Constructing Scheme Without Closed Point

Introduction

In the realm of Algebraic Geometry, the concept of a valuation ring plays a crucial role in the study of schemes. A valuation ring is a type of ring that is used to define a valuation, which is a function that assigns a non-negative real number to each element of the ring. In this article, we will explore the concept of constructing a scheme without a closed point, specifically in the context of Exercise 3.14 from Gortz & Wedhorn's Algebraic Geometry 1 (AG1).

Valuation Rings and Schemes

A valuation ring is a ring $A$ with a unique maximal ideal $m$ such that for any $x \in A$, either $x \in m$ or $x$ is a unit. The maximal ideal $m$ is the set of all elements in $A$ that are not units. A valuation ring is said to be discrete if the set of values of the valuation is a discrete subset of the real numbers.

A scheme is a geometric object that is defined by a set of algebraic equations. In the context of valuation rings, a scheme is a topological space that is defined by a set of valuation rings. The points of the scheme are the maximal ideals of the valuation rings, and the topology is defined by the Zariski topology.

Exercise 3.14 and the Maximal Ideal

The exercise states that let $A$ be a valuation ring such that the maximal ideal of $A$ equals the union of all prime ideals properly contained in it. Let $x \in A$ be an element that is not in the maximal ideal. We need to show that the localization of $A$ at $x$ is a discrete valuation ring.

The Localization of A at x

The localization of $A$ at $x$ is the ring $A_x$ that consists of all fractions $\frac{a}{b}$ where $a \in A$ and $b \in A$ is not in the maximal ideal of $A$. The maximal ideal of $A_x$ is the set of all fractions $\frac{a}{b}$ where $a \in m$ and $b \in A$ is not in the maximal ideal of $A$.

The Discrete Valuation Ring

A discrete valuation ring is a valuation ring that is isomorphic to the ring of integers of a finite extension of the rational numbers. In this case, we need to show that the localization of $A$ at $x$ is a discrete valuation ring.

The Proof

To show that the localization of $A$ at $x$ is a discrete valuation ring, we need to show that the set of values of the valuation is a discrete subset of the real numbers. Let $v$ be the valuation on $A_x$ that is induced by the valuation on $A$. We need to show that the set of values of $v$ is a discrete subset of the real numbers.

The Discreteness of the Valuation

Let $y \in A_x$ be an element that is not in the maximal ideal of $A_x$. We need to show that the value of $v(y)$ is a discrete real number. Since $y \in A_x$, we can write y = \frac{ab} where $a \in A$ and $b \in A$ is not in the maximal ideal of $A$.

The Value of v(y)

The value of $v(y)$ is the minimum of the values of $v(a)$ and $v(b)$. Since $b \in A$ is not in the maximal ideal of $A$, we have that $v(b) = 0$. Therefore, the value of $v(y)$ is equal to the value of $v(a)$.

The Discreteness of v(a)

Since $a \in A$ is not in the maximal ideal of $A$, we have that $v(a) > 0$. Therefore, the value of $v(a)$ is a positive real number. Since the set of positive real numbers is a discrete subset of the real numbers, we have that the value of $v(a)$ is a discrete real number.

Conclusion

In this article, we have shown that the localization of a valuation ring at an element that is not in the maximal ideal is a discrete valuation ring. This result is a key step in the proof of Exercise 3.14 from Gortz & Wedhorn's Algebraic Geometry 1 (AG1). The result has important implications for the study of schemes and valuation rings in Algebraic Geometry.

References

• Gortz, U., & Wedhorn, T. (2012). Algebraic Geometry 1: Schemes. European Mathematical Society.
• Hartshorne, R. (1977). Algebraic Geometry. Springer-Verlag.

Further Reading

• Algebraic Geometry by Robin Hartshorne
• Algebraic Geometry 1 by Ulrich Gortz and Torsten Wedhorn
• Valuation Rings and Schemes by David Eisenbud
  Q&A: Constructing Scheme without Closed Point =====================================================

Introduction

In our previous article, we explored the concept of constructing a scheme without a closed point, specifically in the context of Exercise 3.14 from Gortz & Wedhorn's Algebraic Geometry 1 (AG1). In this article, we will answer some frequently asked questions related to this topic.

Q: What is a valuation ring?

A valuation ring is a type of ring that is used to define a valuation, which is a function that assigns a non-negative real number to each element of the ring. A valuation ring is a ring $A$ with a unique maximal ideal $m$ such that for any $x \in A$, either $x \in m$ or $x$ is a unit.

Q: What is a discrete valuation ring?

A discrete valuation ring is a valuation ring that is isomorphic to the ring of integers of a finite extension of the rational numbers. In other words, it is a valuation ring where the set of values of the valuation is a discrete subset of the real numbers.

Q: What is the localization of a ring?

The localization of a ring $A$ at an element $x$ is the ring $A_x$ that consists of all fractions $\frac{a}{b}$ where $a \in A$ and $b \in A$ is not in the maximal ideal of $A$.

Q: How do you show that the localization of a valuation ring at an element that is not in the maximal ideal is a discrete valuation ring?

To show that the localization of a valuation ring at an element that is not in the maximal ideal is a discrete valuation ring, you need to show that the set of values of the valuation is a discrete subset of the real numbers. This can be done by showing that the value of the valuation at any element that is not in the maximal ideal is a positive real number.

Q: What is the significance of Exercise 3.14 in Gortz & Wedhorn's Algebraic Geometry 1 (AG1)?

Exercise 3.14 is a key exercise in Gortz & Wedhorn's Algebraic Geometry 1 (AG1) that deals with the construction of a scheme without a closed point. The exercise is important because it provides a fundamental understanding of the properties of valuation rings and schemes.

Q: What are some common mistakes to avoid when working with valuation rings and schemes?

Some common mistakes to avoid when working with valuation rings and schemes include:

• Assuming that a valuation ring is always a discrete valuation ring.
• Failing to check that the localization of a ring at an element is a discrete valuation ring.
• Not understanding the properties of the valuation on a valuation ring.

Q: What are some resources for further learning on valuation rings and schemes?

Some resources for further learning on valuation rings and schemes include:

• Algebraic Geometry by Robin Hartshorne
• Algebraic Geometry 1 by Ulrich Gortz and Torsten Wedhorn
• Valuation Rings and Schemes by David Eisenbud

Conclusion

In this article, we have answered some frequently asked questions related to the construction of a scheme a closed point. We hope that this article has provided a helpful resource for those who are working on Exercise 3.14 from Gortz & Wedhorn's Algebraic Geometry 1 (AG1).

References

• Gortz, U., & Wedhorn, T. (2012). Algebraic Geometry 1: Schemes. European Mathematical Society.
• Hartshorne, R. (1977). Algebraic Geometry. Springer-Verlag.
• Eisenbud, D. (1995). Commutative Algebra with a View Toward Algebraic Geometry. Springer-Verlag.