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.

Authors

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 a package.