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Thread: Affine varieties

  1. #1
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    Affine varieties

    1. Show that a map between affine varieties can be continuous for the
    Zariski topology without being regular.
    2. Show that the circle x^{2}+y^{2}=1 is isomorphic (as an affine variety)
    to the hyperbola xy=1, but neither is isomorphic to A^{1}.
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  2. #2
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    i'll assume that the base field is $\displaystyle \mathbb{C}.$

    Quote Originally Posted by temanratm View Post
    1. Show that a map between affine varieties can be continuous for the
    Zariski topology without being regular.[/tex]
    define $\displaystyle f: \mathbb{A}^1 \rightarrow \mathbb{A}^1$ by $\displaystyle f(0)=0,$ and $\displaystyle f(t)=\frac{1}{t}, \ \forall \ t \neq 0.$

    2. Show that the circle x^{2}+y^{2}=1 is isomorphic (as an affine variety)
    to the hyperbola xy=1.
    let $\displaystyle <x^2+y^2-1>=I,$ and $\displaystyle <xy-1>=J.$ also for any $\displaystyle f(x,y) \in \mathbb{C}[x,y]$ let $\displaystyle f^{*}(x,y)=f \left(\frac{x+y}{2}, \frac{i(y-x)}{2} \right).$ define the map

    $\displaystyle \phi: \frac{\mathbb{C}[x,y]}{I} \rightarrow \frac{\mathbb{C}[x,y]}{J},$ by: $\displaystyle \phi(f(x,y) + I)=f^{*}(x,y)+J.$ now if $\displaystyle f(x,y) \in I,$ then $\displaystyle f(x,y)=(x^2+y^2-1)g(x,y),$ for some

    $\displaystyle g(x,y) \in \mathbb{C}[x,y].$ but then: $\displaystyle f^{*}(x,y)=(xy-1)g \left(\frac{x+y}{2}, \frac{i(y-x)}{2} \right),$ i.e. $\displaystyle f^{*}(x,y) \in J.$ so $\displaystyle \phi$ is well-defined.clearly $\displaystyle \phi$ is a surjective

    ring homomorphism. now suppose $\displaystyle f \in \ker \phi.$ then $\displaystyle f^{*}(x,y) \in J,$ i.e. $\displaystyle f^{*}(x,y)= (xy-1)h(x,y),$ for some $\displaystyle h(x,y) \in \mathbb{C}[x,y].$ therefore:

    $\displaystyle f(x,y)=f^{*}(x+iy,x-iy)=(x^2+y^2-1)h(x+iy,x-iy).$ so $\displaystyle f(x,y) \in I.$ hence $\displaystyle \ker \phi =0.$ so $\displaystyle \phi$ is injective and we're done!


    but neither is isomorphic to A^{1}.
    1) $\displaystyle R_1=\frac{\mathbb{C}[x,y]}{I}$ and $\displaystyle \mathbb{C}[t]$ are not isomorphic because $\displaystyle R_1$ is not a UFD: $\displaystyle (y + I)(y+I)=(1-x + I)(1+x + I).$

    2) $\displaystyle R_2=\frac{\mathbb{C}[x,y]}{J}$ and $\displaystyle \mathbb{C}[t]$ are not isomoprphic because, if there was an isomorphism $\displaystyle \psi: R_2 \rightarrow \mathbb{C}[t],$ then assuming that

    $\displaystyle \psi(x+J)=p(t), \ \psi(y+J) = q(t),$ we'll get $\displaystyle 0=\psi(0)=\psi(xy-1+J)=p(t)q(t)-1.$ thus $\displaystyle p(t)q(t)=1,$ and therefore $\displaystyle p(t), q(t)$

    are constant. but then $\displaystyle \psi$ would not be surjective. Q.E.D.
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  3. #3
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    Quote Originally Posted by NonCommAlg View Post
    i'll assume that the base field is $\displaystyle \mathbb{C}.$



    define $\displaystyle f: \mathbb{A}^1 \rightarrow \mathbb{A}^1$ by $\displaystyle f(0)=0,$ and $\displaystyle f(t)=\frac{1}{t}, \ \forall \ t \neq 0.$



    let $\displaystyle <x^2+y^2-1>=I,$ and $\displaystyle <xy-1>=J.$ also for any $\displaystyle f(x,y) \in \mathbb{C}[x,y]$ let $\displaystyle f^{*}(x,y)=f \left(\frac{x+y}{2}, \frac{i(y-x)}{2} \right).$ define the map

    $\displaystyle \phi: \frac{\mathbb{C}[x,y]}{I} \rightarrow \frac{\mathbb{C}[x,y]}{J},$ by: $\displaystyle \phi(f(x,y) + I)=f^{*}(x,y)+J.$ now if $\displaystyle f(x,y) \in I,$ then $\displaystyle f(x,y)=(x^2+y^2-1)g(x,y),$ for some

