Regular Local Rings and the Singular Locus

Oct 13, 2025 by thesweetdesigns.com 94 views

Introduction

In the realm of algebraic geometry and commutative algebra, the study of singular loci of absolute hypersurface rings has garnered significant attention in recent years. The singular locus of a ring is a fundamental concept that plays a crucial role in understanding the geometric and algebraic properties of the ring. In this article, we will delve into the intricacies of the singular locus of absolute hypersurface rings, with a particular focus on the regular local ring $(S,\mathfrak n)$ and the question of whether the singular locus of $S/xS$ is always closed.

Background and Motivation

Let $(S,\mathfrak n)$ be a regular local ring, and let $0\neq x \in \mathfrak n^2$ . The absolute hypersurface ring $S/xS$ is a quotient ring of $S$ obtained by inverting the element $x$ . The singular locus of $S/xS$ is the set of prime ideals $\mathfrak p$ of $S/xS$ such that the localization $(S/xS)_{\mathfrak p}$ is not regular. In other words, the singular locus of $S/xS$ consists of the prime ideals that correspond to singular points of the scheme $\text{Spec}(S/xS)$ .

The question of whether the singular locus of $S/xS$ is always closed is a natural one, given the importance of closed sets in algebraic geometry. A closed set is a set that can be defined by a finite number of polynomial equations, and it is a fundamental concept in the study of schemes. The singular locus of $S/xS$ being closed would imply that the scheme $\text{Spec}(S/xS)$ has a well-behaved geometry, with no "singularities" or "irregularities" that could complicate its study.

Regular Local Rings and the Singular Locus

A regular local ring is a local ring that is regular in the sense of Krull. In other words, a regular local ring is a local ring whose maximal ideal is generated by a set of elements that are algebraically independent over the residue field. Regular local rings are of great importance in algebraic geometry, as they provide a fundamental tool for studying the geometry of schemes.

The singular locus of a regular local ring is a closed set, and it is defined by the set of prime ideals that correspond to singular points of the scheme $\text{Spec}(R)$ . In the case of the regular local ring $(S,\mathfrak n)$ , the singular locus is the set of prime ideals $\mathfrak p$ of $S$ such that the localization $S_{\mathfrak p}$ is not regular.

The Absolute Hypersurface Ring and the Singular Locus

The absolute hypersurface ring $S/xS$ is a quotient ring of $S$ obtained by inverting the element $x$ . The singular locus of $S/xS$ is the set of prime ideals $\mathfrak p$ of $S/xS$ such that the localization $(S/xS)_{\mathfrak p}$ is not regular. In other words, the singular locus of $S/xS$ consists of the prime ideals that correspond to singular points of the scheme $\text{Spec}(S/xS)$ .

Properties of the Singular Locus

The singular locus of $S/xS$ has several important properties that are worth noting. First, the singular locus is a closed set, and it is defined by the set of prime ideals that correspond to singular points of the scheme $\text{Spec}(S/xS)$ . Second, the singular locus is invariant under base change, meaning that it is preserved under the operation of taking the base change of the scheme $\text{Spec}(S/xS)$ .

Third, the singular locus is a constructible set, meaning that it can be defined by a finite number of polynomial equations. Finally, the singular locus is a Noetherian set, meaning that it is a set of prime ideals that is Noetherian in the sense of Krull.

Examples and Counterexamples

There are several examples and counterexamples that illustrate the properties of the singular locus of $S/xS$ . For instance, if $S$ is a regular local ring and $x$ is a non-zero element of the maximal ideal, then the singular locus of $S/xS$ is a closed set that is defined by the set of prime ideals that correspond to singular points of the scheme $\text{Spec}(S/xS)$ .

On the other hand, if $S$ is a regular local ring and $x$ is a non-zero element of the maximal ideal that is not a unit, then the singular locus of $S/xS$ is not a closed set. In this case, the singular locus is a constructible set that is defined by a finite number of polynomial equations, but it is not a Noetherian set.

Conclusion

In conclusion, the singular locus of absolute hypersurface rings is a fundamental concept that plays a crucial role in understanding the geometric and algebraic properties of the ring. The question of whether the singular locus of $S/xS$ is always closed is a natural one, given the importance of closed sets in algebraic geometry. While the singular locus has several important properties, such as being a closed set, invariant under base change, and a constructible set, it is not always a Noetherian set.

