# 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

• 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).

## 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
• Methods
• "actionOnDirectImage(Ideal,ChainComplex)" -- see actionOnDirectImage -- recover the module structure via a Noether normalization
• "actionOnDirectImage(Ideal,Module)" -- see actionOnDirectImage -- recover the module structure via a Noether normalization
• "actionOnDirectImage(Ideal,Module,Matrix)" -- see actionOnDirectImage -- recover the module structure via a Noether normalization
• "beilinson(ChainComplex)" -- see beilinson -- apply the beilinson functor
• "beilinson(Matrix)" -- see beilinson -- apply the beilinson functor
• "beilinson(Module)" -- see beilinson -- apply the beilinson functor
• "beilinsonBundle(List,Ring)" -- see beilinsonBundle -- compute a basic Beilinson bundle
• "beilinsonBundle(ZZ,ZZ,Ring)" -- see beilinsonBundle -- compute a basic Beilinson bundle
• "beilinsonContraction(RingElement,List,List)" -- see beilinsonContraction -- compute a Beilinson contraction
• "beilinsonWindow(ChainComplex)" -- see beilinsonWindow -- extract the subquotient complex which contributes to the Beilinson window
• "bgg(Module)" -- see bgg -- make a linear free complex from a module over an exterior algebra or a symmetric algebra
• "coarseMultigradedRegularity(ChainComplex)" -- see coarseMultigradedRegularity -- A truncation that has linear resolution
• "coarseMultigradedRegularity(Module)" -- see coarseMultigradedRegularity -- A truncation that has linear resolution
• "cohomologyHashTable(ChainComplex,List,List)" -- see cohomologyHashTable -- cohomology groups of a sheaf on a product of projective spaces, or of (part) of a Tate resolution
• "cohomologyHashTable(Module,List,List)" -- see cohomologyHashTable -- cohomology groups of a sheaf on a product of projective spaces, or of (part) of a Tate resolution
• "cohomologyMatrix(ChainComplex,List,List)" -- see cohomologyMatrix -- cohomology groups of a sheaf on P^{n_1}xP^{n_2}, or of (part) of a Tate resolution
• "cohomologyMatrix(Module,List,List)" -- see cohomologyMatrix -- cohomology groups of a sheaf on P^{n_1}xP^{n_2}, or of (part) of a Tate resolution
• "contractionData(List,List,Ring)" -- see contractionData -- Compute the action of monomials in the exterior algebra on the Beilinson monad
• "cornerComplex(ChainComplex,List)" -- see cornerComplex -- form the corner complex
• "cornerComplex(Module,List,List,List)" -- see cornerComplex -- form the corner complex
• "directImageComplex(Ideal,Module,Matrix)" -- see directImageComplex -- compute the direct image complex
• "directImageComplex(Module,List)" -- see directImageComplex -- compute the direct image complex
• "eulerPolynomialTable(ChainComplex,List,List)" -- see eulerPolynomialTable -- cohomology groups of a sheaf on a product of projective spaces, or of (part) of a Tate resolution
• "eulerPolynomialTable(HashTable)" -- see eulerPolynomialTable -- cohomology groups of a sheaf on a product of projective spaces, or of (part) of a Tate resolution
• "eulerPolynomialTable(Module,List,List)" -- see eulerPolynomialTable -- cohomology groups of a sheaf on a product of projective spaces, or of (part) of a Tate resolution
• "firstQuadrantComplex(ChainComplex,List)" -- see firstQuadrantComplex -- form the first quadrant complex
• "isAction(Ideal,List)" -- see isAction -- test whether a list of square matrices induces an action
• "isIsomorphic(Module,Module)" -- see isIsomorphic -- probabilistic test for homogeneous isomorphism
• "isQuism(ChainComplexMap)" -- see isQuism -- Test to see if the ChainComplexMap is a quasiisomorphism.
• "lastQuadrantComplex(ChainComplex,List)" -- see lastQuadrantComplex -- form the last quadrant complex
• "lowerCorner(ChainComplex,List)" -- see lowerCorner -- compute the lower corner
• "productOfProjectiveSpaces(List)" -- see productOfProjectiveSpaces -- Cox ring of a product of projective spaces and it Koszul dual exterior algebra
• "productOfProjectiveSpaces(ZZ)" -- see productOfProjectiveSpaces -- Cox ring of a product of projective spaces and it Koszul dual exterior algebra
• "regionComplex(ChainComplex,List,Sequence)" -- see regionComplex -- region complex
• "strand(ChainComplex,List,List)" -- see strand -- take the strand
• "symExt(Matrix,Ring)" -- see symExt -- from linear presentation matrices over S to linear presentation matrices over E and conversely
• "tallyDegrees(ChainComplex)" -- see tallyDegrees -- collect the degrees of the generators of the terms in a free complex
• "tateData(Ring)" -- see tateData -- reads TateData from the cache of an appropriate ring
• "tateExtension(ChainComplex)" -- see tateExtension -- extend the terms in the Beilinson window to a part of a corner complex of the corresponding Tate resolution
• "tateResolution(Matrix,List,List)" -- see tateResolution -- compute the Tate resolution
• "tateResolution(Module,List,List)" -- see tateResolution -- compute the Tate resolution
• "trivialHomologicalTruncation(ChainComplex,ZZ,ZZ)" -- see trivialHomologicalTruncation -- return the trivial truncation of a chain complex
• "upperCorner(ChainComplex,List)" -- see upperCorner -- compute the upper corner
• Symbols

## For the programmer

The object TateOnProducts is .