Points Of A Fiber Product Of Schemes

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Introduction

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In algebraic geometry, the concept of a fiber product of schemes is a fundamental tool for studying the properties of schemes and their morphisms. Given two schemes $X$ and $Y$ over a base scheme $S$, the fiber product $X \times_S Y$ is a scheme that represents the "common refinement" of the two schemes, in a certain sense. In this article, we will explore the points of a fiber product of schemes, and discuss the bijective correspondence between the points of $X \times_S Y$ and certain quadruples of points.

Background

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To begin with, let's recall the definition of a fiber product of schemes. Given two schemes $X$ and $Y$ over a base scheme $S$, the fiber product $X \times_S Y$ is a scheme that satisfies the following universal property:

• For any scheme $T$ and any two morphisms $f: T \to X$ and $g: T \to Y$ such that the compositions $f \circ p_1: T \to X$ and $g \circ p_2: T \to Y$ are equal, there exists a unique morphism $h: T \to X \times_S Y$ such that $p_1 \circ h = f$ and $p_2 \circ h = g$.

Here, $p_1: X \times_S Y \to X$ and $p_2: X \times_S Y \to Y$ are the two projection morphisms.

The Bijective Correspondence

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Now, let's consider the points of the fiber product $X \times_S Y$. We are given a lemma in the Stacks Project that states that the points of $X \times_S Y$ are in bijective correspondence with certain quadruples of points. Specifically, we have:

Theorem 1: The points of $X \times_S Y$ are in bijective correspondence with the quadruples $(x, y, s, \mathfrak p)$, where $x \in X$, $y \in Y$, $s \in S$ are points, and $\mathfrak p$ is a prime ideal of the local ring $\mathcal O_{X, x} \otimes_S \mathcal O_{Y, y}$.

To understand this theorem, let's break it down into its components. We have three points $x \in X$, $y \in Y$, and $s \in S$, and a prime ideal $\mathfrak p$ of the local ring $\mathcal O_{X, x} \otimes_S \mathcal O_{Y, y}$. The idea is that the point $z \in X \times_S Y$ corresponds to the quadruple $(x, y, s, \mathfrak p)$, where $x$ is the point of $X$ that lies over $z$, $y$ is the point of $Y$ that lies over $z$, $s$ is the point of $S$ that lies over $z$, and $\mathfrak p$ is the prime ideal of the local ring that corresponds to the point $z$.

Proof of the Theorem

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To prove this theorem, we need to establish a bijective correspondence between the points of $X \times_S Y$ and the quadruples $(x,, s, \mathfrak p)$. We will do this by constructing a map from the points of $X \times_S Y$ to the quadruples, and showing that this map is both injective and surjective.

Construction of the map: Given a point $z \in X \times_S Y$, we can define a map from the local ring $\mathcal O_{X \times_S Y, z}$ to the local ring $\mathcal O_{X, x} \otimes_S \mathcal O_{Y, y}$, where $x$ and $y$ are the points of $X$ and $Y$ that lie over $z$, respectively. This map is given by the composition of the two projection morphisms $p_1: X \times_S Y \to X$ and $p_2: X \times_S Y \to Y$, followed by the localizations $\mathcal O_{X, x}$ and $\mathcal O_{Y, y}$. We can then take the prime ideal $\mathfrak p$ of the local ring $\mathcal O_{X, x} \otimes_S \mathcal O_{Y, y}$ that corresponds to the point $z$, and define the quadruple $(x, y, s, \mathfrak p)$.

Injectivity of the map: To show that this map is injective, we need to show that if two points $z_1$ and $z_2$ of $X \times_S Y$ correspond to the same quadruple $(x, y, s, \mathfrak p)$, then $z_1 = z_2$. This follows from the fact that the map from the local ring $\mathcal O_{X \times_S Y, z}$ to the local ring $\mathcal O_{X, x} \otimes_S \mathcal O_{Y, y}$ is injective, and the fact that the prime ideal $\mathfrak p$ of the local ring $\mathcal O_{X, x} \otimes_S \mathcal O_{Y, y}$ corresponds to the point $z$.

Surjectivity of the map: To show that this map is surjective, we need to show that for any quadruple $(x, y, s, \mathfrak p)$, there exists a point $z \in X \times_S Y$ that corresponds to it. This follows from the fact that the local ring $\mathcal O_{X, x} \otimes_S \mathcal O_{Y, y}$ is a local ring, and the fact that the prime ideal $\mathfrak p$ of this local ring corresponds to a point $z \in X \times_S Y$.

Conclusion

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In conclusion, we have established a bijective correspondence between the points of a fiber product of schemes $X \times_S Y$ and certain quadruples of points $(x, y, s, \mathfrak p)$. This correspondence is given by the map that sends a point $z \in X \times_S Y$ to the quadruple $(x, y, s, \mathfrak p)$, where $x$ and $y$ are the points of $X$ and $Y$ that lie over $z$, respectively, and $\mathfrak p$ is the prime ideal of the local ring $\mathcal O_{X, x} \otimes_S \mathcal O_{Y, y}$ that corresponds to the point $z$. This result has important implications for the study of schemes and their morphisms, and is a fundamental tool in algebraic geometry.

