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localComplement -- find the splitting of the target of a map

Usage:

localComplement m
Inputs:
- m, a matrix, representing a homomorphism of free modules over a local ring
Outputs:
- a matrix, the complement of the image of the homomorphism represented by m

Description

The function finds a splitting of the target of m as a direct sum of the image of m and the image of the output.

i1 : R = ZZ/32003[x,y]

o1 = R

o1 : PolynomialRing

i2 : m = matrix{{x,y-1},{0,x}}

o2 = | x y-1 |
     | 0 x   |

             2      2
o2 : Matrix R  <-- R

i3 : setMaxIdeal(ideal(x,y))

o3 = ideal (x, y)

o3 : Ideal of R

i4 : localComplement m

o4 = | 0 |
     | 1 |

             2      1
o4 : Matrix R  <-- R

See also

setMaxIdeal -- set the maximal ideal for local ring methods
localsyz -- find syzygies
localMingens -- finds a minimal set of generators
localModulo -- find the pre-image (pullback) of image of a map over a local ring
localPrune -- find a minimal presentation
localResolution -- find a resolution over a local ring
residueMap -- The class of all local rings
maxIdeal -- The class of all local rings

Ways to use localComplement:

localComplement(Matrix)

For the programmer

The object localComplement is a method function.

The source of this document is in LocalRings/legacy.m2:177:0.