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