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 is called a morphism (or polynomial map or regular map) of algebraic sets if
there are polynomials such that
for all
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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 .
Then is a polynomial in
since are polynomials in .
... ... etc etc
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I am concerned that I do not fully understand exactly how/why .
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:
so then we have that ...
.
Can someone please confirm that the above reasoning and text is logically and notationally correct?
Peter