Question

Let \(\varphi: V \rightarrow W\) be a polynomial map of affine varieties, \(\tilde{\varphi}: \Gamma(W) \rightarrow \Gamma(V)\) the induced map on coordinate rings. Suppose \(P \in V, \varphi(P)=Q\). Show that \(\tilde{\varphi}\) extends uniquely to a ring homomorphism (also written \(\tilde{\varphi}\) ) from \(\mathscr{O}_{Q}(W)\) to \(\mathscr{O}_{P}(V)\). (Note that \(\tilde{\varphi}\) may not extend to all of \(k(W)\).) Show that \(\tilde{\varphi}\left(\mathrm{m}_{Q}(W)\right) \subset \mathfrak{m}_{P}(V)\).

Short answer

Short answer: The induced map \(\tilde{\varphi}\) can be uniquely extended to a ring homomorphism from \(\mathscr{O}_{Q}(W)\) to \(\mathscr{O}_{P}(V)\) by proving that for any \(f \in \mathscr{O}_{Q}(W)\), the function \(\tilde{\varphi}(f)\) belongs to \(\mathscr{O}_{P}(V)\). Additionally, the extended map preserves addition, multiplication, and the identity element. Finally, we showed that \(\tilde{\varphi}(\mathrm{m}_{Q}(W)) \subset \mathfrak{m}_{P}(V)\) by showing that for any \(f \in \mathrm{m}_{Q}(W)\), the function \(\tilde{\varphi}(f)\) belongs to \(\mathfrak{m}_{P}(V)\).

Step-by-step solution

1. Background definitions and notation

Let's first clarify the notation and definitions used in the exercise: - \(\varphi: V \rightarrow W\): A polynomial map of affine varieties (a map that sends points in \(V\) to points in \(W\)). - \(\tilde{\varphi}: \Gamma(W) \rightarrow \Gamma(V)\): The induced map on coordinate rings (maps the coordinate ring of \(W\) to the coordinate ring of \(V\)). - \(P, Q\): Points in \(V\) and \(W\), respectively. - \(\mathscr{O}_{P}(V)\), \(\mathscr{O}_{Q}(W)\): Local rings of the affine varieties at points \(P\) and \(Q\), respectively. They are defined as subrings of their respective function fields \(k(V)\) and \(k(W)\). - \(\mathrm{m}_{P}(V)\), \(\mathrm{m}_{Q}(W)\): Maximal ideals of the local rings \(\mathscr{O}_{P}(V)\) and \(\mathscr{O}_{Q}(W)\), respectively.

2. Extending the induced map to a ring homomorphism

To show that \(\tilde{\varphi}\) extends uniquely to a ring homomorphism from \(\mathscr{O}_{Q}(W)\) to \(\mathscr{O}_{P}(V)\), we need to prove the following: 1. If \(f \in \mathscr{O}_{Q}(W)\), then \(\tilde{\varphi}(f) \in \mathscr{O}_{P}(V)\). 2. The extended map preserves the operations of addition, multiplication, and distributive properties of ring homomorphisms. For the first point, let \(f \in \mathscr{O}_{Q}(W)\). By definition, there exists an open set \(U \subset W\) containing \(Q\) and a regular function \(g \in k[U]\) such that \(f = g|_U\). Since \(\varphi\) is a polynomial map, we have that \(\varphi^{-1}(U) \subset V\) is also an open set containing \(P\). Let \(h = \tilde{\varphi}(g) \in \Gamma(V)\), then we have \(\tilde{\varphi}(f) = h|_{\varphi^{-1}(U)}\). Since the restriction of a regular function is also a regular function, we conclude that \(\tilde{\varphi}(f) \in \mathscr{O}_{P}(V)\). Next, we show that this extension preserves the ring operations. Let \(f, g \in \mathscr{O}_{Q}(W)\). Then, by the properties of the induced map \(\tilde{\varphi}\), we have: 1. \(\tilde{\varphi}(f + g) = \tilde{\varphi}(f) + \tilde{\varphi}(g)\) 2. \(\tilde{\varphi}(f \cdot g) = \tilde{\varphi}(f) \cdot \tilde{\varphi}(g)\) 3. \(\tilde{\varphi}(1) = 1\) (since the identity element should be preserved). These conditions ensure that the extended map is a ring homomorphism between \(\mathscr{O}_{Q}(W)\) and \(\mathscr{O}_{P}(V)\).

