| ### A.3.4 Free resolution In SINGULAR a free resolution of a module or ideal has its own type: `resolution`. It is a structure that stores all information related to free resolutions. This allows partial computations of resolutions via the command `res`. After applying `res`, only a pre-format of the resolution is computed which allows to determine invariants like Betti-numbers or homological dimension. To see the differentials of the complex, a resolution must be converted into the type list which yields a list of modules: the k-th module in this list is the first syzygy-module (module of relations) of the (k-1)st module. There are the following commands to compute a resolution: `res` res computes a free resolution of an ideal or module using a heuristically chosen method. This is the preferred method to compute free resolutions of ideals or modules. `fres` fres improved version of sres, computes a free resolution of an ideal or module using Schreyer's method. The input has to be a standard basis. `lres` lres computes a free resolution of an ideal or module with LaScala's method. The input needs to be homogeneous. `mres` mres computes a minimal free resolution of an ideal or module with the syzygy method. `sres` sres computes a free resolution of an ideal or module with Schreyer's method. The input has to be a standard basis. `nres` nres computes a free resolution of an ideal or module with the standard basis method. `minres` minres minimizes a free resolution of an ideal or module. `syz` syz computes the first syzygy module. `res(i,r)`, `lres(i,r)`, `sres(i,r)`, `mres(i,r)`, `nres(i,r)` compute the first r modules of the resolution of i, resp. the full resolution if r=0 and the basering is not a qring. See the manual for a precise description of these commands. Note: The command `betti` does not require a minimal resolution for the minimal Betti numbers. Now let us take a look at an example which uses resolutions: The Hilbert-Burch theorem says that the ideal i of a reduced curve in has a free resolution of length 2 and that i is given by the 2x2 minors of the 2nd matrix in the resolution. We test this for two transversal cusps in .Afterwards we compute the resolution of the ideal j of the tangent developable of the rational normal curve in from above. Finally we demonstrate the use of the type `resolution` in connection with the `lres` command.
      // Two transversal cusps in (k^3,0):
      ring r2 =0,(x,y,z),ds;
      ideal i =z2-1y3+x3y,xz,-1xy2+x4,x3z;
      resolution rs=mres(i,0);   // computes a minimal resolution
      rs;                        // the standard representation of complexes
    ==>   1       3       2
    ==> r2 <--  r2 <--  r2
    ==>
    ==> 0       1       2
    ==>
        list resi=rs;            // conversion to a list
      print(resi[1]);            // the 1st module is i minimized
    ==> xz,
    ==> z2-y3+x3y,
    ==> xy2-x4
      print(resi[2]);            // the 1st syzygy module of i
    ==> -z,-y2+x3,
    ==> x, 0,
    ==> y, z
      resi[3];                   // the 2nd syzygy module of i
    ==> _[1]=0
      ideal j=minor(resi[2],2);
      reduce(j,std(i));          // check whether j is contained in i
    ==> _[1]=0
    ==> _[2]=0
    ==> _[3]=0
      size(reduce(i,std(j)));    // check whether i is contained in j
    ==> 0
      // size(<ideal>) counts the non-zero generators
      // ---------------------------------------------
      // The tangent developable of the rational normal curve in P^4:
      ring P = 0,(a,b,c,d,e),dp;
      ideal j= 3c2-4bd+ae, -2bcd+3ad2+3b2e-4ace,
               8b2d2-9acd2-9b2ce+9ac2e+2abde-1a2e2;
      resolution rs=mres(j,0);
      rs;
    ==>  1      2      1
    ==> P <--  P <--  P
    ==>
    ==> 0      1      2
    ==>
      list L=rs;
      print(L[2]);
    ==> 2bcd-3ad2-3b2e+4ace,
    ==> -3c2+4bd-ae
      // create an intmat with graded Betti numbers
      intmat B=betti(rs);
      // this gives a nice output of Betti numbers
      print(B,"betti");
    ==>            0     1     2
    ==> ------------------------
    ==>     0:     1     -     -
    ==>     1:     -     1     -
    ==>     2:     -     1     -
    ==>     3:     -     -     1
    ==> ------------------------
    ==> total:     1     2     1
    ==>
      // the user has access to all Betti numbers
      // the 2-nd column of B:
      B[1..4,2];
    ==> 0 1 1 0
      ring cyc5=32003,(a,b,c,d,e,h),dp;
      ideal i=
      a+b+c+d+e,
      ab+bc+cd+de+ea,
      abc+bcd+cde+dea+eab,
      abcd+bcde+cdea+deab+eabc,
      h5-abcde;
      resolution rs=lres(i,0);   //computes the resolution according LaScala
      rs;                        //the shape of the minimal resolution
    ==>     1         5         10         10         5         1
    ==> cyc5 <--  cyc5 <--  cyc5 <--   cyc5 <--   cyc5 <--  cyc5
    ==>
    ==> 0         1         2          3          4         5
    ==>
      print(betti(rs),"betti");  //shows the Betti-numbers of cyclic 5
    ==>            0     1     2     3     4     5
    ==> ------------------------------------------
    ==>     0:     1     1     -     -     -     -
    ==>     1:     -     1     1     -     -     -
    ==>     2:     -     1     1     -     -     -
    ==>     3:     -     1     2     1     -     -
    ==>     4:     -     1     2     1     -     -
    ==>     5:     -     -     2     2     -     -
    ==>     6:     -     -     1     2     1     -
    ==>     7:     -     -     1     2     1     -
    ==>     8:     -     -     -     1     1     -
    ==>     9:     -     -     -     1     1     -
    ==>    10:     -     -     -     -     1     1
    ==> ------------------------------------------
    ==> total:     1     5    10    10     5     1
    ==>
      dim(rs);                   //the homological dimension
    ==> 4
      size(list(rs));            //gets the full (non-reduced) resolution
    ==> 6
      minres(rs);                //minimizes the resolution
    ==>     1         5         10         10         5         1
    ==> cyc5 <--  cyc5 <--  cyc5 <--   cyc5 <--   cyc5 <--  cyc5
    ==>
    ==> 0         1         2          3          4         5
    ==>
      size(list(rs));            //gets the minimized resolution
    ==> 6

User manual for Singular version 4.4.0, 2024, generated by texi2html.