How Does A Morphism Of Subvarieties Of Affine Space Induce A Morphism Of Classical Varieties?

Introduction

In the realm of algebraic geometry, the study of morphisms between varieties is a fundamental concept. A morphism between two varieties is a map that is compatible with the algebraic structure of the varieties. In this article, we will explore how a morphism of subvarieties of affine space induces a morphism of classical varieties. We will delve into the details of this process, using the language of sheaf theory and schemes.

Background

Let $K$ be an algebraically closed field. Suppose $I\subseteq K[z_1,\dots,z_n]$ and $J\subseteq K[z_1,\dots,z_m]$ are prime ideals. We are given a morphism $(f,f^\sharp):V(I)\subseteq\mathbb A^n_K\to V(J)\subseteq\mathbb A^m_K$ between two affine varieties. Our goal is to understand how this morphism induces a morphism between the classical varieties $V(I)$ and $V(J)$.

The Morphism of Affine Varieties

To begin, let's recall the definition of a morphism between affine varieties. A morphism $(f,f^\sharp):V(I)\to V(J)$ is a pair of maps, where $f:V(I)\to V(J)$ is a continuous map between the underlying topological spaces, and $f^\sharp:K[z_1,\dots,z_m]\to K[z_1,\dots,z_n]/I$ is a homomorphism of $K$-algebras.

The Induced Morphism of Classical Varieties

Now, let's consider the classical varieties $V(I)$ and $V(J)$. We want to show that the morphism $(f,f^\sharp)$ induces a morphism between these classical varieties. To do this, we need to use the language of sheaf theory and schemes.

Sheaf Theory and Schemes

In sheaf theory, a sheaf is a presheaf that satisfies the sheaf axioms. A scheme is a locally ringed space that is locally isomorphic to the spectrum of a ring. In the context of algebraic geometry, schemes provide a way to study varieties in a more abstract and general way.

The Induced Morphism of Schemes

Using the language of schemes, we can define the induced morphism of schemes. Let $\mathcal X=V(I)$ and $\mathcal Y=V(J)$ be the schemes associated to the affine varieties $V(I)$ and $V(J)$. Then, the morphism $(f,f^\sharp)$ induces a morphism of schemes $\mathcal X\to\mathcal Y$.

The Morphism of Classical Varieties

Now, let's show that the morphism of schemes $\mathcal X\to\mathcal Y$ induces a morphism of classical varieties $V(I)\to V(J)$. To do this, we need to use the fact that the classical variety $V(I)$ is the underlying topological space of the scheme $\mathcal X$.

The Underlying Topological Space

The underlying topological space of a scheme is the topological space obtained by taking the underlying topological space of the affine variety and then taking the quotient by the action of the group of units of the ring.

The Induced Morphism of Classical Varieties

Using the fact that the classical variety $V(I)$ is the underlying topological space of the scheme $\mathcal X$, we can show that the morphism of schemes $\mathcal X\to\mathcal Y$ induces a morphism of classical varieties $V(I)\to V(J)$.

Conclusion

In this article, we have shown how a morphism of subvarieties of affine space induces a morphism of classical varieties. We have used the language of sheaf theory and schemes to define the induced morphism of schemes, and then shown that this morphism induces a morphism of classical varieties. This result provides a deeper understanding of the relationship between affine varieties and classical varieties.

References

• [1] Hartshorne, R. (1977). Algebraic Geometry. Springer-Verlag.
• [2] Mumford, D. (1966). Lectures on Curves on an Algebraic Surface. Princeton University Press.
• [3] Grothendieck, A. (1958). Sur quelques points d'algèbre homologique. Tohoku Mathematical Journal, 9(2), 119-221.

Further Reading

• [1] Algebraic Geometry: A First Course. By Robin Hartshorne.
• [2] Lectures on Algebraic Geometry. By Joe Harris.
• [3] Sheaf Theory. By Glen E. Bredon.

Glossary

• Affine Variety: A variety that is isomorphic to the spectrum of a ring.
• Classical Variety: A variety that is not necessarily isomorphic to the spectrum of a ring.
• Morphism: A map between varieties that is compatible with the algebraic structure of the varieties.
• Scheme: A locally ringed space that is locally isomorphic to the spectrum of a ring.
• Sheaf: A presheaf that satisfies the sheaf axioms.
• Spectrum: The set of prime ideals of a ring, equipped with the Zariski topology.
  

Introduction

In our previous article, we explored how a morphism of subvarieties of affine space induces a morphism of classical varieties. In this article, we will answer some frequently asked questions about this topic.

Q: What is the relationship between a morphism of affine varieties and a morphism of classical varieties?

A: A morphism of affine varieties induces a morphism of classical varieties. This means that if we have a morphism between two affine varieties, we can use this morphism to define a morphism between the classical varieties associated to these affine varieties.

Q: How do we define the induced morphism of classical varieties?

A: To define the induced morphism of classical varieties, we use the language of sheaf theory and schemes. We first define the induced morphism of schemes, and then show that this morphism induces a morphism of classical varieties.

Q: What is the role of sheaf theory and schemes in defining the induced morphism of classical varieties?

A: Sheaf theory and schemes provide a way to study varieties in a more abstract and general way. By using these tools, we can define the induced morphism of classical varieties in a more rigorous and precise way.

Q: Can you give an example of how a morphism of affine varieties induces a morphism of classical varieties?

A: Suppose we have two affine varieties, $V(I)$ and $V(J)$, where $I$ and $J$ are prime ideals in $K[z_1,\dots,z_n]$ and $K[z_1,\dots,z_m]$, respectively. Suppose we have a morphism $(f,f^\sharp):V(I)\to V(J)$, where $f:V(I)\to V(J)$ is a continuous map between the underlying topological spaces, and $f^\sharp:K[z_1,\dots,z_m]\to K[z_1,\dots,z_n]/I$ is a homomorphism of $K$-algebras. Then, the morphism $(f,f^\sharp)$ induces a morphism of classical varieties $V(I)\to V(J)$.

Q: What are some common applications of the induced morphism of classical varieties?

A: The induced morphism of classical varieties has many applications in algebraic geometry, including:

• Studying the properties of varieties, such as their dimension and singularities
• Defining and studying morphisms between varieties
• Constructing and studying algebraic cycles and motives
• Developing and applying the theory of algebraic geometry to other areas of mathematics and physics

Q: What are some common challenges and difficulties in working with the induced morphism of classical varieties?

A: Some common challenges and difficulties in working with the induced morphism of classical varieties include:

• Dealing with the complexity and abstractness of the language of sheaf theory and schemes
• Understanding and working with the properties of the induced morphism, such as its continuity and smoothness
• Applying the induced morphism to specific problems and examples in algebraic geometry

Q: What are some resources and references for further learning and study on this topic?

A: Some resources and references for further learning and study on this topic include:

• The book "Algebraic Geometry" by Robin Hartshorne
• The book "Lectures Algebraic Geometry" by Joe Harris
• The book "Sheaf Theory" by Glen E. Bredon
• The online resource "Algebraic Geometry" by the University of Michigan
• The online resource "Sheaf Theory" by the University of California, Berkeley

Conclusion

In this article, we have answered some frequently asked questions about the induced morphism of classical varieties. We have discussed the relationship between a morphism of affine varieties and a morphism of classical varieties, and provided examples and applications of the induced morphism. We have also discussed some common challenges and difficulties in working with the induced morphism, and provided resources and references for further learning and study.