idealizer -- compute Hom(I,I) as a quotient ring

Synopsis

• Usage:

  (F,G) = idealizer(I,f)

• Inputs:

  • I, an ideal, in a domain R
  • f, a ring element, an element of the ideal I
  • Variable, a symbol
  • Index, an integer
• Outputs:

  • F, a ring map, The inclusion map from R into S = Hom_R(I,I)
  • G, a ring map, frac S →frac R, giving the fractions corresponding to each generator of S.
• Optional inputs:

  • Index => ..., -- Sets the starting index on the new variables used to build the endomorphism ring Hom(J,J). If the program idealizer is used independently, the user will generally want to use the default value of 0. However, when used as part of the integralClosure computation the number needs to start higher depending on the level of recursion involved.
  • Strategy => ... (missing documentation),
  • Variable => ..., -- Sets the name of the indexed variables introduced in computing the endomorphism ring Hom(J,J).

Description

i1 : R = QQ[x,y]/(y^3-x^7)

o1 = R

o1 : QuotientRing

i2 : I = ideal(x^2,y^2)

             2   2
o2 = ideal (x , y )

o2 : Ideal of R

i3 : (F,G) = idealizer(I,x^2);

i4 : target F

                 QQ[w   , x, y]
                     0,0
o4 = -------------------------------------
               5       2    2   2      3
     (w   y - x , w   x  - y , w    - x y)
       0,0         0,0          0,0

o4 : QuotientRing

i5 : first entries G.matrix

       2
      y
o5 = {--, x, y}
       2
      x

o5 : List

See also

• ringFromFractions -- find presentation for f.g. ring
• integralClosure -- integral closure of an ideal or a domain