# PruneComplex -- Pruning chain complexes over polynomial and local rings

## Description

This package includes various methods for pruning chain complexes over polynomial and local rings. In particular, in the local or graded case the output is guaranteed to be a minimal free resolution.

Algorithms in this package are also implemented using C++ in e/mutablecomplex.hpp for speed.

 i1 : R = ZZ/32003[vars(0..17)]; i2 : m1 = genericMatrix(R,a,3,3) o2 = | a d g | | b e h | | c f i | 3 3 o2 : Matrix R <--- R i3 : m2 = genericMatrix(R,j,3,3) o3 = | j m p | | k n q | | l o r | 3 3 o3 : Matrix R <--- R i4 : I = ideal(m1*m2-m2*m1) o4 = ideal (d*k + g*l - b*m - c*p, b*j - a*k + e*k + h*l - b*n - c*q, c*j + ------------------------------------------------------------------------ f*k - a*l + i*l - b*o - c*r, - d*j + a*m - e*m + d*n + g*o - f*p, - d*k ------------------------------------------------------------------------ + b*m + h*o - f*q, - d*l + c*m + f*n - e*o + i*o - f*r, - g*j - h*m + ------------------------------------------------------------------------ a*p - i*p + d*q + g*r, - g*k - h*n + b*p + e*q - i*q + h*r, - g*l - h*o ------------------------------------------------------------------------ + c*p + f*q) o4 : Ideal of R

Here we produce an intentionally nonminimal resolution:

 i5 : C = res(I, FastNonminimal=>true) 1 26 108 208 221 132 41 5 o5 = R <-- R <-- R <-- R <-- R <-- R <-- R <-- R <-- 0 0 1 2 3 4 5 6 7 8 o5 : ChainComplex

Now we prune the resolution above to get a minimal resolution:

 i6 : D = pruneComplex(C, UnitTest => isScalar) 1 8 33 60 61 32 5 o6 = R <-- R <-- R <-- R <-- R <-- R <-- R 0 1 2 3 4 5 6 o6 : ChainComplex i7 : isCommutative D.cache.pruningMap o7 = true i8 : betti D == betti res I o8 = true

## Caveat

Only supports localization at prime ideals.

• LocalRings -- Localizations of polynomial rings at prime ideals

## Version

This documentation describes version 1.0 of PruneComplex.

## Source code

The source code from which this documentation is derived is in the file PruneComplex.m2. The auxiliary files accompanying it are in the directory PruneComplex/.

## Exports

• Functions and commands
• isScalar -- check whether a ring element is a scalar
• pruneComplex -- Prunes a chain complex or list of mutable matrices
• pruneDiff -- Prunes a single differential in a chain complex or list of mutable matrices
• pruneUnit -- Prunes a unit of a differential in a list of mutable matrices
• toChainComplex -- Converts a list of mutable matrices into a ChainComplex.
• toMutableComplex -- Converts a chain complex into a list of mutable matrices.
• Methods
• "isScalar(RingElement)" -- see isScalar -- check whether a ring element is a scalar
• "pruneComplex(ChainComplex)" -- see pruneComplex -- Prunes a chain complex or list of mutable matrices
• "pruneComplex(ChainComplex,ZZ)" -- see pruneComplex -- Prunes a chain complex or list of mutable matrices
• "pruneComplex(List)" -- see pruneComplex -- Prunes a chain complex or list of mutable matrices
• "pruneComplex(List,ZZ)" -- see pruneComplex -- Prunes a chain complex or list of mutable matrices
• "pruneDiff(ChainComplex,ZZ)" -- see pruneDiff -- Prunes a single differential in a chain complex or list of mutable matrices
• "pruneDiff(ChainComplex,ZZ,List)" -- see pruneDiff -- Prunes a single differential in a chain complex or list of mutable matrices
• "pruneDiff(List,ZZ)" -- see pruneDiff -- Prunes a single differential in a chain complex or list of mutable matrices
• "pruneDiff(List,ZZ,List)" -- see pruneDiff -- Prunes a single differential in a chain complex or list of mutable matrices
• "pruneUnit(List,ZZ,Sequence,List)" -- see pruneUnit -- Prunes a unit of a differential in a list of mutable matrices
• "toChainComplex(List)" -- see toChainComplex -- Converts a list of mutable matrices into a ChainComplex.
• "toChainComplex(List,Module)" -- see toChainComplex -- Converts a list of mutable matrices into a ChainComplex.
• "toMutableComplex(ChainComplex)" -- see toMutableComplex -- Converts a chain complex into a list of mutable matrices.
• Symbols
• Direction -- Determines the direction with which the matrices in the complex is pruned
• PruningMap -- Whether to compute a morphism of complexes
• UnitTest -- Limit which units are to be pruned

## For the programmer

The object PruneComplex is .