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DifferentialModule -- The class of differential modules.

Description

A differential module is just a module with a square zero endomorphism. Given a module map $f: M \rightarrow M$ of degree $a$, we represent a differential module from $f$ as a 3-term chain complex in homological degrees $-1, 0$, and $1$. If $a \neq 0$, then since the source and target of $f$ are required to be equal, we must specify the degree of the differential to be $a$ in order for the differential to be homogeneous.

See also

Functions and methods returning a differential module:

  • differentialModule -- converts a square zero matrix into a differential module
  • foldComplex(Complex,ZZ) -- see foldComplex -- converts a chain complex into a differential module

Methods that use a differential module:

  • degree(DifferentialModule) -- returns the degree of the differential
  • differential(DifferentialModule) -- see differential -- returns the differential of a differential module
  • HH DifferentialModule -- computes the homology of a differential module
  • image(DifferentialModule) -- computes the image of the differential of a differential module
  • kernel(DifferentialModule) -- computes the kernel of the differential in a differential module.
  • minimizeDM(DifferentialModule) -- see minimizeDM -- minimizes a square matrix or a differential module
  • module(DifferentialModule) -- computes the underlying module of a differential module.
  • resDM(DifferentialModule) -- see resDM -- uses a "killing cycles"-style construction to find a free resolution of a differential module
  • resDM(DifferentialModule,ZZ) -- see resDM -- uses a "killing cycles"-style construction to find a free resolution of a differential module
  • resMinFlag(DifferentialModule,ZZ) -- see resMinFlag -- gives a minimal free flag resolution of a differential module of degree 0
  • ring(DifferentialModule) -- returns the ring of a differential module.
  • unfold(DifferentialModule,ZZ,ZZ) -- see unfold -- converts a differential module into a 1-periodic complex

For the programmer

The object DifferentialModule is a type, with ancestor classes Complex < MutableHashTable < HashTable < Thing.

The source of this document is in MultigradedBGG.m2:788:0.