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5.1.93 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 $coker(A)=F_0/M$

\begin{displaymath}...\longrightarrow F_2 \buildrel{A_2}\over{\longrightarrow} F... ...er{\longrightarrow} F_0\longrightarrow F_0/M \longrightarrow 0,\end{displaymath}

where the columns of the matrix $A_1$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 $M_i={\tt module} (A_i)$, 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 ${\tt L[i]}\neq 0$ for $i \le p$, 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 some partial computations 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.