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Math Help - Composition of Functions - in the context of morphisms in algebraic geometry

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    Super Member Bernhard's Avatar
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    Composition of Functions - in the context of morphisms in algebraic geometry

    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 .

    ... ... etc etc

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

    I am concerned that I do not fully understand exactly how/why  F \circ \phi = F({\phi}_1, {\phi}_2, .......... , {\phi}_m)   .

    I may be obsessively over-thinking the validity of this matter (that may be just a notational matter) ... but anyway my understanding is as follows:

     F \circ \phi (( a_1, a_2, ... a_n))

      = F( \phi (( a_1, a_2, ... a_n))

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

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

    so then we have that ...

     F \circ \phi = F({\phi}_1, {\phi}_2, .......... , {\phi}_m)   .

    Can someone please confirm that the above reasoning and text is logically and notationally correct?

    Peter
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