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5.1.103 nres

Syntax:
nres ( ideal_expression, int_expression )
nres ( module_expression, int_expression )
Type:
resolution
Purpose:
computes a free resolution of an ideal or module M which is minimized from the second module on (by the standard basis method).

More precisely, let $A_1$=matrix(M),then nres computes a free resolution of $coker(A_1)=F_0/M$

\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 the given set of generators of M. If the int expression k is not zero then the computation stops after k steps and returns a list of modules $M_i={\tt module}(A_i),\;i = 1,\ldots,k$.
nres(M,0) returns a list of n modules where n is the number of variables of the basering. Let list L=nres(M,0); then L[1]=M is identical to the input, L[2] is a minimal set of generators for the first syzygy module of L[1], etc. ( ${\tt L[i]}=M_i$in the notations from above).
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=nres(M,0);
  L;
==>  1      4      15      18      7      1      
==> r <--  r <--  r <--   r <--   r <--  r
==> 
==> 0      1      2       3       4      5      
==> resolution not minimized yet
==>
See fres; hres; ideal; lres; module; mres; res; resolution; sres.