Elementary Algebraic Geometry - D&F Section 15.1 - Exercise 15

[Math Amateur]

Dummit and Foote (D&F), Ch15, Section 15.1, Exercise 15 reads as follows:

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If $k = \mathbb{F}_2$ and $V = \{ (0,0), (1,1) \} \subset \mathbb{A}^2$,

show that $\mathcal{I} (V)$ is the product ideal $m_1m_2$

where $m_1 = (x,y)$ and $m_2 = (x -1, y-1)$.

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I am having trouble getting started on this problem.

One issue/problem I have is - what is the exact nature of $m_1, m_2$ and $m_1m_2$. What (explicitly) are the nature of the elements of these ideals.

I would appreciate some help and guidance.

Peter



Note: D&F define $\mathcal{I} (V)$ as follows:

$\mathcal{I} (V) = \{ f \in k(x_1, x_2, ... , x_n) \ | \ f(a_1, a_2, ... , a_n) = 0 \ \ \forall \ \ (a_1, a_2, ... , a_n) \in V \}$

 

[R136a1]

The ideal $m_1m_2$ is generated by $\{xy~\vert~x\in m_1,y\in m_2\}$. So $m_1m_2$ consists of sums of these elements.

Is this what you wanted to know?

 

[Math Amateur]

Thanks R136a1

Can you also specify what the elements of $m_1$ and $m_2$ look like?

Thanks Again,

Peter

 

[R136a1]

Elements of $m_1$ have the form

$$\alpha x + \beta y$$

for $\alpha,\beta\in \mathbb{F}_2$. Analogous for $m_2$.

 

[Math Amateur]

Thanks R136a1

Just reflecting on your reply.

Given the context of this exercise (algebraic geometry) and the definition of

$\mathcal{I} (V)$ as follows:

$\mathcal{I} (V) = \{ f \in k(x_1, x_2, ... , x_n) \ | \ f(a_1, a_2, ... , a_n) = 0 \ \ \forall \ \ (a_1, a_2, ... , a_n) \in V \}$

would it be more accurate to (following our lead) to define the elements of (x,y) as

$f_1x + f_2y$

where $f_1, f_2 \in \mathbb{F}_2[x,y]$

What do you think?

Peter

 

[R136a1]

Yes, of course. What was I thinking...