3. Proving the property of maximal ideals

Finally, we need to show that \(\tilde{\varphi}(\mathrm{m}_{Q}(W)) \subset \mathfrak{m}_{P}(V)\). Let \(f \in \mathrm{m}_{Q}(W)\), then by definition \(f(Q) = 0\). Since \(\tilde{\varphi}(f) \in \mathscr{O}_{P}(V)\) and \(\varphi(P) = Q\), we have \(\tilde{\varphi}(f)(P) = f(\varphi(P)) = f(Q) = 0\). This implies that \(\tilde{\varphi}(f) \in \mathfrak{m}_{P}(V)\). Thus, we have shown that \(\tilde{\varphi}(\mathrm{m}_{Q}(W)) \subset \mathfrak{m}_{P}(V)\). In conclusion, we have shown that the induced map \(\tilde{\varphi}\) can be uniquely extended to a ring homomorphism from \(\mathscr{O}_{Q}(W)\) to \(\mathscr{O}_{P}(V)\), and that \(\tilde{\varphi}(\mathrm{m}_{Q}(W)) \subset \mathfrak{m}_{P}(V)\).

Key concepts

Affine Varieties

In algebraic geometry, affine varieties are fundamental objects of study. They are essentially the solutions to systems of algebraic equations. To better understand this, consider that an affine variety is a subset of affine space. Affine space can be imagined as the space of all possible solutions that sets of polynomials might produce.

When we talk about affine varieties, we are often interested in maps between them. These maps are polynomial in nature, meaning they are defined by polynomials much like the ones defining the variety itself. For instance, the map \(\varphi: V \rightarrow W\) is a polynomial map indicating that each point in \(V\) is associated with a point in \(W\).

This fundamental concept serves as the basis for examining more complex structures in algebraic geometry, including the behavior of rings associated with these varieties.

Ring Homomorphism

A ring homomorphism is a function that respects the operations of addition and multiplication in rings, structures that encompass elements and two operations commonly known as addition and multiplication.

In the context of algebraic geometry, a ring homomorphism connects coordinate rings associated with affine varieties. When we consider the induced map \(\tilde{\varphi}: \Gamma(W) \rightarrow \Gamma(V)\), we observe that this is a kind of ring homomorphism. It reflects the transformations between the coordinate rings of the varieties \(W\) and \(V\).

• Preservation is key: It retains the inherent structure of rings during transformation.
• Addition: \(\tilde{\varphi}(a + b) = \tilde{\varphi}(a) + \tilde{\varphi}(b)\)
• Multiplication: \(\tilde{\varphi}(a \cdot b) = \tilde{\varphi}(a) \cdot \tilde{\varphi}(b)\).

This property ensures that the algebraic structure remains consistent across transformations, allowing us to extend mathematical behavior between different geometric objects.

Coordinate Rings

Coordinate rings provide a bridge between geometry and algebra by associating algebraic objects with the geometric structures of affine varieties. For any affine variety \(V\), the coordinate ring, denoted \(\Gamma(V)\), is formed by polynomials that vanish on \(V\).

These rings encapsulate vital properties of the variety. They are often examined through ring homomorphisms which map these properties from one variety to another. In our example, the induced map \(\tilde{\varphi}: \Gamma(W) \rightarrow \Gamma(V)\) transports information encoded in \(W\)'s coordinate ring to \(V\)'s.

Understanding coordinate rings is crucial for working with affine varieties as they harbor the algebraic backbone of these geometric entities. They reflect the solutions of polynomial equations defining the varieties.

Maximal Ideals

Maximal ideals play an important role in ring theory, providing insights into the structure and properties of rings by identifying elements that do not become larger except by including the whole ring.

Here's how they factor into our context:

• For any point \(P\) in an affine variety \(V\), \(\mathfrak{m}_P(V)\) signifies the maximal ideal within its local ring \(\mathscr{O}_P(V)\).
• Maximal ideals help by isolating the structure at smaller, more focused scales, like specific points on a variety.
• In the example from the exercise, proving that the induced map \(\tilde{\varphi}(\mathrm{m}_Q(W)) \subset \mathfrak{m}_P(V)\) shows how these ideals relate when mapped across varieties.

These ideals help isolate and analyze points' algebraic properties within the whole variety's scope, making them pivotal for deeper exploration of algebraic geometry.