# modulo(Matrix,Matrix) -- find the pre-image (pullback) of image of a map (low level version)

## Synopsis

• Function: modulo
• Usage:
modulo(f,g)
• Inputs:
• f,
• g,
• Outputs:
• , whose image is the pre-image (pullback) of the image of g under f

## Description

The maps f and g must have the same target, and their sources and targets must be free. If f is null, then it is taken to be the identity. If g is null, it is taken to be zero.

This function is mainly for internal use.

 i1 : R = QQ[x,y,z] o1 = R o1 : PolynomialRing i2 : f = matrix {{x,y}} o2 = | x y | 1 2 o2 : Matrix R <--- R i3 : g = matrix {{y,z}} o3 = | y z | 1 2 o3 : Matrix R <--- R i4 : modulo(f,g) o4 = {1} | 0 z y | {1} | 1 0 0 | 2 3 o4 : Matrix R <--- R i5 : kernel( inducedMap(coker g, target g) * f ) o5 = image {1} | 0 z y | {1} | 1 0 0 | 2 o5 : R-module, submodule of R

## Ways to use this method:

• modulo(Matrix,Matrix) -- find the pre-image (pullback) of image of a map (low level version)
• "modulo(Matrix,Nothing)"
• "modulo(Nothing,Matrix)"