Category: D. Cox, J. Little, and D. O’Shea’s Using Algebraic Geometry (1998)

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202110121137 Exercise 1.1.1

(a) Show that x^2\in\langle x-y^2,xy\rangle in k[x,y] (k any field).

(b) Show that \langle x-y^2,xy,y^2\rangle = \langle x,y^2\rangle.

(c) Is \langle x-y^2,xy\rangle = \langle x^2,xy\rangle? Why or why not?

Definition.

Let f_1,\dots ,f_s\in k[x_1,\dots ,x_n]. We let \langle f_1,\dots ,f_s\rangle denote the collection \langle f_1,\dots ,f_s\rangle = \{ p_1f_1+\cdots +p_sf_s:p_i\in k[x_1,\dots ,x_n]\} for i= 1,\dots ,s.

(a)

\begin{aligned} x^2 & = (x)(x-y^2) + (y)(xy) \\ & \in \langle x-y^2,xy\rangle \\ \end{aligned}

(b)

First, we want to show

\langle x-y^2,xy,y^2\rangle \in \langle x,y^2\rangle.

For any p_1,p_2,p_3\in k[x,y],

\begin{aligned} & \quad \langle x-y^2,xy,y^2\rangle \\ & = (p_1)(x-y^2)+(p_2)(xy)+(p_3)(y^2) \\ & = p_1x-p_1y^2+p_2xy+p_3y^2 \\ & = (p_1+p_2y)(x)+(p_3-p_1)(y^2) \\ & \in \langle x,y^2\rangle \\ \end{aligned}

Next, we want to show

\langle x,y^2\rangle\in\langle x-y^2,xy,y^2\rangle.

For any p_1,p_2\in k[x,y],

\begin{aligned} &\quad \langle x,y^2\rangle \\ & = (p_1)(x)+(p_2)(y^2) \\ & = (p_1)(x-y^2)+(0)(xy)+(p_1+p_2)(y^2) \\ & \in \langle x-y^2,xy,y^2\rangle \\ \end{aligned}

All in all,

\langle x-y^2,xy,y^2\rangle = \langle x,y^2\rangle.

(c) I guess \textrm{\scriptsize{NOT}}.