### 5.1.98 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

  where the columns of the matrix 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 , 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 for , 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.
`Note:`
  `mres` honours `degBound`.
`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 degBound; fres; hres; ideal; lres; module; res; sres.

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