Composition of Functions - in the context of morphisms in algebraic ge

[Math Amateur]

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

 

[R136a1]

Your reasoning is correct.

The reasoning you're confused is because the notation $F(\varphi_1,...,\varphi_n)$ is really bad. I would never use it that way.