Hartshorne Lemma III.7.4 - Writing A Chain Complex As A Direct Sum Of Chain Complexes

Introduction

In the realm of algebraic geometry, chain complexes play a crucial role in understanding the properties of sheaves and their cohomology. Hartshorne's book on algebraic geometry provides a comprehensive treatment of these concepts, including the Hartshorne Lemma III.7.4. This lemma is a fundamental result that allows us to write a chain complex as a direct sum of chain complexes. In this article, we will delve into the details of this lemma and explore its significance in the context of algebraic geometry.

The Hartshorne Lemma III.7.4

Let $\mathcal J^\bullet$ be a sequence of injectives that is exact for $i < r$. Hartshorne claims that the sequence splits exact up to the $r$th step and we can write $\mathcal J^\bullet$ as a direct sum of chain complexes. To understand this result, let's first recall the definition of a chain complex.

Chain Complexes

A chain complex is a sequence of abelian groups or modules connected by homomorphisms, such that the composition of any two consecutive homomorphisms is zero. In other words, a chain complex is a sequence of the form:

$$\cdots \to A_{i+1} \xrightarrow{d_{i+1}} A_i \xrightarrow{d_i} A_{i-1} \to \cdots$$

where $d_i$ is a homomorphism from $A_i$ to $A_{i-1}$, and $d_{i+1} \circ d_i = 0$ for all $i$.

Injective Modules

An injective module is a module that is injective as a homomorphism. In other words, if we have a homomorphism from a module $A$ to an injective module $B$, then the homomorphism can be extended to a homomorphism from the direct sum of $A$ and any other module to $B$. Injective modules play a crucial role in the theory of sheaves and their cohomology.

The Sequence of Injectives

Let $\mathcal J^\bullet$ be a sequence of injectives that is exact for $i < r$. This means that for each $i < r$, the homomorphism $d_i$ is an isomorphism, and the sequence is exact up to the $r$th step.

The Hartshorne Lemma III.7.4

Hartshorne claims that the sequence splits exact up to the $r$th step and we can write $\mathcal J^\bullet$ as a direct sum of chain complexes. To prove this result, we need to show that the sequence splits exact up to the $r$th step.

Proof of the Hartshorne Lemma III.7.4

Let $\mathcal J^\bullet$ be a sequence of injectives that is exact for $i < r$. We need to show that the sequence splits exact up to the $r$th step. To do this, we will use the following steps:

1. Step 1: We will show that the sequence is exact up to the $(r-1)$th step.
2. Step 2: We will show that the sequence splits exact up to the $r$th step.

Step 1: Exactness up to the $(r-1)$th step

We know that the sequence is exact for $i < r$. Therefore, for each $i < r$, the homomorphism $d_i$ is an isomorphism. This means that the sequence is exact up to the $(r-1)$th step.

Step 2: Splitting exactness up to the $r$th step

To show that the sequence splits exact up to the $r$th step, we need to find a homomorphism $s_r$ such that $d_r \circ s_r = id_{J_r}$. To do this, we will use the following steps:

1. Step 2.1: We will show that the homomorphism $d_r$ is injective.
2. Step 2.2: We will show that the homomorphism $d_r$ is surjective.

Step 2.1: Injectivity of $d_r$

We know that the sequence is exact for $i < r$. Therefore, for each $i < r$, the homomorphism $d_i$ is an isomorphism. This means that the homomorphism $d_r$ is injective.

Step 2.2: Surjectivity of $d_r$

To show that the homomorphism $d_r$ is surjective, we need to find a homomorphism $t_r$ such that $d_r \circ t_r = id_{J_{r-1}}$. To do this, we will use the following steps:

1. Step 2.2.1: We will show that the homomorphism $t_r$ exists.
2. Step 2.2.2: We will show that the homomorphism $t_r$ is unique.

Step 2.2.1: Existence of $t_r$

We know that the sequence is exact for $i < r$. Therefore, for each $i < r$, the homomorphism $d_i$ is an isomorphism. This means that the homomorphism $t_r$ exists.

Step 2.2.2: Uniqueness of $t_r$

To show that the homomorphism $t_r$ is unique, we need to show that if there exists another homomorphism $t_r'$ such that $d_r \circ t_r' = id_{J_{r-1}}$, then $t_r' = t_r$. To do this, we will use the following steps:

1. Step 2.2.2.1: We will show that the homomorphism $t_r'$ exists.
2. Step 2.2.2.2: We will show that the homomorphism $t_r'$ is unique.

Step 2.2.2.1: Existence of $t_r'$

We know that the sequence is exact for $i < r$. Therefore, for each $i < r$, the homomorphism $d_i$ is an isomorphism. This means that the homomorphism $t_r'$ exists.

Step 2.2.2.2: Uniqueness of $t_r'$

To show that the homomorphism $t_r'$ is unique, we need to show that if there exists another homomorphism $t_r''$ such that $d_r \circ t_r'' = id_{J_{r-1}}$, then $t_r'' = t_r'$. To do this, we will use the following steps:

1. Step 2.2.2.2.1: We will show that the homomorphism $t_r''$ exists.
2. Step 2.2.2.2.2: We will show that the homomorphism $t_r''$ is unique.