    $\displaystyle g(x,y) \in \mathbb{C}[x,y].$ but then: $\displaystyle f^{*}(x,y)=(xy-1)g \left(\frac{x+y}{2}, \frac{i(y-x)}{2} \right),$ i.e. $\displaystyle f^{*}(x,y) \in J.$ so $\displaystyle \phi$ is well-defined.clearly $\displaystyle \phi$ is a surjective

    ring homomorphism. now suppose $\displaystyle f \in \ker \phi.$ then $\displaystyle f^{*}(x,y) \in J,$ i.e. $\displaystyle f^{*}(x,y)= (xy-1)h(x,y),$ for some $\displaystyle h(x,y) \in \mathbb{C}[x,y].$ therefore:

    $\displaystyle f(x,y)=f^{*}(x+iy,x-iy)=(x^2+y^2-1)h(x+iy,x-iy).$ so $\displaystyle f(x,y) \in I.$ hence $\displaystyle \ker \phi =0.$ so $\displaystyle \phi$ is injective and we're done!




    1) $\displaystyle R_1=\frac{\mathbb{C}[x,y]}{I}$ and $\displaystyle \mathbb{C}[t]$ are not isomorphic because $\displaystyle R_1$ is not a UFD: $\displaystyle (y + I)(y+I)=(1-x + I)(1+x + I).$

    2) $\displaystyle R_2=\frac{\mathbb{C}[x,y]}{J}$ and $\displaystyle \mathbb{C}[t]$ are not isomoprphic because, if there was an isomorphism $\displaystyle \psi: R_2 \rightarrow \mathbb{C}[t],$ then assuming that

    $\displaystyle \psi(x+J)=p(t), \ \psi(y+J) = q(t),$ we'll get $\displaystyle 0=\psi(0)=\psi(xy-1+J)=p(t)q(t)-1.$ thus $\displaystyle p(t)q(t)=1,$ and therefore $\displaystyle p(t), q(t)$

    are constant. but then $\displaystyle \psi$ would not be surjective. Q.E.D.
    Thank you a lot
    I've one more problem

    Let q be a power of a prime p and let F_{q} be the field with q elements.
    S is a subset of F_{q}[X_{1},...,X_{n}] and let V be ots zero set in k^{n}.
    where k is alg-c closure of F_{q}.
    Show that the map (a_{1}, .... a_{n})->((a_{1})^q,...,(a_{n})^q) is a regular map
    f:V->V (i.e., f(v)subset of V)
    Verify that the set of fixed points of f(x) is the set of zeros of the elements
    of S with coordinates in F_{q}.

    I would be very grateful
    thx in advance
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  4. #4
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    Quote Originally Posted by temanratm View Post
    Thank you a lot
    I've one more problem

    Let q be a power of a prime p and let F_{q} be the field with q elements.
    S is a subset of F_{q}[X_{1},...,X_{n}] and let V be its zero set in k^{n}.
    where k is alg-c closure of F_{q}.
    Show that the map (a_{1}, .... a_{n})->((a_{1})^q,...,(a_{n})^q) is a regular map
    f:V->V (i.e., f(v)subset of V)
    let $\displaystyle v = (a_1, \ ... \ ,a_n) \in V,$ and $\displaystyle g(x_1, \ ... \ ,x_n)=\sum \beta_{i_1, \ ... , \ i_n}x_1^{i_1} \ ... \ x_n^{i_n} \in S.$ since $\displaystyle \beta_{i_1, \ ... \ ,i_n} \in \mathbb{F}_q,$

    we'll have: $\displaystyle (g(x_1, \ ... \ ,x_n))^q=\sum \beta_{i_1, \ ... \ ,i_n} x_1^{qi_1} \ ... \ , x_n^{qi_n}=g(x_1^q, \ ... \ , x_n^q).$ therefore:

    $\displaystyle g(a_1^q, \ ... \ , a_n^q) = (g(a_1, \ ... \ , a_n))^q = (g(v))^q=0,$ i.e. $\displaystyle f(v)=(a_1^q, \ ... \ ,a_n^q) \in V. \ \ \ \square$

    Verify that the set of fixed points of f(x) is the set of zeros of the elements
    of S with coordinates in F_{q}.
    well, this is pretty obvious: $\displaystyle (a_1, \ ... \ , a_n)$ is a fixed point of $\displaystyle f$ if and only if $\displaystyle a_j^q=a_j, \ j=1, \ ... \ , n.$

    so each $\displaystyle a_j$ is a root of the polynomial $\displaystyle h(x)=x^q - x \in k[x].$ but every element of $\displaystyle \mathbb{F}_q$ is also a root

    of $\displaystyle h(x),$ and obviously $\displaystyle h(x)$ has exactly $\displaystyle q=|\mathbb{F}_q|$ roots in $\displaystyle k.$ thus $\displaystyle a_j \in \mathbb{F}_q, \ j=1, \ ... \ , n. \ \ \ \square$
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