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TateOnProducts :: TateOnProducts

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:

Beilinson monads

Numerical Information

From graded modules to Tate resolutions

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).

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
  • 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 a package.