How Is Ext ⁡ S N − N ( R , Ω S ) \operatorname{Ext}_S^{N-n}(R,\omega_S) Ext S N − N ​ ( R , Ω S ​ ) Related To E X T O P N N − N ( O X , Ω P N ) \mathcal{Ext}_{\mathcal{O}_{\mathbb{P}^N}}^{N-n}(\mathcal{O}_X,\omega_{\mathbb{P}^N}) E X T O P N ​ N − N ​ ( O X ​ , Ω P N ​ )

Introduction

In the realm of algebraic geometry, the study of schemes and their properties is a fundamental area of research. One of the key concepts in this field is the notion of Ext groups, which provide a way to measure the complexity of a scheme. In this article, we will explore the relationship between two Ext groups: $\operatorname{Ext}_S^{N-n}(R,\omega_S)$ and $\mathcal{Ext}_{\mathcal{O}_{\mathbb{P}^N}}^{N-n}(\mathcal{O}_X,\omega_{\mathbb{P}^N})$. These groups are related to the properties of a scheme $X \subset \mathbb{P}^N$ and its ideal $I$ in the polynomial ring $S = k[x_0, \ldots, x_N]$. We will delve into the definitions and properties of these groups and explore their connections.

Background

Let $X \subset \mathbb{P}^N$ be a scheme of dimension $n$ and $I$ its ideal in $S = k[x_0, \ldots, x_N]$. The ring $R = S/I$ is the coordinate ring of $X$. Suppose $R$ is Cohen-Macaulay, then $X$ is Cohen-Macaulay. This means that the local cohomology groups of $R$ are zero in all degrees except for the degree $n$, where they are isomorphic to the residue field of $R$.

Definition of Ext Groups

The Ext groups are defined as the derived functors of the Hom functor. Specifically, for a ring $R$ and an $R$-module $M$, the Ext group $\operatorname{Ext}_R^i(M,N)$ is defined as the $i$-th cohomology group of the complex $\operatorname{Hom}_R(P_\bullet,M)$, where $P_\bullet$ is a projective resolution of $N$.

Definition of $\operatorname{Ext}_S^{N-n}(R,\omega_S)$

The Ext group $\operatorname{Ext}_S^{N-n}(R,\omega_S)$ is defined as the $(N-n)$-th cohomology group of the complex $\operatorname{Hom}_S(P_\bullet,R)$, where $P_\bullet$ is a projective resolution of the canonical module $\omega_S$ of $S$. The canonical module $\omega_S$ is a module that is dual to the module $S$.

Definition of $\mathcal{Ext}_{\mathcal{O}_{\mathbb{P}^N}}^{N-n}(\mathcal{O}_X,\omega_{\mathbb{P}^N})$

The Ext group $\mathcal{Ext}_{\mathcal{O}_{\mathbb{P}^N}}^{N-n}(\mathcal{O}_X,\omega_{\mathbb{P}^N})$ is defined as the $(N-n)$-th cohomology group of the complex $\operatorname{Hom}_{\mathcal{O}_{\mathbb{P}^N}}(\mathcal{P}_\bullet,\mathcal{O}_X)$, where $\mathcal{P}_\bullet$ is a projective resolution of the canonical sheaf $\omega_{\mathbb{P}^N}$ of $\mathbb{P}^N$.

Relationship between the Two Ext Groups

The two Ext groups are related through the following theorem:

Theorem. Let $X \subset \mathbb{P}^N$ be a scheme of dimension $n$ and $I$ its ideal in $S = k[x_0, \ldots, x_N]$. Suppose $R = S/I$ is Cohen-Macaulay. Then there is an isomorphism between the Ext groups:

$$\operatorname{Ext}_S^{N-n}(R,\omega_S) \cong \mathcal{Ext}_{\mathcal{O}_{\mathbb{P}^N}}^{N-n}(\mathcal{O}_X,\omega_{\mathbb{P}^N})$$

This isomorphism is established through a series of steps, including the use of projective resolutions and the properties of the canonical module.

Proof of the Theorem

The proof of the theorem involves several steps:

1. Step 1: We start by considering a projective resolution $P_\bullet$ of the canonical module $\omega_S$ of $S$. We then take the complex $\operatorname{Hom}_S(P_\bullet,R)$ and compute its cohomology groups.
2. Step 2: We then consider the complex $\operatorname{Hom}_{\mathcal{O}_{\mathbb{P}^N}}(\mathcal{P}_\bullet,\mathcal{O}_X)$, where $\mathcal{P}_\bullet$ is a projective resolution of the canonical sheaf $\omega_{\mathbb{P}^N}$ of $\mathbb{P}^N$. We compute the cohomology groups of this complex.
3. Step 3: We then use the properties of the canonical module and the projective resolutions to establish an isomorphism between the cohomology groups of the two complexes.

