The Inverse Image Presheaf as a Sheaf

Oct 13, 2025 by talynt.org 56 views

Introduction

Sheaf theory is a fundamental concept in mathematics, particularly in algebraic geometry and topology. It provides a way to study the properties of spaces and their functions in a rigorous and abstract manner. One of the key concepts in sheaf theory is the inverse image presheaf, which is a presheaf that is associated with a given sheaf and a continuous map between spaces. In this article, we will explore the conditions under which the inverse image presheaf is already a sheaf.

Background

Let $\mathcal F$ be a sheaf over a space $X$ and let $S$ be a closed subset of $X$ . The inverse image presheaf $\mathcal F_S$ is defined as follows:

\mathcal F_S(U) = \mathcal F(U \cap S)

for any open subset $U$ of $X$ . The restriction maps are defined as the restriction maps of $\mathcal F$ .

The Inverse Image Presheaf as a Sheaf

The inverse image presheaf $\mathcal F_S$ is a presheaf, but it may not be a sheaf. In fact, it is a sheaf if and only if the following condition is satisfied:

Separation: For any two distinct sections $s, t \in \mathcal F_S(U)$ , there exists an open covering $\{U_i\}$ of $U$ such that $s|_{U_i} \neq t|_{U_i}$ for all $i$ .
Gluing: For any open covering $\{U_i\}$ of $U$ and any collection of sections $s_i \in \mathcal F_S(U_i)$ such that $s_i|_{U_i \cap U_j} = s_j|_{U_i \cap U_j}$ for all $i, j$ , there exists a section $s \in \mathcal F_S(U)$ such that $s|_{U_i} = s_i$ for all $i$ .

The Condition for the Inverse Image Presheaf to be a Sheaf

The inverse image presheaf $\mathcal F_S$ is a sheaf if and only if the following condition is satisfied:

The space $S$ is locally closed: For any point $x \in S$ , there exists an open neighborhood $U$ of $x$ such that $U \cap S$ is closed in $U$ .

This condition is known as the local closedness condition.

Proof of the Condition

Let $\mathcal F_S$ be the inverse image presheaf associated with the sheaf $\mathcal F$ and the closed subset $S$ of $X$ . Suppose that the local closedness condition is satisfied. We need to show that the inverse image presheaf $\mathcal F_S$ is a sheaf.

Let $U$ be an open subset of $X$ and let $s, t \in \mathcal F_S(U)$ be two distinct sections. We need to show that there exists an open covering $\{U_i\}$ of $U$ such that $s|_{U_i} \neq t|_{U_i}$ for all $i$ .

Since $s$ and $t$ are distinct sections, there exists an open subset $V$ of $U$ such that $s|_V \neq t|_V$ . Let $x \in V$ be a point such that $s(x) \neq t(x)$ . Since the local closedness condition is satisfied, there exists an open neighborhood $W$ of $x$ such that $W \cap S$ is closed in $W$ .

Since $s|_V \neq t|_V$ , there exists an open subset $U_i$ of $V$ such that $s|_{U_i} \neq t|_{U_i}$ . Since $W \cap S$ is closed in $W$ , we have $U_i \cap (W \cap S) = \emptyset$ . Therefore, $U_i \cap S = \emptyset$ .

Since $s|_{U_i} \neq t|_{U_i}$ , we have $s|_{U_i} \neq t|_{U_i}$ for all $i$ . Therefore, the inverse image presheaf $\mathcal F_S$ is a sheaf.

Conversely

Suppose that the inverse image presheaf $\mathcal F_S$ is a sheaf. We need to show that the local closedness condition is satisfied.

Let $x \in S$ be a point. We need to show that there exists an open neighborhood $U$ of $x$ such that $U \cap S$ is closed in $U$ .

Since $\mathcal F_S$ is a sheaf, the separation and gluing conditions are satisfied. Let $U$ be an open neighborhood of $x$ such that $U \cap S$ is not closed in $U$ . We need to show that this is impossible.

Since $U \cap S$ is not closed in $U$ , there exists a point $y \in U \setminus (U \cap S)$ . Let $s \in \mathcal F_S(U)$ be a section such that $s(y) \neq 0$ . Since the gluing condition is satisfied, there exists a section $t \in \mathcal F_S(U)$ such that $t|_{U \cap S} = s|_{U \cap S}$ .

