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5.1.94 modulo

Syntax:
modulo ( ideal_expression, ideal_expression )
modulo ( module_expression, module_expression )
modulo ( ideal_expression, ideal_expression{,} string_expression )
modulo ( module_expression, module_expression{,} string_expression )
modulo ( ideal_expression, ideal_expression{,} matrix_name )
Type:
module
Purpose:
modulo(h1,h2) represents $h_1/(h_1 \cap h_2) \cong (h_1+h_2)/h_2$where $h_1$ and $h_2$are considered as submodules of the same free module $R^l$(l=1 for ideals). Let $H_1$, resp. $H_2$,be the matrices of size $l \times k$, resp. $l \times m$, having the generators of $h_1$, resp. $h_2$,as columns. Then $h_1/(h_1 \cap h_2) \cong R^k / ker(\overline{H_1})$where $\overline{H_1}: R^k \rightarrow R^l/Im(H_2)=R^l/h_2$ is the induced map.
modulo(h1,h2) returns generators of the kernel of this induced map.
An optional third string argument give the GB algorithm to use. (default: "std", else: "groebner", "modstd", "slimgb").
An optional third argument (a name of a matrix) gives the matrix to store the transformation of the input to the GB (see liftstd).
Note:
If for at least one of h1 or h2 the attribute "isHomog" is set, modulo(h1,h2) also sets the attribute "isHomog" (if possible, that is, if the weights are compatible).
Example:
 
  ring r;
  ideal h1=x,y,z;
  ideal h2=x;
  module m=modulo(h1,h2);
  print(m);
==> 1,0, 0,0,
==> 0,-z,x,0,
==> 0,y, 0,x
See hom_kernel; syz.
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