Is This Morphism O Y → F ∗ O X \mathcal O_Y\to F_*\mathcal O_X O Y ​ → F ∗ ​ O X ​ The Same As F ♯ F^\sharp F ♯ ?

Is this morphism $\mathcal O_Y\to f_*\mathcal O_X$ the same as $f^\sharp$?

In the realm of algebraic geometry, sheaf theory plays a crucial role in understanding the properties of schemes and their morphisms. A morphism of schemes is a fundamental concept that enables us to study the relationships between different geometric objects. In this discussion, we will delve into the properties of a morphism of schemes and explore the relationship between the induced morphism $\mathcal O_Y\to f_*\mathcal O_X$ and the given morphism $f^\sharp: \mathcal O_Y\to f_*\mathcal O_X$.

A morphism of schemes is a pair $(f,f^\sharp): (X,\mathcal O_X)\to (Y,\mathcal O_Y)$, where $f: X\to Y$ is a continuous map between topological spaces, and $f^\sharp: \mathcal O_Y\to f_*\mathcal O_X$ is a ring homomorphism between the structure sheaves of $X$ and $Y$. The map $f$ is often referred to as the underlying map, while $f^\sharp$ is called the structure map.

Given a morphism of schemes $(f,f^\sharp): (X,\mathcal O_X)\to (Y,\mathcal O_Y)$, we can induce a morphism between the sheaves of rings $f_*\mathcal O_X$ and $\mathcal O_Y$. This induced morphism is given by the composition $f_* f^{-1} \mathcal O_Y\to f_*f^{-1}f_*\mathcal O_X\to f_*\mathcal O_X$. We can also consider the morphism $\mathcal O_Y\to f_*\mathcal O_X$ induced by the structure map $f^\sharp$.

The question we want to address is whether the morphism $\mathcal O_Y\to f_*\mathcal O_X$ induced by the structure map $f^\sharp$ is the same as the morphism $f_* f^{-1} \mathcal O_Y\to f_*f^{-1}f_*\mathcal O_X\to f_*\mathcal O_X$ induced by the composition of maps.

To answer this question, we need to examine the properties of the induced morphism and the structure map. We can start by considering the following commutative diagram:

$$\begin{CD} \mathcal O_Y @>f^\sharp>> f_*\mathcal O_X\\ @VVV @VVV\\ f_* f^{-1} \mathcal O_Y @>f_* f^{-1} f^\sharp>> f_*f^{-1}f_*\mathcal O_X\\ @VVV @VVV\\ f_*\mathcal O_X @= f_*\mathcal O_X \end{CD}$$

In this diagram, the vertical maps are induced by the composition of maps, and the horizontal maps are given by the structure map $f^\sharp$. We can see that the diagram commutes, which means that the morphism $\mathcal O_Y\to f_*\mathcal O_X$ induced by the structure map $f^\sharp$ is the same as the morphism $f_* f^{-1} \mathcal O_Y\to f_*f^{-1}f_*\mathcal O_X\to f_*\mathcal O_X$ induced by the composition of maps.

In conclusion, the morphism $\mathcal O_Y\to f_*\mathcal O_X$ induced by the structure map $f^\sharp$ is the same as the morphism $f_* f^{-1} \mathcal O_Y\to f_*f^{-1}f_*\mathcal O_X\to f_*\mathcal O_X$ induced by the composition of maps. This result highlights the importance of understanding the properties of morphisms of schemes and their induced morphisms.

The discussion of morphisms of schemes and their induced morphisms is a fundamental aspect of algebraic geometry. The properties of these morphisms have far-reaching implications for the study of schemes and their geometric objects. In particular, the study of morphisms of schemes has led to the development of important concepts such as the Picard group and the Brauer group.

• [1] Hartshorne, R. (1977). Algebraic Geometry. Springer-Verlag.
• [2] Grothendieck, A. (1963). Éléments de Géométrie Algébrique. Springer-Verlag.
• [3] Mumford, D. (1976). Algebraic Geometry I: Complex Projective Varieties. Springer-Verlag.

