Question

Let \(\varphi: V \rightarrow W\) be a polynomial map of affine varieties, \(V^{\prime} \subset V, W^{\prime} \subset W\) subvarieties. Suppose \(\varphi\left(V^{\prime}\right) \subset W^{\prime}\). (a) Show that \(\tilde{\varphi}\left(I_{W}\left(W^{\prime}\right)\right) \subset I_{V}\left(V^{\prime}\right)\) (see Problems 2.3). (b) Show that the restriction of \(\varphi\) gives a polynomial map from \(V^{\prime}\) to \(W^{\prime}\).

Short answer

In summary, given a polynomial map between two affine varieties and two subvarieties such that the map is compatible with the inclusion, we have shown that the pullback of the ideal of the subvariety in the codomain is included in the ideal of the subvariety in the domain, and that the restriction of the map is a polynomial map between the subvarieties. This result is useful in the study of algebraic geometry, as it relates geometric objects, such as affine varieties and subvarieties, to algebraic objects, such as ideals in polynomial rings, and highlights the importance of polynomial maps in the study of these relationships.

Step-by-step solution

Part (a): Show that \(\tilde{\varphi}\left(I_{W}\left(W^{\prime}\right)\right) \subset I_{V}\left(V^{\prime}\right)\)

Let's recall that for a polynomial map \(\varphi: V \rightarrow W\), we can define its corresponding ring map \(\tilde{\varphi}: k[W] \rightarrow k[V]\) by \(\tilde{\varphi}(g) = g \circ \varphi\) for \(g \in k[W]\). Now, let \(g \in I_{W}(W^{\prime})\). This means that \(g(P) = 0\) for all \(P \in W^{\prime}\). We want to show that \(\tilde{\varphi}(g) \in I_{V}(V^{\prime})\). First, let's consider \(\tilde{\varphi}(g)\). By the definition, \(\tilde{\varphi}(g) = g \circ \varphi\). Now, let \(P^{\prime} \in V^{\prime}\). We know that \(\varphi(P^{\prime}) \in W^{\prime}\). Since \(g \in I_{W}(W^{\prime})\), we have \(g(\varphi(P^{\prime})) = 0\). But this is exactly \(\tilde{\varphi}(g)(P^{\prime})\). Thus, \(\tilde{\varphi}(g)(P^{\prime}) = 0\) for all \(P^{\prime} \in V^{\prime}\), which implies that \(\tilde{\varphi}(g) \in I_{V}(V^{\prime})\). So, \(\tilde{\varphi}\left(I_{W}\left(W^{\prime}\right)\right) \subset I_{V}\left(V^{\prime}\right)\), as desired.

Part (b): Show that the restriction of \(\varphi\) gives a polynomial map from \(V^{\prime}\) to \(W^{\prime}\)

To show that the restriction of \(\varphi\) gives a polynomial map from \(V^{\prime}\) to \(W^{\prime}\), we need to show that it is well-defined and polynomial. Let \(\varphi^{\prime}: V^{\prime} \rightarrow W^{\prime}\) be the restriction of \(\varphi\). This means that for any point \(P \in V^{\prime}\), we have \(\varphi^{\prime}(P) = \varphi(P)\). Since \(\varphi\left(V^{\prime}\right) \subset W^{\prime}\), it follows that the \(\varphi^{\prime}(P) \in W^{\prime}\) for all \(P \in V^{\prime}\) - so, \(\varphi^{\prime}\) is well-defined. Now, let's show that \(\varphi^{\prime}\) is polynomial. Recall that \(\varphi: V \rightarrow W\) is a polynomial map. To show that \(\varphi^{\prime}\) is also a polynomial map, we need to show that each coordinate function of \(\varphi^{\prime}\) is polynomial. Let \(f_i\) be the \(i\)-th coordinate function of \(\varphi^{\prime}\). Then, by definition, \(f_i(P) = \varphi^{\prime}_i(P)\) for all \(P \in V^{\prime}\). We need to show that \(f_i\) is a polynomial function on \(V^{\prime}\). Since \(\varphi_i\) is a polynomial function on \(V\), we can write \(\varphi_i = \sum_{j = 1}^m c_jg_j\) where \(c_j \in k\) and \(g_j\) are polynomial functions on \(V\). We can therefore write $$f_i(P) = \varphi^{\prime}_i(P) = \sum_{j=1}^m c_jg_j(P)$$ for all \(P \in V^{\prime}\). Since the sum of polynomial functions is also polynomial, \(f_i\) is a polynomial function on \(V^{\prime}\). This holds for all coordinate functions of \(\varphi^{\prime}\). Hence, we have shown that \(\varphi^{\prime}\) is polynomial, and therefore the restriction of \(\varphi\) gives a polynomial map from \(V^{\prime}\) to \(W^{\prime}\).