Since $s(y) \neq 0$ , we have $t(y) \neq 0$ . Therefore, $t \neq 0$ in $\mathcal F_S(U)$ . However, this is impossible since $y \notin U \cap S$ .

Therefore, the local closedness condition is satisfied.

Conclusion

In this article, we have shown that the inverse image presheaf $\mathcal F_S$ is a sheaf if and only if the local closedness condition is satisfied. This condition is a necessary and sufficient condition for the inverse image presheaf to be a sheaf.

References

Wells, R. O. (1973). Differential Analysis on Complex Manifolds. Springer-Verlag.

Q: What is the inverse image presheaf?

A: The inverse image presheaf $\mathcal F_S$ is a presheaf that is associated with a given sheaf $\mathcal F$ and a closed subset $S$ of a space $X$ . It is defined as follows:

\mathcal F_S(U) = \mathcal F(U \cap S)

for any open subset $U$ of $X$ . The restriction maps are defined as the restriction maps of $\mathcal F$ .

Q: What is the condition for the inverse image presheaf to be a sheaf?

A: The inverse image presheaf $\mathcal F_S$ is a sheaf if and only if the local closedness condition is satisfied. This condition states that for any point $x \in S$ , there exists an open neighborhood $U$ of $x$ such that $U \cap S$ is closed in $U$ .

Q: What is the local closedness condition?

A: The local closedness condition is a condition that states that for any point $x \in S$ , there exists an open neighborhood $U$ of $x$ such that $U \cap S$ is closed in $U$ . This means that the intersection of $S$ with any open neighborhood of $x$ is a closed subset of that neighborhood.

Q: Why is the local closedness condition necessary for the inverse image presheaf to be a sheaf?

A: The local closedness condition is necessary for the inverse image presheaf to be a sheaf because it ensures that the sections of the presheaf can be glued together in a consistent way. If the local closedness condition is not satisfied, then it may not be possible to glue together the sections of the presheaf in a way that is consistent with the sheaf axioms.

Q: What are the implications of the result?

A: The result has several implications for the study of sheaves and their properties. For example, it provides a necessary and sufficient condition for the inverse image presheaf to be a sheaf, which can be used to study the properties of sheaves and their behavior under various operations.

Q: Can you provide an example of a situation where the inverse image presheaf is not a sheaf?

A: Yes, consider the following example. Let $X$ be a topological space and let $S$ be a closed subset of $X$ . Let $\mathcal F$ be a sheaf on $X$ and let $\mathcal F_S$ be the inverse image presheaf associated with $\mathcal F$ and $S$ . Suppose that $S$ is not locally closed, i.e., there exists a point $x \in S$ such that for any open neighborhood $U$ of $x$ , $U \cap S$ is not closed in $U$ . Then the inverse image presheaf $\mathcal F_S$ is not a sheaf.

Q: Can you provide an example of a situation where the inverse image presheaf is a sheaf?

A: Yes, consider the following example. Let $X$ be a topological space and let $S$ be a closed subset of $X$ . Let $\mathcal F$ be a sheaf on $X$ and let $\mathcal F_S$ be the inverse image presheaf associated with $\mathcal F$ and $S$ . Suppose that $S$ is locally closed, i.e., for any point $x \in S$ , there exists an open neighborhood $U$ of $x$ such that $U \cap S$ is closed in $U$ . Then the inverse image presheaf $\mathcal F_S$ is a sheaf.

Q: What are some applications of the result?

A: The result has several applications in mathematics and physics, including:

Algebraic geometry: The result can be used to study the properties of sheaves on algebraic varieties and their behavior under various operations.
Topology: The result can be used to study the properties of sheaves on topological spaces and their behavior under various operations.
Physics: The result can be used to study the properties of sheaves on manifolds and their behavior under various operations, which is important in the study of quantum field theory and other areas of physics.

Q: What are some open problems related to the result?

A: Some open problems related to the result include:

Characterizing the sheaf property: Can we characterize the sheaf property in terms of the local closedness condition?
Generalizing the result: Can we generalize the result to other types of presheaves and sheaves?
Applications to other areas of mathematics: Can we apply the result to other areas of mathematics, such as differential geometry and representation theory?