Top
Back: 5.1.77 mpresmat
Forward: 5.1.79 mstd
FastBack: 5 Functions and system variables
FastForward: 6 Tricks and pitfalls
Up: 5.1 Functions
Top: 1 Preface
Contents: Table of Contents
Index: F Index
About: About This Document

5.1.78 mres

Syntax:

mres ( ideal_expression, int_expression )
mres ( module_expression, int_expression )

Type:

resolution

Purpose:

computes a minimal free resolution of an ideal or module M with the standard basis method. More precisely, let A=matrix(M), then mres computes a free resolution of coker(A) = F0∕M

         A2    A1
...−→  F2− → F1 −→ F0 −→ F0∕M  −→ 0,
where the columns of the matrix A1 are a minimal set of generators of M if the basering is local or if M is homogeneous. If the int expression k is not zero then the computation stops after k steps and returns a list of modules Mi = module(Ai), i=1...k.
mres(M,0) returns a resolution consisting of at most n+2 modules, where n is the number of variables of the basering. Let list L=mres(M,0); then L[1] consists of a minimal set of generators of the input, L[2] consists of a minimal set of generators for the first syzygy module of L[1], etc., until L[p+1], such that L[i]0 for i p, but L[p+1], the first syzygy module of L[p], is 0 (if the basering is not a qring).

Note:

Accessing single elements of a resolution may require that some partial computations have to be finished and may therefore take some time.

Example:
  ring r=31991,(t,x,y,z,w),ls;
  ideal M=t2x2+tx2y+x2yz,t2y2+ty2z+y2zw,
          t2z2+tz2w+xz2w,t2w2+txw2+xyw2;
  resolution L=mres(M,0);
  L;
→  1      4      15      18      7      1      
→ r <--  r <--  r <--   r <--   r <--  r
→ 
→ 0      1      2       3       4      5      
→ 
  // projective dimension of M is 5

See hres; ideal; lres; module; res; sres.


Top Back: 5.1.77 mpresmat Forward: 5.1.79 mstd FastBack: 5 Functions and system variables FastForward: 6 Tricks and pitfalls Up: 5.1 Functions Top: 1 Preface Contents: Table of Contents Index: F Index About: About This Document
            User manual for Singular version 2-0-4, October 2002, generated by texinfo.