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

In summary: Your Name]In summary, D&F's Section 15.1 defines a morphism or polynomial map of algebraic sets as a map between two algebraic sets V and W, where there exist polynomials {\phi}_1, {\phi}_2, ..., {\phi}_m in k[x_1, x_2, ..., x_n] such that the map \phi maps any point (a_1, a_2, ..., a_n) \in V to ({\phi}_1(a_1, a_2, ..., a_n), {\phi}_2(a_1, a_2, ..., a_n), ..., {\phi}_m(a_1, a_2, ..., a_n))
  • #1
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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 [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]

-------------------------------------------------------------------------------------------------------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 [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]
 
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  • #2


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.
 

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

What is a composition of functions in the context of morphisms in algebraic geometry?

In algebraic geometry, a composition of functions refers to the process of combining two or more functions to create a new function. In the context of morphisms, this means applying a function to the image of another function, resulting in a new function that maps from one set to another set.

How is the composition of functions related to the concept of morphisms?

Morphisms in algebraic geometry are functions that preserve the structure of algebraic objects, such as polynomials or algebraic varieties. The composition of functions allows us to define new morphisms by applying one function to the image of another, preserving the underlying structure of the objects.

Can the composition of functions be applied to any type of function in algebraic geometry?

Yes, the composition of functions can be applied to any type of function in algebraic geometry, as long as the domain and codomain of the functions are compatible. This means that the output of one function must match the input of the other function, for the composition to make sense.

What is the difference between the composition of functions and the multiplication of functions in algebraic geometry?

The composition of functions refers to the process of combining two or more functions to create a new function. On the other hand, the multiplication of functions refers to the process of multiplying two or more functions together, resulting in a new function. While both operations can be applied to functions in algebraic geometry, they serve different purposes and have different effects on the resulting function.

How is the composition of functions used in algebraic geometry?

The composition of functions is a fundamental concept in algebraic geometry and is used in various aspects of the field. It is used to define and construct new morphisms, to prove theorems, and to study the properties of algebraic objects. It is also a useful tool for understanding the structure and behavior of algebraic varieties and their maps.

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