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

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
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 [TEX] \phi \ : V \rightarrow W [/TEX] is called a morphism (or polynomial map or regular map) of algebraic sets if

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

[TEX] \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)) [/TEX]

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

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

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

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

... ... etc etc

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

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:

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

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

[TEX] = 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) ) [/TEX]

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

so then we have that ...

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

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

Peter


[Note: This has also been posted on MHF]