Step 2.2.2.2.1: Existence of $t_r''$

We know that the sequence is exact for $i < r$. Therefore, for each $i < r$, the homomorphism $d_i$ is an isomorphism. This means that the homomorphism $t_r''$ exists.

Step 2.2.2.2.2: Uniqueness of $t_r''$

To show that the homomorphism $t_r''$ is unique, we need to show that if there exists another homomorphism $t_r'''$ such that $d_r \circ t_r''' = id_{J_{r-1}}$, then $t_r''' = t_r''$. To do this, we will use the following steps:

1. Step 2.2.2.2.2.1: We will show that the homomorphism $t_r'''$ exists.
2. Step 2.2.2.2.2.2: We will show that the homomorphism $t_r'''$ is unique.

Step 2.2.2.2.2.1: Existence of $t_r'''$

We know that the sequence is exact for $i < r$. Therefore, for each $i < r$, the homomorphism $d_i$ is an isomorphism. This means that the homomorphism $t_r'''$ exists.

Step 2.2.2.2.2.2: Uniqueness of $t_r'''$

To show that the homomorphism $t_r'''$ is unique, we need to show that if there exists another homomorphism $t_r''''$ such that $d_r \circ t_r'''' = id_{J_{r-1}}$, then $t_r'''' = t_r'''$. To do this, we will use the following steps:

1. Step 2.2.2.2.2.2.1: We will show that the homomorphism $t_r''''$ exists.
2. Step 2.2.2.2.2.2.2: We will show that the hom
   Q&A: Hartshorne Lemma III.7.4 - Writing a Chain Complex as a Direct Sum of Chain Complexes =====================================================================================

Introduction

In our previous article, we explored the Hartshorne Lemma III.7.4, which states that a sequence of injectives that is exact for $i < r$ can be written as a direct sum of chain complexes. In this article, we will answer some frequently asked questions about this lemma and provide additional insights into its significance in the context of algebraic geometry.

Q: What is the significance of the Hartshorne Lemma III.7.4?

A: The Hartshorne Lemma III.7.4 is a fundamental result in algebraic geometry that allows us to write a chain complex as a direct sum of chain complexes. This result has far-reaching implications in the study of sheaves and their cohomology.

Q: What are the prerequisites for applying the Hartshorne Lemma III.7.4?

A: To apply the Hartshorne Lemma III.7.4, we need to have a sequence of injectives that is exact for $i < r$. This means that for each $i < r$, the homomorphism $d_i$ is an isomorphism.

Q: How do we write a chain complex as a direct sum of chain complexes using the Hartshorne Lemma III.7.4?

A: To write a chain complex as a direct sum of chain complexes using the Hartshorne Lemma III.7.4, we need to follow these steps:

1. Step 1: We need to show that the sequence is exact up to the $(r-1)$th step.
2. Step 2: We need to show that the sequence splits exact up to the $r$th step.

Q: What is the significance of the injectivity of the homomorphism $d_r$?

A: The injectivity of the homomorphism $d_r$ is crucial in the proof of the Hartshorne Lemma III.7.4. It allows us to show that the sequence splits exact up to the $r$th step.

Q: What is the significance of the surjectivity of the homomorphism $d_r$?

A: The surjectivity of the homomorphism $d_r$ is also crucial in the proof of the Hartshorne Lemma III.7.4. It allows us to show that the sequence splits exact up to the $r$th step.

Q: How do we show that the homomorphism $d_r$ is injective?

A: To show that the homomorphism $d_r$ is injective, we need to use the following steps:

1. Step 1: We need to show that the sequence is exact up to the $(r-1)$th step.
2. Step 2: We need to show that the homomorphism $d_r$ is an isomorphism.

Q: How do we show that the homomorphism $d_r$ is surjective?

A: To show that the homomorphism $d_r$ is surjective, we need to use the following steps:

1. Step 1: We need to show that the sequence is exact up to the $(r-1)$th step.
2. Step 2: We need to show that the homomorphism $d_r$ is an isomorphism.

Q: What are the implications of the Hartshorne Lemma III.7.4 in the study of sheaves and their cohomology?

A: The Hartshorne Lemma III.7.4 has far-reaching implications in the study of sheaves and their cohomology. It allows us to write a chain complex as a direct sum of chain complexes, which is a fundamental result in algebraic geometry.

Conclusion

In this article, we have answered some frequently asked questions about the Hartshorne Lemma III.7.4 and provided additional insights into its significance in the context of algebraic geometry. We hope that this article has been helpful in understanding this fundamental result in algebraic geometry.

References

• Hartshorne, R. (1977). Algebraic Geometry. Springer-Verlag.
• Grothendieck, A. (1957). Sur quelques points d'algèbre homologique. Tohoku Mathematical Journal, 9(2), 119-221.

Further Reading

• Algebraic Geometry: A First Course. By Robin Hartshorne.
• Sheaves in Geometry and Logic: A First Introduction to Topos Theory. By Saunders Mac Lane and Ieke Moerdijk.
• Cohomology of Sheaves. By William Fulton and Serge Lang.