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

vendredi 1 novembre 2013

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 [itex] \phi \ : V \rightarrow W [/itex] is called a morphism (or polynomial map or regular map) of algebraic sets if

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

[itex] \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)) [/itex]

for all [itex] ( a_1, a_2, ... a_n) \in V [/itex]

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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 [itex] k[x_1, x_2, ... ... x_n] [/itex].

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

since [itex] {\phi}_1, {\phi}_2, .......... , {\phi}_m [/itex] are polynomials in [itex] x_1, x_2, ... ... , x_n [/itex].

... ... etc etc

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

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:

[itex] F \circ \phi (( a_1, a_2, ... a_n)) [/itex]

[itex] = F( \phi (( a_1, a_2, ... a_n)) [/itex]

[itex] = 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) ) [/itex]

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

so then we have that ...

[itex] F \circ \phi = F({\phi}_1, {\phi}_2, .......... , {\phi}_m) [/itex].

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

Peter

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