Computing minimal resolutions of graded modules

Hanno · Tue Aug 06, 2013 8:18 am

Hello,

First of all: thank you very much for developing this great software!

I'm having trouble handling graded modules and computing their minimal resolutions:

1) How can I assign gradings to the generators in a free module over a graded polynomial ring?
2) I tried to compute the minimal resolution of the graded module Q[a]*e + Q[a] * f / (a*e+f) (with deg(a)=deg(f)=1, deg(e)=0), but didn't know how to define the gradings, and the ungraded approach

Code:

ring r = 0,a,lp;
matrix A[2][1] = 1,a;
module M = module(A);
resolution RES = mres(M,0);
RES;

suggested ranks 2 and 1 in the minimal resolution. Applying the function prune to the module, however, does eliminate the irrelevant generator f.

What am I doing wrong?

Thank you!
Hanno

Re: Computing minimal resolutions of graded modules

malex · Tue Aug 06, 2013 1:44 pm

you could try using the following new feature (probably in the development version of Singular - spielwiese)

New orderings for modules

Quote:

One can assign weights to module components, when defining a monomial ordering. For example

Code:

ring R = 0, (x,y,z), (am(1,2,3,   10,20,30,40), dp, C);
deg(x*gen(1));
==> 11

will assign weights 1,2,3 to x,y,z respectively, and weights 10,20,30,40,0,0,... to components of any free module defined over R. This ordering will first sort by this weighted degree, then by dp on the ring monomials and then will give priority to the large component index.