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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 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 $f:X -> P^{n_2}$.

Authors

Version

This documentation describes version 0.2 of TateOnProducts.

Source code

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

Exports

  • Functions and commands
    • appendZeroMap -- append a zero map to chain complex
    • beilinsonWindow -- extract the subquotient complex which contributes to the Beilinson window
    • boxDegrees (missing documentation)
    • chainComplexMap -- Defines a ChainComplexMap via a list of matrices.
    • cohomologyTable -- compute the the cohomology groups of a (part) of a Tate resolution or sheaf on products of projective spaces
    • cornerCohomologyTablesOfUa -- cohomology tables of Ua and related complexes - Example 3.6
    • cornerComplex -- form the corner complex
    • cornerComplex1 (missing documentation)
    • firstQuadrantComplex -- form the first quadrant complex
    • isChainComplex -- checks whether the differentials compose to zero
    • isMinimalChainComplex -- tests for minimality
    • isQuism -- Test to see if the ChainComplexMap is a quasiisomorphism.
    • lastQuadrantComplex -- form the last quadrant complex
    • lowerCorner -- compute the lower corner
    • minimize -- minimal quotient complex of a free ChainComplex
    • nonzeroMax -- computes the homological position of the last non-zero module in a ChainComplex
    • nonzeroMin -- computes the homological position of the first non-zero module in a ChainComplex
    • numFactors (missing documentation)
    • prependZeroMap -- prepend a zero map to chain complex
    • pushAboveWindow -- push a projective resolution of the Beilinson complex out of the window
    • regionComplex -- region complex
    • removeZeroTrailingTerms -- remove trailing zero terms of a chain complex
    • resolutionOfChainComplex -- free resolution of a chain complex
    • ringData (missing documentation)
    • setupRings -- setup the Cox ring of a product of t projective space, and its exterior dual
    • sloppyTateExtension -- extend the terms in the Beilinson window to a part of a corner complex of the corresponding Tate resolution
    • 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
    • trivialHomologicalTruncation -- return the trivial truncation of a chain complex
    • truncateInE (missing documentation)
    • upperCorner -- compute the upper corner
  • Symbols
    • InitialDegree (missing documentation)
    • tateExtension (missing documentation)