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D.4.9.3 primeClosure
Procedure from library reesclos.lib (see reesclos_lib).
- Usage:
primeClosure(L [,c]); L a list of a ring containing a prime ideal ker, c an optional integer
- Return:
a list L consisting of rings L[1],...,L[n] such that
- L[1] is a copy of (not a reference to!) the input ring L[1] - all rings L[i] contain ideals ker, L[2],...,L[n] contain ideals phi such that
L[1]/ker –> ... –> L[n]/ker
are injections given by the corresponding ideals phi, and L[n]/ker is the integral closure of L[1]/ker in its quotient field. - all rings L[i] contain a polynomial nzd such that elements of L[i]/ker are quotients of elements of L[i-1]/ker with denominator nzd via the injection phi.- Note:
- L is constructed by recursive calls of primeClosure itself. - c determines the choice of nzd:
- c not given or equal to 0: first generator of the ideal SL, the singular locus of Spec(L[i]/ker)
- c<>0: the generator of SL with least number of monomials.
Example:
LIB "reesclos.lib"; ring R=0,(x,y),dp; ideal I=x4,y4; def K=ReesAlgebra(I)[1]; // K contains ker such that K/ker=R[It] list L=primeClosure(K); def R(1)=L[1]; // L[4] contains ker, L[4]/ker is the def R(4)=L[4]; // integral closure of L[1]/ker setring R(1); R(1); → // characteristic : 0 → // number of vars : 4 → // block 1 : ordering dp → // : names x y U(1) U(2) → // block 2 : ordering C ker; → ker[1]=y^4*U(1)-x^4*U(2) setring R(4); R(4); → // characteristic : 0 → // number of vars : 7 → // block 1 : ordering a → // : names T(1) T(2) T(3) T(4) T(5) T(6) T(7) → // : weights 1 1 1 1 1 1 1 → // block 2 : ordering dp → // : names T(1) T(2) T(3) T(4) T(5) T(6) T(7) → // block 3 : ordering C ker; → ker[1]=T(2)*T(5)-T(1)*T(7) → ker[2]=T(1)*T(5)-T(2)*T(6) → ker[3]=T(5)*T(6)-T(3)*T(7) → ker[4]=T(4)*T(6)-T(5)*T(7) → ker[5]=T(5)^2-T(6)*T(7) → ker[6]=T(4)*T(5)-T(7)^2 → ker[7]=T(3)*T(5)-T(6)^2 → ker[8]=T(2)^2*T(6)-T(1)^2*T(7) → ker[9]=T(3)*T(4)-T(6)*T(7) → ker[10]=T(1)*T(4)-T(2)*T(7) → ker[11]=T(2)*T(3)-T(1)*T(6) → ker[12]=T(2)^2*T(6)^2-T(1)^2*T(6)*T(7) |