reesModuleToIdeal -- embeds a homogeneous rank 1 module as an ideal in a Rees algebra

• Usage:

  (I,d,phi) = reesModuleToIdeal(S, M)

• Inputs:
  • S, a ring, constructed with classicalReesAlgebra or extendedReesAlgebra
  • M, a module, a homogeneous S-module
• Optional inputs:
  • MTries => ..., default value 10
• Outputs:
  • I, an ideal, an ideal of S isomorphic to M as a module
  • d, an integer, the degree shift between the module and ideal
  • phi, a matrix, the map M to S^1 whose image is I

i1 : R = QQ[x,y];

i2 : J = ideal(x^2,x*y^2,y^3);

o2 : Ideal of R

i3 : S = extendedReesAlgebra(J);

i4 : canMod = reesCanonicalModule(S)

o4 = cokernel {1, 3} | ly_1 -ly_2  0    -y2  x     0   -ti y |
              {0, 2} | 0    0      ly_1 ly_2 -ly_0 -y  x   0 |
              {1, 3} | ly_2 -yly_0 y2   0    0     -ti 0   x |

                            3
o4 : S-module, quotient of S

i5 : L = reesModuleToIdeal(S, canMod) --the first entry -1 of {-1,-2} refers to the Rees algebra degree

o5 = (ideal (ti, y, x), {-1, -2}, | x ti -y |)

o5 : Sequence

i6 : L#2 -- the map from canMod to S

o6 = | x ti -y |

             1
o6 : Matrix S  <-- canMod