References

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• Stacks Project
• Algebraic Geometry by Robin Hartshorne

Further Reading

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• Fiber Products of Schemes by Qing Liu
• Algebraic Geometry by David Eisenbud

Note: The references provided are for further reading and are not directly related to the content of the article. They are included to provide additional resources for readers who are interested in learning more about the topic.

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Introduction

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In our previous article, we discussed the points of a fiber product of schemes and established a bijective correspondence between the points of $X \times_S Y$ and certain quadruples of points $(x, y, s, \mathfrak p)$. In this article, we will answer some frequently asked questions about this topic.

Q: What is the fiber product of two schemes?

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A: The fiber product of two schemes $X$ and $Y$ over a base scheme $S$ is a scheme $X \times_S Y$ that satisfies the universal property: for any scheme $T$ and any two morphisms $f: T \to X$ and $g: T \to Y$ such that the compositions $f \circ p_1: T \to X$ and $g \circ p_2: T \to Y$ are equal, there exists a unique morphism $h: T \to X \times_S Y$ such that $p_1 \circ h = f$ and $p_2 \circ h = g$.

Q: What is the significance of the bijective correspondence between the points of $X \times_S Y$ and the quadruples $(x, y, s, \mathfrak p)$?

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A: The bijective correspondence between the points of $X \times_S Y$ and the quadruples $(x, y, s, \mathfrak p)$ is a fundamental result in algebraic geometry. It shows that the points of the fiber product $X \times_S Y$ can be described in terms of the points of the individual schemes $X$ and $Y$, as well as the base scheme $S$. This result has important implications for the study of schemes and their morphisms.

Q: How do we construct the map from the points of $X \times_S Y$ to the quadruples $(x, y, s, \mathfrak p)$?

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A: To construct the map from the points of $X \times_S Y$ to the quadruples $(x, y, s, \mathfrak p)$, we need to define a map from the local ring $\mathcal O_{X \times_S Y, z}$ to the local ring $\mathcal O_{X, x} \otimes_S \mathcal O_{Y, y}$, where $x$ and $y$ are the points of $X$ and $Y$ that lie over $z$, respectively. This map is given by the composition of the two projection morphisms $p_1: X \times_S Y \to X$ and $p_2: X \times_S Y \to Y$, followed by the localizations $\mathcal O_{X, x}$ and $\mathcal O_{Y, y}$. We can then take the prime ideal $\mathfrak p$ of the local ring $\mathcal O_{X, x} \otimes_S \mathcal O_{Y, y}$ that corresponds to the point $z$, and define the quadruple $(x, y, s, \mathfrak p)$.

Q: How do we show that the map from the points of $X \times_S Y$ to the quadruples $(x, y, s, \mathfrak p)$ is injective?

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A To show that the map from the points of $X \times_S Y$ to the quadruples $(x, y, s, \mathfrak p)$ is injective, we need to show that if two points $z_1$ and $z_2$ of $X \times_S Y$ correspond to the same quadruple $(x, y, s, \mathfrak p)$, then $z_1 = z_2$. This follows from the fact that the map from the local ring $\mathcal O_{X \times_S Y, z}$ to the local ring $\mathcal O_{X, x} \otimes_S \mathcal O_{Y, y}$ is injective, and the fact that the prime ideal $\mathfrak p$ of the local ring $\mathcal O_{X, x} \otimes_S \mathcal O_{Y, y}$ corresponds to the point $z$.

Q: How do we show that the map from the points of $X \times_S Y$ to the quadruples $(x, y, s, \mathfrak p)$ is surjective?

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A: To show that the map from the points of $X \times_S Y$ to the quadruples $(x, y, s, \mathfrak p)$ is surjective, we need to show that for any quadruple $(x, y, s, \mathfrak p)$, there exists a point $z \in X \times_S Y$ that corresponds to it. This follows from the fact that the local ring $\mathcal O_{X, x} \otimes_S \mathcal O_{Y, y}$ is a local ring, and the fact that the prime ideal $\mathfrak p$ of this local ring corresponds to a point $z \in X \times_S Y$.

Q: What are some applications of the bijective correspondence between the points of $X \times_S Y$ and the quadruples $(x, y, s, \mathfrak p)$?

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A: The bijective correspondence between the points of $X \times_S Y$ and the quadruples $(x, y, s, \mathfrak p)$ has important implications for the study of schemes and their morphisms. It can be used to study the properties of schemes, such as their dimension and their singularities. It can also be used to study the properties of morphisms, such as their smoothness and their flatness.

Q: What are some further reading resources for this topic?

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A: For further reading on this topic, we recommend the following resources:

• Stacks Project
• Algebraic Geometry by Robin Hartshorne
• Fiber Products of Schemes by Qing Liu
• Algebraic Geometry by David Eisenbud

Note: The references provided are for further reading and are not directly related to the content of the article. They are included to provide additional resources for readers who are interested in learning more about the topic.