- test whether a list of square matrices induces an action
- isIsomorphic -- probabilistic test for homogeneous isomorphism
- isQuism -- Test to see if the ChainComplexMap is a quasiisomorphism.
- lastQuadrantComplex -- form the last quadrant complex
- lowerCorner -- compute the lower corner
- productOfProjectiveSpaces -- Cox ring of a product of projective spaces and it Koszul dual exterior algebra
- regionComplex -- region complex
- strand -- take the strand
- symExt -- from linear presentation matrices over S to linear presentation matrices over E and conversely
- tallyDegrees -- collect the degrees of the generators of the terms in a free complex
- tateData -- reads TateData from the cache of an appropriate ring
- tateExtension -- extend the terms in the Beilinson window to a part of a corner complex of the corresponding Tate resolution
- tateResolution -- compute the Tate resolution
- trivialHomologicalTruncation -- return the trivial truncation of a chain complex
- upperCorner -- compute the upper corner
Symbols
- BundleType -- Option in beilinson with values PrunedQuotient, QuotientBundle, DummyQuotientBundle, SubBundle, FreeBundle, or MapsBetweenFreeBundles
- CoefficientField -- Option for productOfProjectiveSpaces
- CohomologyVariables -- Option for productOfProjectiveSpaces
- ContractionData -- name of a cached datum
- DummyQuotientBundle -- value for the option BundleType in beilinson
- FreeBundle -- value for the option BundleType in beilinson
- InitialDegree -- Option for chainComplexMap
- MapsBetweenFreeBundles -- value for the option BundleType in beilinson
- PrunedQuotient -- value for the option BundleType in beilinson
- QuotientBundle -- value for the option BundleType in beilinson
- Rings -- Option for productOfProjectiveSpaces
- SubBundle -- value for the option BundleType in beilinson
- TateData -- symbol used in beilinsonBundle