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## 2.2 Standard bases for submodules of free modules

We consider also module orderings <m on the set of monomials'' of which are compatible with the ordering < on . That is for all monomials and we have: f <m f' implies pf <m pf' and p < q implies pf <m qf.

We now fix an ordering <m on compatible with < and denote it also with <. Again we have the notion of coefficient c(f) and leading monomial L(f). < has the important property:

Definition 2..4
1)
denotes the submodule of generated by .

2)
is called a standard basis of I if generates the submodule .

In SINGULAR submodules of free modules are defined by a set of generators. These sets are of type module.

SINGULAR example (see [PS]):

// ===========Poincare complex ==========================
// counterexample to a possible generalization of a theorem of Kyoji
// Saito. A complete intersection with exact Poincare complex at 0
// but which is in no coordinate system weighted homogeneous
// see [PS] for an exeact decription.
//
// define (Z/32003)[[x,y,z]]
ring Rp=32003,(x,y,z),(c,ds);
LIB "sing.lib";
// select an example, parametrized by n and m
int n=883; int m=937;
poly f1=xy+z^(n-1);
poly f2=xz+y^(m-1)+yz2;
ideal f=f1,f2;
// define the basering as Rp/f and fetch the data
qring R=std(f);
ideal f=fetch(Rp,f);
poly f1,f2=fetch(Rp,f1),fetch(Rp,f2);
// the module Omega2:
module omega2=
[diff(f1,y),diff(f1,z),0],
[diff(f1,x),0,-diff(f1,z)],
[0,diff(f1,x),diff(f1,y)],
[diff(f2,y),diff(f2,z),0],
[diff(f2,x),0,-diff(f2,z)],
[0,diff(f2,x),diff(f2,y)];
//it can be shown, that the Poincare complex is exact, if (in this case)
//Milnor number(f)+1 = multiplicity(omega2)
omega2=std(omega2);
multiplicity(omega2);
// The Milnor number of the complete intersection f";
milnor(f);
// The Tjurina number of the complete intersection f
tjurina(f);
//since the Milnor number and the Tjurina number do not coincide,
//the singularity is not weighted homogeneous


Next: 2.3 Basic properties Up: 2. Standard bases Previous: 2.1 Definition
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