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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);
}
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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
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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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