### A.3.7 Depth

We compute the depth of the module of Kaehler differentials D(R)of the variety defined by the -minors of a generic symmetric -matrix.We do this by computing the resolution over the polynomial ring. Then, by the Auslander-Buchsbaum formula, the depth is equal to the number of variables minus the length of a minimal resolution. This example was suggested by U. Vetter in order to check whether his bound could be improved.

      LIB "matrix.lib"; LIB "sing.lib";
      int n = 4;
      int m = 3;
      int N = n*(n+1) div 2;       // will become number of variables
      ring R = 32003,x(1..N),dp;
      matrix X = symmat(n);        // proc from matrix.lib
                                   // creates the symmetric generic nxn matrix
      print(X);
    ==> x(1),x(2),x(3),x(4),
    ==> x(2),x(5),x(6),x(7),
    ==> x(3),x(6),x(8),x(9),
    ==> x(4),x(7),x(9),x(10)
      ideal J = minor(X,m);
      J=std(J);
      // Kaehler differentials D_k(R)
      // of R=k[x1..xn]/J:
      module D = J*freemodule(N)+transpose(jacob(J));
      ncols(D);
    ==> 110
      nrows(D);
    ==> 10
      //
      // Note: D is a submodule with 110 generators of a free module
      // of rank 10 over a polynomial ring in 10 variables.
      // Compute a full resolution of D with sres.
      // This takes about 17 sec on a Mac PB 520c and 2 sec an a HP 735
      int time = timer;
      module sD = std(D);
      list Dres = sres(sD,0);                // the full resolution
      timer-time;                            // time used for std + sres
    ==> 0
      intmat B = betti(Dres);
      print(B,"betti");
    ==>            0     1     2     3     4     5     6
    ==> ------------------------------------------------
    ==>     0:    10     -     -     -     -     -     -
    ==>     1:     -    10     -     -     -     -     -
    ==>     2:     -    84   144    60     -     -     -
    ==>     3:     -     -    35    80    60    16     1
    ==> ------------------------------------------------
    ==> total:    10    94   179   140    60    16     1
    ==>
      N-ncols(B)+1;                          // the desired depth
    ==> 4

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