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A.3.6 Computation of Ext

We start by showing how to calculate the n-th Ext group of an ideal. The ingredients to do this are by the definition of Ext the following: calculate a (minimal) resolution at least up to length n, apply the Hom functor, and calculate the n-th homology group, that is, form the quotient ker/im in the resolution sequence.

The Hom functor is given simply by transposing (hence dualizing) the module or the corresponding matrix with the command transpose. The image of the (n-1)-st map is generated by the columns of the corresponding matrix. To calculate the kernel apply the command syz at the (n-1)-st transposed entry of the resolution. Finally, the quotient is obtained by the command modulo, which gives for two modules A = ker, B = Im the module of relations of in the usual way. As we have a chain complex, this is obviously the same as ker/Im.

We collect these statements in the following short procedure:

 
proc ext(int n, ideal I)
{
  resolution rs = mres(I,n+1);
  module tAn    = transpose(rs[n+1]);
  module tAn_1  = transpose(rs[n]);
  module ext_n  = modulo(syz(tAn),tAn_1);
  return(ext_n);
}

Now consider the following example:

 
ring r5 = 32003,(a,b,c,d,e),dp;
ideal I = a2b2+ab2c+b2cd, a2c2+ac2d+c2de,a2d2+ad2e+bd2e,a2e2+abe2+bce2;
print(ext(2,I));
==> 1,0,0,0,0,0,0,
==> 0,1,0,0,0,0,0,
==> 0,0,1,0,0,0,0,
==> 0,0,0,1,0,0,0,
==> 0,0,0,0,1,0,0,
==> 0,0,0,0,0,1,0,
==> 0,0,0,0,0,0,1
ext(3,I);   // too big to be displayed here

The library homolog.lib contains several procedures for computing Ext-modules and related modules, which are much more general and sophisticated than the above one. They are used in the following example:

If is a module, then are the modules of infinitesimal deformations, respectively of obstructions, of (like T1 and T2 for a singularity). Similar to the treatment of singularities, the semiuniversal deformation of can be computed (if is finite dimensional) with the help of and the cup product. There is an extra procedure for if is an ideal in , since this is faster than the general Ext.

We compute

  • the infinitesimal deformations and obstructions of the residue field of an ordinary cusp, To compute we have to apply Ext(1,syz(m),syz(m)) with syz(m) the first syzygy module of , which is isomorphic to
  • for some ideal and with an extra option.

 


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