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Nonnormal Locus
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Non-Normal Locus
Theorem: Let R be a reduced Noetherian ring, J an ideal in R and let a be any element of J which is a non-zerodivisor for R. Then the non-normal locus of R is given by
AnnR ( HomR(J,J) / R )   =   <a> : (aJ:J) .

We compute the non-normal locus of the ring   R := Q[x,y,z] / < xy2-xz3-z6 > :
LIB"primdec.lib";
ring A = 0,(x,y,z),dp;

ideal I = xy2-xz3-z6;
ideal sing = I+jacob(I);
ideal J = radical(sing);

qring R = std(I);     // quotient ring A/I
ideal J = fetch(A,J);
ideal a = J[1];       // a non-zero element of J

quotient(a,quotient(a*J,J));   // <a>:(aJ:J)
==>
_[1]=z
_[2]=y
From the output, we read that the non-normal locus is the x-axis (the zero-set of <y,z>).


Sao Carlos, 08/02 http://www.singular.uni-kl.de