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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

\begin{displaymath}A/(A \cap B)\end{displaymath}

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);

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;
==> 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 $M$ is a module, then $\hbox{Ext}^1(M,M)$, resp. $\hbox{Ext}^2(M,M)$,are the modules of infinitesimal deformations, respectively of obstructions, of $M$ (like T1 and T2 for a singularity). Similar to the treatment of singularities, the semiuniversal deformation of $M$ can be computed (if $\hbox{Ext}^1$is finite dimensional) with the help of $\hbox{Ext}^1$, $\hbox{Ext}^2$and the cup product. There is an extra procedure for $\hbox{Ext}^k(R/J,R)$if $J$ is an ideal in $R$, since this is faster than the general Ext.

We compute

  • the infinitesimal deformations ( $=\hbox{Ext}^1(K,K)$)and obstructions ( $=\hbox{Ext}^2(K,K)$)of the residue field $K=R/m$ of an ordinary cusp, $R=K[x,y]_m/(x^2-y^3)$, $m=(x,y)$.To compute $\hbox{Ext}^1(m,m)$we have to apply Ext(1,syz(m),syz(m)) with syz(m) the first syzygy module of $m$, which is isomorphic to $\hbox{Ext}^2(K,K)$.
  • $\hbox{Ext}^k(R/i,R)$for some ideal $i$ and with an extra option.

  LIB "homolog.lib";
  ring R=0,(x,y),ds;
  ideal i=x2-y3;
  qring q = std(i);      // defines the quotient ring k[x,y]_m/(x2-y3)
  ideal m = maxideal(1);
  module T1K = Ext(1,m,m);  // computes Ext^1(R/m,R/m)
==> // dimension of Ext^1:  0
==> // vdim of Ext^1:       2
==> 0,x,0,y,
==> x,0,y,0 
  printlevel=2;             // gives more explanation
  module T2K=Ext(2,m,m);    // computes Ext^2(R/m,R/m)
==> // Computing Ext^2 (help Ext; gives an explanation):
==> // Let 0<--coker(M)<--F0<--F1<--F2<--... be a resolution of coker(M),
==> // and 0<--coker(N)<--G0<--G1 a presentation of coker(N),
==> // then Hom(F2,G0)-->Hom(F3,G0) is given by:
==> y2,x,
==> x, y 
==> // and Hom(F1,G0) + Hom(F2,G1)-->Hom(F2,G0) is given by:
==> -y,x,  x,0,y,0,
==> x, -y2,0,x,0,y 
==> // dimension of Ext^2:  0
==> // vdim of Ext^2:       2
==> 0,x,0,y,
==> x,0,y,0 
  module E = Ext(1,syz(m),syz(m));
==> // dimension of Ext^1:  0
==> // vdim of Ext^1:       2
==> x, 0,-y2,x,0,y,
==> -y,0,x,  0,y,0,
==> 0, 1,0,  0,0,0,
==> 1, 0,0,  0,0,0 
  //The matrices which we have just computed are presentation matrices
  //of the modules T2K and E. Hence we may ignore those columns
  //containing 1 as an entry and see that T2K and E are isomorphic
  //as expected, but differently presented.
  ring S=0,(x,y,z),dp;
  ideal  i = x2y,y2z,z3x;
  module E = Ext_R(2,i);
==> // dimension of Ext^2:  1
==> 0,y,0,z2,
==> z,0,0,-x,
==> 0,0,x,-y 
  // if a 3-rd argument of type int is given,
  // a list of Ext^k(R/i,R), a SB of Ext^k(R/i,R) and a vector space basis
  // is returned:
  list LE = Ext_R(3,i,0);
==> // dimension of Ext^3:  0
==> // vdim of Ext^3:       2
==> [1]:
==>    _[1]=y*gen(1)
==>    _[2]=x*gen(1)
==>    _[3]=z2*gen(1)
==> [2]:
==>    _[1]=y*gen(1)
==>    _[2]=x*gen(1)
==>    _[3]=z2*gen(1)
==> [3]:
==>    _[1,1]=z
==>    _[1,2]=1
==> y,x,z2
==> z,1