Key concepts

Subvarieties

Within the context of affine varieties, subvarieties play a crucial role. Imagine you have a garden, and in this garden, there is a specific section dedicated to roses. Similarly, subvarieties can be understood as smaller, defined sections within a larger variety. These sections or subsets maintain a consistent structure with the overarching variety they belong to.
Subvarieties are subsets that satisfy polynomial equations of the larger variety. This concept allows mathematicians to work on more manageable sections without losing the essence of the entire variety. For example:

• If you have an affine variety described by certain polynomial equations, a subvariety will be a solution set satisfying a subset of those equations.
• Subvarieties allow for the study of properties and characteristics within these 'sub-gardens' to draw conclusions about the entirety of the 'garden'.

Recognizing and working with subvarieties is essential for delving deeper into the nature of complex polynomial maps.

Polynomial Functions

Polynomial functions provide the essential link between different affine varieties. Think of polynomial functions as the recipes that define the behavior and characteristics of points within a variety. These functions can describe transformations or mappings, such as moving points from one affine space to another through a series of defined steps or directions.
In the context of affine varieties, a map defined by polynomial functions retains the algebraic structure. For instance:

• Consider a polynomial map \(\varphi: V \rightarrow W\), where each coordinate of a point in \(V\) is transformed to a new coordinate in \(W\) via a polynomial function.
• Polynomial functions ensure that operations such as addition and multiplication within the variety are preserved through these mappings.

This preservation of structure and operations is fundamental, allowing complicated algebraic systems to be studied and understood in a consistent manner.

Ring Map

The concept of a ring map is central to understanding polynomial maps of affine varieties. A ring map can be thought of as a bridge connecting the polynomial rings of different varieties. Imagine two cities that are connected by a series of roads; a ring map ensures that there's a clear route for traveling between them mathematically.
For polynomial maps, ring maps translate the algebraic relationships from one variety to another:

• Given a polynomial map \(\varphi: V \rightarrow W\), the corresponding ring map is \(-\tilde{\varphi}: k[W] \rightarrow k[V]\).
• This map sends a polynomial function \(g\) in \(k[W]\) to \(g \circ \varphi\) in \(k[V]\), maintaining the relationships defined by the original polynomial functions.

Ring maps preserve the algebraic structure and ensure that equations hold true when transitioning between different spaces, just as the roads must remain intact to connect the cities effectively.

Coordinate Functions

Coordinate functions are the building blocks of polynomial maps that allow for the specific and detailed transformation between varieties. It's similar to how a GPS provides exact coordinates to navigate a route; coordinate functions indicate how each feature of the map is altered individually.
For an affine variety transformation, each point in the domain variety is assigned a new position in the target variety through these coordinate functions:

• Each coordinate function is a polynomial, which means it transforms a point via a specific mathematical formula.
• The transformation is consistent for every point, ensuring that all algebraic properties of the variety are retained throughout the mapping process.

By understanding coordinate functions, one grasps how a polynomial map reshapes a variety's features and retains its essence across different mathematical landscapes.