A Question Related To Exactness And Surjection In An Abelian Category

Introduction

In the realm of Homological Algebra, Abelian Categories play a crucial role in understanding the properties of complexes and their behavior under various morphisms. One of the fundamental concepts in this area is the notion of exactness, which is closely related to the concept of surjection. In this article, we will delve into a question related to exactness and surjection in an Abelian category, exploring the conditions under which a complex is exact and the implications of this exactness on the morphisms involved.

Background

Let $\mathcal A$ be an Abelian category, and consider a complex $X' \xrightarrow{f} X \xrightarrow{g} X''$. A complex is a sequence of objects and morphisms in $\mathcal A$ such that the composition of consecutive morphisms is zero. In this case, we have a sequence of objects $X'$, $X$, and $X''$ and morphisms $f: X' \to X$ and $g: X \to X''$.

Exactness and Surjection

A complex is said to be exact if the image of each morphism is equal to the kernel of the next morphism. In other words, a complex is exact if for any object $X$ in the complex, the image of the morphism $f: X' \to X$ is equal to the kernel of the morphism $g: X \to X''$. This condition is equivalent to the statement that for any morphism $S \xrightarrow{h} X$, satisfying $gh = 0$, there exists a morphism $S \xrightarrow{k} X'$ such that $fk = h$.

The Question

Given a complex $X' \xrightarrow{f} X \xrightarrow{g} X''$ in an Abelian category $\mathcal A$, we want to determine the conditions under which this complex is exact. Specifically, we are interested in the following question:

• Let $X' \xrightarrow{f} X \xrightarrow{g} X''$ be a complex in an Abelian category $\mathcal A$. Then this complex is exact if and only if for any morphism $S \xrightarrow{h} X$, satisfying $gh = 0$, there exists a morphism $S \xrightarrow{k} X'$ such that $fk = h$.

The Answer

To answer this question, we need to explore the properties of Abelian categories and the behavior of complexes under various morphisms. We will use the following result:

• Let $\mathcal A$ be an Abelian category, and let $X' \xrightarrow{f} X \xrightarrow{g} X''$ be a complex in $\mathcal A$. Then this complex is exact if and only if for any morphism $S \xrightarrow{h} X$, satisfying $gh = 0$, there exists a morphism $S \xrightarrow{k} X'$ such that $fk = h$.

Proof

To prove this result, we need to show that the complex is exact if and only if for any morphism $S \xrightarrow{h} X$, satisfying $gh = 0$, there exists a morphism $S \xrightarrow{k} X'$ such that $fk = h$.

• Suppose that the complex is exact. Then for any morphism $S \xrightarrow{h} X$, satisfying $gh = 0$, we have that the image of the morphism $f: X' \to X$ is equal to the kernel of the morphism $g: X \to X''$. This implies that there exists a morphism $S \xrightarrow{k} X'$ such that $fk = h$.
• Conversely, suppose that for any morphism $S \xrightarrow{h} X$, satisfying $gh = 0$, there exists a morphism $S \xrightarrow{k} X'$ such that $fk = h$. Then we can show that the complex is exact by using the following argument:
  • Let $X$ be an object in the complex, and let $h: S \to X$ be a morphism satisfying $gh = 0$. Then there exists a morphism $k: S \to X'$ such that $fk = h$.
  • Since the complex is a sequence of objects and morphisms, we can use the morphism $k$ to construct a morphism $X' \to X$ that is a right inverse of the morphism $f: X' \to X$.
  • Using this right inverse, we can show that the image of the morphism $f: X' \to X$ is equal to the kernel of the morphism $g: X \to X''$, which implies that the complex is exact.

Conclusion

In this article, we have explored a question related to exactness and surjection in an Abelian category. We have shown that a complex is exact if and only if for any morphism $S \xrightarrow{h} X$, satisfying $gh = 0$, there exists a morphism $S \xrightarrow{k} X'$ such that $fk = h$. This result has important implications for the behavior of complexes under various morphisms and has applications in various areas of mathematics, including Homological Algebra and Algebraic Geometry.

References

• [1] Weibel, C. (1994). An Introduction to Homological Algebra. Cambridge University Press.
• [2] Mac Lane, S. (1998). Categories for the Working Mathematician. Springer-Verlag.
• [3] Hartshorne, R. (1977). Algebraic Geometry. Springer-Verlag.

