# Composition of Functions - in the context of morphisms in algebraic geometry

• Nov 1st 2013, 03:49 PM
Bernhard
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:

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

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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 $k[x_1, x_2, ... ... x_n]$.

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

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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