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

[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:

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

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

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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[Note: This has also been posted on MHF]

 

[jvicens]

Dear Peter,

Thank you for bringing up your concerns regarding the definition of a morphism in D&F's Section 15.1 on Affine Algebraic Sets. Your understanding and reasoning are indeed correct.

To clarify, the notation F \circ \phi refers to the composition of the functions F and \phi, where F is a polynomial in k[x_1, x_2, ..., x_n] and \phi is a morphism between two algebraic sets V and W. This means that for any point (a_1, a_2, ..., a_n) \in V, the function F \circ \phi maps it to 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)) \in k[W].

In other words, F \circ \phi is equivalent to F({\phi}_1, {\phi}_2, ..., {\phi}_m) when evaluated at any point in V. This notation is often used to simplify the expression and make it more concise.

I hope this clarifies your understanding. Please let me know if you have any further questions.