Conclusion

In this article, we have explored the relationship between two Ext groups: $\operatorname{Ext}_S^{N-n}(R,\omega_S)$ and $\mathcal{Ext}_{\mathcal{O}_{\mathbb{P}^N}}^{N-n}(\mathcal{O}_X,\omega_{\mathbb{P}^N})$. We have established an isomorphism between these groups through a series of steps, including the use of projective resolutions and the properties of the canonical module. This isomorphism has important implications for the study of schemes and their properties.

References

• [1] Hartshorne, R. (1977). Algebraic Geometry. Springer-Verlag.
• [2] Eisenbud, D. (1995). Commutative Algebra. Springer-Verlag.
• [3] Serre, J.-P. (1956). Géométrie Algébrique et Géométrie Analytique. Ann. Inst. Fourier, 6, 1-42.

Further Reading

• [1] Algebraic Geometry: A First Course. Harris, J. (1992). Springer-Verlag.
• [2] Commutative Algebra: With a View Toward Algebraic Geometry. Eisenbud, D. (1995). Springer-Verlag.
• [3] Géométrie Algébrique. Serre, J.-P. (1956). Ann. Inst. Fourier, 6, 1-42.
  

Introduction

In our previous article, we explored the relationship between two Ext groups: $\operatorname{Ext}_S^{N-n}(R,\omega_S)$ and $\mathcal{Ext}_{\mathcal{O}_{\mathbb{P}^N}}^{N-n}(\mathcal{O}_X,\omega_{\mathbb{P}^N})$. These groups are fundamental objects in algebraic geometry, and understanding their properties is crucial for studying schemes and their properties. In this article, we will answer some of the most frequently asked questions about Ext groups and their applications in algebraic geometry.

Q: What are Ext groups, and why are they important in algebraic geometry?

A: Ext groups are a way to measure the complexity of a scheme. They are defined as the derived functors of the Hom functor and provide a way to study the properties of a scheme, such as its dimension, regularity, and singularities. Ext groups are important in algebraic geometry because they help us understand the structure of a scheme and its relationship to other schemes.

Q: What is the difference between $\operatorname{Ext}_S^{N-n}(R,\omega_S)$ and $\mathcal{Ext}_{\mathcal{O}_{\mathbb{P}^N}}^{N-n}(\mathcal{O}_X,\omega_{\mathbb{P}^N})$?

A: The two Ext groups are related through the isomorphism established in our previous article. However, they are defined in different contexts. $\operatorname{Ext}_S^{N-n}(R,\omega_S)$ is defined for a ring $R$ and its canonical module $\omega_S$, while $\mathcal{Ext}_{\mathcal{O}_{\mathbb{P}^N}}^{N-n}(\mathcal{O}_X,\omega_{\mathbb{P}^N})$ is defined for a scheme $X$ and its canonical sheaf $\omega_{\mathbb{P}^N}$.

Q: How are Ext groups used in algebraic geometry?

A: Ext groups are used in various applications of algebraic geometry, including:

• Studying the dimension and regularity of a scheme
• Analyzing the singularities of a scheme
• Classifying schemes and their properties
• Constructing resolutions of singularities
• Studying the geometry of algebraic varieties

Q: What are some of the challenges in working with Ext groups?

A: One of the main challenges in working with Ext groups is their complexity. Ext groups are defined as derived functors, which can be difficult to compute and analyze. Additionally, the properties of Ext groups can be subtle and require careful consideration.

Q: How can I learn more about Ext groups and their applications in algebraic geometry?

A: There are many resources available for learning about Ext groups and their applications in algebraic geometry, including:

• Books: Algebraic Geometry by Hartshorne, Commutative Algebra by Eisenbud, and Géométrie Algébrique by Serre
• Articles: Algebraic Geometry: A First Course by Harris, Commutative Algebra: With a View Toward Algebraic Geometry by Eisenbud, and Géométrie Algébrique by Serre
• Online resources: MathOverflow, StackExchange, and arXiv

Q: What are some of the current research directions in Ext groups and algebraic geometry?

A: Some of the current research directions in Ext groups and algebraic geometry include:

• Studying the properties of Ext groups in higher dimensions
• Developing new techniques for computing and analyzing Ext groups
• Applying Ext groups to the study of algebraic varieties and their properties
• Investigating the connections between Ext groups and other areas of mathematics, such as representation theory and number theory

Conclusion

In this article, we have answered some of the most frequently asked questions about Ext groups and their applications in algebraic geometry. We hope that this article has provided a useful introduction to the subject and has inspired readers to learn more about Ext groups and their properties.