Singular Manual: freerank

| #### D.2.8.4 freerank Procedure from library `polylib.lib` (see polylib_lib). Usage: freerank(M[,any]); M=poly/ideal/vector/module/matrix Compute: rank of module presented by M in case it is free. By definition this is vdim(coker(M)/m*coker(M)) if coker(M) is free, where m is the maximal ideal of the variables of the basering and M is considered to be a matrix. (the 0-module is free of rank 0) Return: rank of coker(M) if coker(M) is free and -1 else; in case of a second argument return a list: L[1] = rank of coker(M) or -1 L[2] = minbase(M) Note: freerank(syz(M)); computes the rank of M if M is free (and -1 else) Example:
    LIB "polylib.lib";
    ring r;
    ideal i=x;
    module M=[x,0,1],[-x,0,-1];
    freerank(M);          // should be 2, coker(M) is not free
    ==> 2
    freerank(syz (M),"");
    ==> [1]:
    ==>    1
    ==> [2]:
    ==>    _[1]=gen(2)+gen(1)
    // [1] should be 1, coker(syz(M))=M is free of rank 1
    // [2] should be gen(2)+gen(1) (minimal relation of M)
    freerank(i);
    ==> -1
    freerank(syz(i));     // should be 1, coker(syz(i))=i is free of rank 1
    ==> 1

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