reesMap -- compute the Rees map of $I$

Synopsis

Usage:

reesMap I
Inputs:
- I, an ideal, a homogeneous ideal of polynomial ring $S = \mathbb{K}[x_1,\ldots,x_n]$, $\mathbb{K}$ a field
Outputs:
- g, a ring map, the Rees map of $I$

Description

Given a homogeneous ideal $I$, with minimal homogeneous generating set $\{f_1,\dots,f_m\}$, this method yields the bigraded map $g: \mathbb{K}[x_1,\dots,x_n,y_1,\dots,y_m] \to \mathbb{K}[x_1,\dots,x_n,t]$ defined by setting $g(x_i) = x_i$ for $1 \le i \le n$ and $g(y_j) = f_j t$ for $1 \le j \le m$ and with $\text{bideg}(x_i) = (1,0)$ and $\text{bideg}(y_j) = (\text{deg}(f_j), 1)$.

i1 : S = QQ[x_1..x_3];

i2 : I = ideal(x_1*x_2,x_1*x_3,x_2*x_3);

o2 : Ideal of S

i3 : reesMap I

o3 = map (QQ[x ..x , t], QQ[x ..x , y ..y ], {x , x , x , x x t, x x t, x x t})
              1   3          1   3   1   3     1   2   3   1 2    1 3    2 3

o3 : RingMap QQ[x ..x , t] <-- QQ[x ..x , y ..y ]
                 1   3             1   3   1   3

i4 : S = QQ[x_1..x_4];

i5 : I = ideal(x_1^3*x_2+x_3^4,x_1+x_2+x_4,x_3^3);

o5 : Ideal of S

i6 : reesMap I

                                                               3       4                     3
o6 = map (QQ[x ..x , t], QQ[x ..x , y ..y ], {x , x , x , x , x x t + x t, x t + x t + x t, x t})
              1   4          1   4   1   3     1   2   3   4   1 2     3    1     2     4    3

o6 : RingMap QQ[x ..x , t] <-- QQ[x ..x , y ..y ]
                 1   4             1   4   1   3

Ways to use reesMap:

reesMap(Ideal)

For the programmer

The object reesMap is a method function.