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Thread: Composition of Functions - in the context of morphisms in algebraic geometry

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

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

    $\displaystyle \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 $\displaystyle ( a_1, a_2, ... a_n) \in V $

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


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    Suppose F is a polynomial in $\displaystyle k[x_1, x_2, ... ... x_n] $.

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

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

    ... ... etc etc

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

    I am concerned that I do not fully understand exactly how/why $\displaystyle 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:

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

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

    $\displaystyle = 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) ) $

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

    so then we have that ...

    $\displaystyle 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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