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Math Help - Algebriac Geometry - Morphisms of Algebraic Sets

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    Algebriac Geometry - Morphisms of Algebraic Sets

    I am reading Dummit and Foote (D&F) Section 15.1 on Affine Algebraic Sets.

    On page 662 (see attached) D&F define a morphism or polynomial map of algebraic sets as follows:

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

    Definition. A map  \phi \ : V \rightarrow W is called a morphism (or polynomial map or regular map) of algebraic sets if

    there are polynomials  {\phi}_1, {\phi}_2, .......... , {\phi}_m \in k[x_1, x_2, ... ... x_n] such that

      \phi(( a_1, a_2, ... a_n)) = ( {\phi}_1 ( a_1, a_2, ... a_n) , {\phi}_2 ( a_1, a_2, ... a_n), ... ... ... {\phi}_m ( a_1, a_2, ... a_n))

    for all  ( a_1, a_2, ... a_n) \in V

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


    D&F then go on to define a map between the quotient rings k[W] and k[V] as follows: (see attachment page 662)


    -------------------------------------------------------------------------------------------------------
    Suppose F is a polynomial in  k[x_1, x_2, ... ... x_n] .

    Then  F \circ \phi = F({\phi}_1, {\phi}_2, .......... , {\phi}_m) is a polynomial in  k[x_1, x_2, ... ... x_n]

    since  {\phi}_1, {\phi}_2, .......... , {\phi}_m  are polynomials in  x_1, x_2, ... ... x_n .

    If  F \in \mathcal{I}(W), then  F \circ \phi (( a_1, a_2, ... a_n)) = 0  for every  ( a_1, a_2, ... a_n) \in V

    since   \phi (( a_1, a_2, ... a_n)) \in W  .

    Thus   F \circ \phi \in  \mathcal{I}(V)

    It follows that  \phi  induces a well defined map from the quotient ring   k[x_1, x_2, ... ... x_n]/\mathcal{I}(W)

    to the quotient ring   k[x_1, x_2, ... ... x_n]/\mathcal{I}(V)  :

     \widetilde{\phi} \ : \ k[W] \rightarrow k[V]

      f \rightarrow f \circ \phi

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

    My problem is, how exactly does it follow (and why?) that  \phi  induces a well defined map from the quotient ring   k[x_1, x_2, ... ... x_n]/\mathcal{I}(W)  to the quotient ring   k[x_1, x_2, ... ... x_n]/\mathcal{I}(V)  ?

    Can someone (explicitly) show me the logic of this - why exactly does it follow?

    Peter
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