• Morphism of schemes: A pair $(f,f^\sharp): (X,\mathcal O_X)\to (Y,\mathcal O_Y)$, where $f: X\to Y$ is a continuous map between topological spaces, and $f^\sharp: \mathcal O_Y\to f_*\mathcal O_X$ is a ring homomorphism between the structure sheaves of $X$ and $Y$.
• Structure map: The ring homomorphism $f^\sharp: \mathcal O_Y\to f_*\mathcal O_X$ associated with a morphism of schemes.
• Induced morphism: The morphism between the sheaves of rings $f_*\mathcal O_X$ and $\mathcal O_Y$ induced by a morphism of schemes.
• Picard group: The group of isomorphism classes of line bundles on a scheme.
• Brauer group: The group of isomorphism classes of Azumaya algebras on a scheme.
  Q&A: Morphisms of Schemes and Sheaf Theory

In our previous discussion, we explored the properties of morphisms of schemes and their induced morphisms. We also examined the relationship between the structure map and the induced morphism. In this article, we will continue to delve into the world of algebraic geometry and sheaf theory, answering some of the most frequently asked questions about morphisms of schemes and their induced morphisms.

A morphism of schemes is a pair $(f,f^\sharp): (X,\mathcal O_X)\to (Y,\mathcal O_Y)$, where $f: X\to Y$ is a continuous map between topological spaces, and $f^\sharp: \mathcal O_Y\to f_*\mathcal O_X$ is a ring homomorphism between the structure sheaves of $X$ and $Y$.

The structure map is the ring homomorphism $f^\sharp: \mathcal O_Y\to f_*\mathcal O_X$ associated with a morphism of schemes. It is a crucial component of the morphism of schemes, as it enables us to study the properties of the scheme $X$ in terms of the scheme $Y$.

The induced morphism is the morphism between the sheaves of rings $f_*\mathcal O_X$ and $\mathcal O_Y$ induced by a morphism of schemes. It is a way of studying the properties of the scheme $X$ in terms of the scheme $Y$.

To determine if two morphisms of schemes are isomorphic, you need to check if the underlying maps are isomorphic and if the structure maps are isomorphic. This involves checking if the two morphisms have the same properties, such as being surjective or injective.

The structure map and the induced morphism are closely related. In fact, the induced morphism is the composition of the structure map and the inverse of the underlying map. This means that the induced morphism is a way of studying the properties of the scheme $X$ in terms of the scheme $Y$.

To compute the induced morphism, you need to use the following formula:

$$f_* f^{-1} \mathcal O_Y\to f_*f^{-1}f_*\mathcal O_X\to f_*\mathcal O_X$$

This formula involves the composition of the inverse of the underlying map, the structure map, and the inverse of the underlying map again.

Some common mistakes to avoid when working with morphisms of schemes include:

• Not checking if the underlying maps are isomorphic
• Not checking if the structure maps are isomorphic
• Not using the correct formula to compute the induced morphism

Some advanced topics in algebraic geometry and sheaf theory include:

• The Picard group and the Brauer group
• The study of line bundles and vector bundles
• The study of algebraic cycles and motives

In conclusion, morphisms of schemes and their induced morphisms are fundamental concepts in algebraic geometry and sheaf theory. By understanding these concepts, you can gain a deeper insight into the properties of schemes and their geometric objects. We hope that this Q&A article has been helpful in answering some of the most frequently asked questions about morphisms of schemes and their induced morphisms.

• Morphism of schemes: A pair $(f,f^\sharp): (X,\mathcal O_X)\to (Y,\mathcal O_Y)$, where $f: X\to Y$ is a continuous map between topological spaces, and $f^\sharp: \mathcal O_Y\to f_*\mathcal O_X$ is a ring homomorphism between the structure sheaves of $X$ and $Y$.
• Structure map: The ring homomorphism $f^\sharp: \mathcal O_Y\to f_*\mathcal O_X$ associated with a morphism of schemes.
• Induced morphism: The morphism between the sheaves of rings $f_*\mathcal O_X$ and $\mathcal O_Y$ induced by a morphism of schemes.
• Picard group: The group of isomorphism classes of line bundles on a scheme.
• Brauer group: The group of isomorphism classes of Azumaya algebras on a scheme.