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

Synopsis

• Usage:

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

• Inputs:
  • I, an ideal, whose endomorphism ring we'll compute
  • f, a ring element, a nonzerodivisor in $I$
  • Variable, a symbol,
  • Index, an integer,
• Optional inputs:
  • Strategy => a list, default value {}, possible elements include ``Vasconcelos''
  • Verbosity => an integer, default value 0, larger numbers give more information
  • Index => ..., default value 0, Sets the starting index on the new variables used to build the endomorphism ring Hom(J,J)
  • Variable => ..., default value w, Sets the name of the indexed variables introduced in computing the endomorphism ring Hom(J,J).
• Outputs:
  • F, a ring map, The inclusion map from $R$ into $S = Hom_R(I,I)$
  • G, a ring map, $frac S \rightarrow frac R$, giving the fractions corresponding to each generator of $S$.

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 = -------------------------------------
           2    2   2      3            5
     (w   x  - y , w    - x y, w   y - x )
       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