D.4.6.2 HomJJ

Procedure from library normal.lib (see normal_lib).

Usage:

HomJJ (Li); Li = list: ideal SBid, ideal id, ideal J, poly p

Assume:

R = P/id, P = basering, a polynomial ring, id an ideal of P,
SBid = standard basis of id,
J = ideal of P containing the polynomial p,
p = nonzero divisor of R

Compute:

Endomorphism ring End_R(J)=Hom_R(J,J) with its ring structure as affine ring, together with the canonical map R –> Hom_R(J,J), where R is the quotient ring of P modulo the standard basis SBid.

Return:

a list l of two objects

l[1] : a polynomial ring, containing two ideals, ’endid’ and ’endphi’ such that l[1]/endid = Hom_R(J,J) and endphi describes the canonical map R -> Hom_R(J,J) l[2] : an integer which is 1 if phi is an isomorphism, 0 if not l[3] : an integer, the contribution to delta

Note:

printlevel >=1: display comments (default: printlevel=0)

Example:

LIB "normal.lib";
ring r   = 0,(x,y),wp(2,3);
ideal id = y^2-x^3;
ideal J  = x,y;
poly p   = x;
list Li = std(id),id,J,p;
list L   = HomJJ(Li);
def end = L[1];    // defines ring L[1], containing ideals endid, endphi
setring end;       // makes end the basering
end;
→ //   characteristic : 0
→ //   number of vars : 1
→ //        block   1 : ordering dp
→ //                  : names    T(1) 
→ //        block   2 : ordering C
endid;             // end/endid is isomorphic to End(r/id) as ring
→ endid[1]=0
map psi = r,endphi;// defines the canonical map r/id -> End(r/id)
psi;
→ psi[1]=T(1)^2
→ psi[2]=T(1)^3