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5.1.69 lres

Syntax:

lres ( ideal_expression, int_expression )

Type:

resolution

Purpose:

computes a free resolution of an ideal using La Scala’s algorithm.

More precisely, let R be the basering and I be the given ideal. Then lres computes a minimal free resolution of R/I

         A2    A1
...−→  F2− → F1 −→ R −→  R∕I −→ 0.
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.

list L=lres(I,0); returns a list L of n modules (where n is the number of variables of the basering) such that L[i] = Mi in the above notation.

Note:

The ideal_expression has to be homogeneous.
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=0,(x,y,z),dp;
  ideal I=xz,yz,x3-y3;
  def L=lres(I,0);
  print(betti(L),"betti");
→            0     1     2
→ ------------------------
→     0:     1     -     -
→     1:     -     2     1
→     2:     -     1     1
→ ------------------------
→ total:     1     3     2
  L[2];     // the first syzygy module of r/I
→ _[1]=-x*gen(1)+y*gen(2)
→ _[2]=-x2*gen(2)+y2*gen(1)+z*gen(3)

See betti; hres; ideal; int; minres; module; mres; res; sres.


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