BGG -- Bernstein-Gel'fand-Gel'fand correspondence

Description

The Bernstein-Gel'fand-Gel'fand correspondence is an isomorphism between the derived category of bounded complexes of finitely generated modules over a polynomial ring and the derived category of bounded complexes of finitely generated module over an exterior algebra (or of certain Tate resolutions). This package implements routines for investigating the BGG correspondence.

More details can be found in Sheaf Algorithms Using Exterior Algebra.

Authors

Version

This documentation describes version 1.4.2 of BGG, released Jan 11, 2016.

Citation

If you have used this package in your research, please cite it as follows:

@misc{BGGSource,
  title = {{BGG: Bernstein-Gelfand-Gelfand correspondence. Version~1.4.2}},
  author = {Hirotachi Abo and Wolfram Decker and David Eisenbud and Frank-Olaf Schreyer and Gregory G. Smith and Mike Stillman},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}}
}

Exports

Functions and commands
- beilinson -- Vector bundle map associated to the Beilinson monad
- bgg -- the ith differential of the complex R(M)
- cohomologyTable -- dimensions of cohomology groups
- directImageComplex -- direct image complex
- projectiveProduct -- Makes a product of projective spaces and a system of parameters
- pureResolution -- creates a pure resolution as an iterated direct image
- symExt -- the first differential of the complex R(M)
- tateResolution -- finite piece of the Tate resolution
- universalExtension -- Universal extension of vector bundles on P^1
Methods
- beilinson(Matrix,PolynomialRing) -- see beilinson -- Vector bundle map associated to the Beilinson monad
- bgg(ZZ,Module,PolynomialRing) -- see bgg -- the ith differential of the complex R(M)
- cohomologyTable(Matrix,PolynomialRing,ZZ,ZZ) -- see cohomologyTable -- dimensions of cohomology groups
- directImageComplex(Complex) -- direct image of a chain complex
- directImageComplex(Matrix) -- map of direct image complexes
- directImageComplex(Module) -- Complex representing the direct image
- projectiveProduct(Matrix,List) -- see projectiveProduct -- Makes a product of projective spaces and a system of parameters
- projectiveProduct(Ring,List) -- see projectiveProduct -- Makes a product of projective spaces and a system of parameters
- pureResolution(Matrix,List) -- see pureResolution -- creates a pure resolution as an iterated direct image
- pureResolution(Ring,List) -- see pureResolution -- creates a pure resolution as an iterated direct image
- pureResolution(ZZ,List) -- see pureResolution -- creates a pure resolution as an iterated direct image
- pureResolution(ZZ,ZZ,List) -- see pureResolution -- creates a pure resolution as an iterated direct image
- symExt(Matrix,PolynomialRing) -- see symExt -- the first differential of the complex R(M)
- tateResolution(Matrix,PolynomialRing,ZZ,ZZ) -- see tateResolution -- finite piece of the Tate resolution
- universalExtension(List,List) -- see universalExtension -- Universal extension of vector bundles on P^1
Symbols
- Exterior -- dual exterior algebra cached in a polynomial ring
- Regularity -- Option for directImageComplex

For the programmer

The object BGG is a package, defined in BGG.m2.

The source of this document is in BGG.m2:669:0.