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Math Help - Algebraic Geometry

  1. #1
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    Algebraic Geometry

    Take two affine varieties in A^2(Q)
    V_1 = {(x,y): x^2 = y^3}
    V_2 = {(u,v) : u^3 = y^4}
    with function fields Q(V_1) and Q(V_2)

    a.) Show the function fields are isomorphic as Q-algebras
    b.) Construct an explicit Birational map V_1 --> V_2 (dotted line)

    Also,

    Consider the lines
    V_1={x=y=0}
    V_2={y=z=0}
    V_3={z=x=0}

    Show that the product Ideal I(V_1)I(V_2)I(V_3) is smaller than the intersection of I(V_1), I(V_2), I(V_3), even though they define the same variety

    b.) Suppose I,J \subset R are ideals such that I + J= R. Show that IJ = the intersection of I and J.
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  2. #2
    MHF Contributor kalagota's Avatar
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    Taguig City, Philippines
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    Quote Originally Posted by terr13 View Post

    b.) Suppose I,J \subset R are ideals such that I + J= R. Show that IJ = the intersection of I and J.
    well, i will assume that R is a commutative ring..

    note that I\cap J \subset J and I\cap J \subset I

    also, I\cdot R = I for any ideal I of R.

    thus I\cap J = (I\cap J) \cdot R = (I\cap J) \cdot (I+J) = (I\cap J) \cdot I + (I\cap J) \cdot J \subset JI + IJ = IJ

    for the other direction, if a \in IJ, then

    a = \sum x_iy_i (finite sum) where x_i \in I and y_i \in J

    properties of ideals will show you that a is also in I\cap J
    Last edited by kalagota; October 19th 2008 at 10:50 PM.
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