### 7.3.18 nres (plural)

`Syntax:`
  `nres (` ideal_expression`,` int_expression `)`
  `nres (` module_expression`,` int_expression `)`
`Type:`
  resolution
`Purpose:`
  computes a free resolution of an ideal or module which is minimized from the second module on (by the Groebner basis method).
`Note: Assigning a resolution to a list is the best choice of usage. The resolution may be minimized by using the`
  command `minres`. Use the command `betti` to compute Betti numbers.
`Example:`

      LIB "ncalg.lib";
      def A = makeUsl2();
      setring A; // this algebra is U(sl_2)
      option(redSB);
      option(redTail);
      ideal i = e,f,h;
      i = std(i);
      resolution F=nres(i,0); F;
      ==>  1      3      3      1
      ==> A <--  A <--  A <--  A
      ==>
      ==> 0      1      2      3
      ==> resolution not minimized yet
      ==>
      list l = F; l;
      ==> [1]:
      ==>    _[1]=h
      ==>    _[2]=f
      ==>    _[3]=e
      ==> [2]:
      ==>    _[1]=f*gen(1)-h*gen(2)-2*gen(2)
      ==>    _[2]=e*gen(1)-h*gen(3)+2*gen(3)
      ==>    _[3]=e*gen(2)-f*gen(3)-gen(1)
      ==> [3]:
      ==>    _[1]=e*gen(1)-f*gen(2)+h*gen(3)
      ==> [4]:
      ==>    _[1]=0
      // see the exactness at this point:
      size(ideal(transpose(matrix(l[2]))*transpose(matrix(l[1]))));
      ==> 0
      // see the exactness at this point:
      size(ideal(transpose(matrix(l[3]))*transpose(matrix(l[2]))));
      ==> 0
      print(betti(l), "betti");
      ==>            0     1     2     3
      ==> ------------------------------
      ==>     0:     1     -     3     1
      ==> ------------------------------
      ==> total:     1     0     3     1
      ==>
      print(betti(minres(l)), "betti");
      ==>            0     1     2     3
      ==> ------------------------------
      ==>     0:     1     -     -     -
      ==>     1:     -     -     2     1
      ==> ------------------------------
      ==> total:     1     0     2     1
      ==>

See ideal (plural); minres (plural); module (plural); mres (plural).

User manual for Singular version 4.4.1, 2025, generated by texi2html.