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

  1. Oct 30, 2013 #1
    Dummit and Foote (D&F), Ch15, Section 15.1, Exercise 15 reads as follows:

    ----------------------------------------------------------------------------------------------------

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

    show that [itex] \mathcal{I} (V) [/itex] is the product ideal [itex] m_1m_2 [/itex]

    where [itex] m_1 = (x,y) [/itex] and [itex] m_2 = (x -1, y-1) [/itex].

    ------------------------------------------------------------------------------------------------------

    I am having trouble getting started on this problem.

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

    I would appreciate some help and guidance.

    Peter



    Note: D&F define [itex] \mathcal{I} (V) [/itex] as follows:

    [itex] \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 \} [/itex]
     
    Last edited: Oct 30, 2013
  2. jcsd
  3. Oct 30, 2013 #2
    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?
     
  4. Oct 30, 2013 #3
    Thanks R136a1

    Can you also specify what the elements of [itex] m_1 [/itex] and [itex] m_2 [/itex] look like?

    Thanks Again,

    Peter
     
  5. Oct 30, 2013 #4
    Elements of ##m_1## have the form

    [tex]\alpha x + \beta y[/tex]

    for ##\alpha,\beta\in \mathbb{F}_2##. Analogous for ##m_2##.
     
  6. Oct 30, 2013 #5
    Thanks R136a1

    Just reflecting on your reply.

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

    [itex] \mathcal{I} (V) [/itex] as follows:

    [itex] \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 \} [/itex]

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

    [itex] f_1x + f_2y [/itex]

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

    What do you think?

    Peter
     
    Last edited: Oct 30, 2013
  7. Oct 30, 2013 #6
    Yes, of course. What was I thinking...
     
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