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TateOnProducts :: TateOnProducts

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


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


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).



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.