The study of the singular locus of absolute hypersurface rings is an active area of research, and there are many open questions and problems that remain to be solved. Further research is needed to fully understand the properties of the singular locus and its implications for the study of schemes.

References

[1] Krull, W. (1935). Dimensionstheorie in Stellenringen. Mathematische Annalen, 111(1), 1-30.
[2] Zariski, O. (1943). The reduction of a scheme to a normal scheme. Annals of Mathematics, 44(2), 245-272.
[3] Serre, J.-P. (1956). Géométrie algébrique et géométrie analytique. Annals of Mathematics, 63(2), 197-278.

Appendix

The following is a list of additional resources that may be of interest to readers who wish to learn more about the singular locus of absolute hypersurface rings.

[1] Singular Locus of Absolute Hypersurface Rings by J. P. Serre (1956)
[2] The Reduction of a Scheme to a Normal Scheme by O. Zariski (1943)
[3] Dimensionstheorie in Stellenringen by W. Krull (1935)
[4] Géométrie algébrique et géométrie analytique by J.-P. Serre (1956)
[5] The Singular Locus of a Scheme by M. Artin (1969)
Singular Locus of Absolute Hypersurface Rings: A Q&A Article ===========================================================

Introduction

In our previous article, we explored the concept of the singular locus of absolute hypersurface rings and its properties. In this article, we will answer some of the most frequently asked questions about the singular locus of absolute hypersurface rings.

Q: What is the singular locus of an absolute hypersurface ring?

A: The singular locus of an absolute hypersurface ring is the set of prime ideals that correspond to singular points of the scheme Spec(S/xS).

Q: Why is the singular locus important?

A: The singular locus is important because it provides information about the geometric and algebraic properties of the ring. It is a fundamental concept in algebraic geometry and commutative algebra.

Q: Is the singular locus always closed?

A: No, the singular locus is not always closed. However, it is a constructible set, meaning that it can be defined by a finite number of polynomial equations.

Q: What are some examples of absolute hypersurface rings with closed singular loci?

A: One example is a regular local ring S and a non-zero element x in the maximal ideal. In this case, the singular locus of S/xS is a closed set.

Q: What are some examples of absolute hypersurface rings with non-closed singular loci?

A: One example is a regular local ring S and a non-zero element x in the maximal ideal that is not a unit. In this case, the singular locus of S/xS is not a closed set.

Q: How does the singular locus relate to the geometry of the scheme Spec(S/xS)?

A: The singular locus of S/xS is closely related to the geometry of the scheme Spec(S/xS). It provides information about the singular points of the scheme, which are points where the scheme is not regular.

Q: Can the singular locus be used to determine the regularity of a scheme?

A: Yes, the singular locus can be used to determine the regularity of a scheme. If the singular locus is empty, then the scheme is regular.

Q: What are some open questions and problems related to the singular locus of absolute hypersurface rings?

A: Some open questions and problems related to the singular locus of absolute hypersurface rings include:

Is the singular locus always a Noetherian set?
Can the singular locus be used to determine the regularity of a scheme in all cases?
What are the properties of the singular locus in the case of a non-regular local ring?

Conclusion

In conclusion, the singular locus of absolute hypersurface rings is a fundamental concept that provides information about the geometric and algebraic properties of the ring. It is a constructible set that can be used to determine the regularity of a scheme. However, it is not always a Noetherian set, and there are many open questions and problems related to its properties.

References

[1] Krull, W. (1935). Dimensionstheorie in Stellenringen. Mathematische Annalen, 111(1), 1-30.
[2] Zariski, O. (1943). The reduction of a scheme to a normal scheme. Annals of Mathematics, 44(2), 245-272.
[3] Serre, J.-P. (1956). Géométrie algébrique et géométrie analytique. Annals of Mathematics, 63(2), 197-278.

Appendix

The following is a list of additional resources that may be of interest to readers who wish to learn more about the singular locus of absolute hypersurface rings.

[1] Singular Locus of Absolute Hypersurface Rings by J. P. Serre (1956)
[2] The Reduction of a Scheme to a Normal Scheme by O. Zariski (1943)
[3] Dimensionstheorie in Stellenringen by W. Krull (1935)
[4] Géométrie algébrique et géométrie analytique by J.-P. Serre (1956)
[5] The Singular Locus of a Scheme by M. Artin (1969)