Further Reading

• Homological Algebra: This is a branch of mathematics that studies the properties of complexes and their behavior under various morphisms.
• Abelian Categories: These are categories that satisfy certain properties, including the existence of kernels and cokernels.
• Surjection: A surjection is a morphism that is onto, meaning that it maps every element of the domain to an element of the codomain.

Glossary

• Complex: A sequence of objects and morphisms in a category.
• Exactness: A complex is exact if the image of each morphism is equal to the kernel of the next morphism.
• Surjection: A morphism that is onto, meaning that it maps every element of the domain to an element of the codomain.
• Abelian Category: A category that satisfies certain properties, including the existence of kernels and cokernels.
  A Question Related to Exactness and Surjection in an Abelian Category: Q&A ====================================================================

Introduction

In our previous article, we explored a question related to exactness and surjection in an Abelian category. We showed that a complex is exact if and only if for any morphism $S \xrightarrow{h} X$, satisfying $gh = 0$, there exists a morphism $S \xrightarrow{k} X'$ such that $fk = h$. In this article, we will provide a Q&A section to further clarify the concepts and answer any questions that readers may have.

Q: What is an Abelian category?

A: An Abelian category is a category that satisfies certain properties, including the existence of kernels and cokernels. This means that for any morphism $f: X \to Y$ in the category, there exists a kernel $Ker(f)$ and a cokernel $Coker(f)$.

Q: What is a complex in an Abelian category?

A: A complex in an Abelian category is a sequence of objects and morphisms in the category such that the composition of consecutive morphisms is zero. In other words, a complex is a sequence of objects $X_0, X_1, \ldots, X_n$ and morphisms $d_i: X_i \to X_{i+1}$ such that $d_{i+1} \circ d_i = 0$ for all $i$.

Q: What is exactness in a complex?

A: A complex is exact if the image of each morphism is equal to the kernel of the next morphism. In other words, a complex is exact if for any object $X$ in the complex, the image of the morphism $d_i: X_i \to X_{i+1}$ is equal to the kernel of the morphism $d_{i+1}: X_{i+1} \to X_{i+2}$.

Q: What is a surjection in an Abelian category?

A: A surjection in an Abelian category is a morphism that is onto, meaning that it maps every element of the domain to an element of the codomain. In other words, a morphism $f: X \to Y$ is a surjection if for every element $y \in Y$, there exists an element $x \in X$ such that $f(x) = y$.

Q: How does exactness relate to surjection?

A: Exactness and surjection are closely related in an Abelian category. A complex is exact if and only if for any morphism $S \xrightarrow{h} X$, satisfying $gh = 0$, there exists a morphism $S \xrightarrow{k} X'$ such that $fk = h$. This means that if a complex is exact, then for any morphism that satisfies a certain condition, there exists a morphism that is a right inverse of the original morphism.

Q: What are some examples of Abelian categories?

A: Some examples of Abelian categories include:

• The category of abelian groups
• The category of modules over a ring
• The category of vector spaces over a field
• The category of sheaves on a topological space

Q: What are some applications of Abelian categories?

A: Abelian categories have many applications in mathematics and computer science, including:

• Homological algebra
• Algebraic geometry
• Representation theory
• Computer science (e.g. category theory, type theory)

Conclusion

In this article, we have provided a Q&A section to further clarify the concepts and answer any questions that readers may have. We hope that this article has been helpful in understanding the concepts of exactness and surjection in an Abelian category.

References

• [1] Weibel, C. (1994). An Introduction to Homological Algebra. Cambridge University Press.
• [2] Mac Lane, S. (1998). Categories for the Working Mathematician. Springer-Verlag.
• [3] Hartshorne, R. (1977). Algebraic Geometry. Springer-Verlag.

Further Reading

• Homological Algebra: This is a branch of mathematics that studies the properties of complexes and their behavior under various morphisms.
• Abelian Categories: These are categories that satisfy certain properties, including the existence of kernels and cokernels.
• Surjection: A surjection is a morphism that is onto, meaning that it maps every element of the domain to an element of the codomain.

Glossary

• Complex: A sequence of objects and morphisms in a category.
• Exactness: A complex is exact if the image of each morphism is equal to the kernel of the next morphism.
• Surjection: A morphism that is onto, meaning that it maps every element of the domain to an element of the codomain.
• Abelian Category: A category that satisfies certain properties, including the existence of kernels and cokernels.