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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].
If [itex] F \in \mathcal{I}(W)[/itex], then [itex] F \circ \phi (( a_1, a_2, ... a_n)) = 0 [/itex] for every [itex] ( a_1, a_2, ... a_n) \in V [/itex]
since [itex] \phi (( a_1, a_2, ... a_n)) \in W [/itex].
Thus [itex] F \circ \phi \in \mathcal{I}(V) [/itex]
It follows that [itex] \phi [/itex] induces a well defined map from the quotient ring [itex] k[x_1, x_2, ... ... x_n]/\mathcal{I}(W) [/itex]
to the quotient ring [itex] k[x_1, x_2, ... ... x_n]/\mathcal{I}(V) [/itex] :
[itex] \widetilde{\phi} \ : \ k[W] \rightarrow k[V] [/itex]
[itex] f \rightarrow f \circ \phi [/itex]
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My problem is, how exactly does it follow (and why?) that [itex] \phi [/itex] induces a well defined map from the quotient ring [itex] k[x_1, x_2, ... ... x_n]/\mathcal{I}(W) [/itex] to the quotient ring [itex] k[x_1, x_2, ... ... x_n]/\mathcal{I}(V) [/itex] ?
Can someone (explicitly) show me the logic of this - why exactly does it follow?
Peter
On page 662 (see attached) D&F define a morphism or polynomial map of algebraic sets as follows:
----------------------------------------------------------------------------------------------
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)
-------------------------------------------------------------------------------------------------------
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].
If [itex] F \in \mathcal{I}(W)[/itex], then [itex] F \circ \phi (( a_1, a_2, ... a_n)) = 0 [/itex] for every [itex] ( a_1, a_2, ... a_n) \in V [/itex]
since [itex] \phi (( a_1, a_2, ... a_n)) \in W [/itex].
Thus [itex] F \circ \phi \in \mathcal{I}(V) [/itex]
It follows that [itex] \phi [/itex] induces a well defined map from the quotient ring [itex] k[x_1, x_2, ... ... x_n]/\mathcal{I}(W) [/itex]
to the quotient ring [itex] k[x_1, x_2, ... ... x_n]/\mathcal{I}(V) [/itex] :
[itex] \widetilde{\phi} \ : \ k[W] \rightarrow k[V] [/itex]
[itex] f \rightarrow f \circ \phi [/itex]
-------------------------------------------------------------------------------------------------------------------
My problem is, how exactly does it follow (and why?) that [itex] \phi [/itex] induces a well defined map from the quotient ring [itex] k[x_1, x_2, ... ... x_n]/\mathcal{I}(W) [/itex] to the quotient ring [itex] k[x_1, x_2, ... ... x_n]/\mathcal{I}(V) [/itex] ?
Can someone (explicitly) show me the logic of this - why exactly does it follow?
Peter