Support of a sheaf

Definition

The support of a sheaf is the closed set where the sheaf is non-zero, capturing the essential locations where the sheaf has relevant information. Understanding the support helps in analyzing cohomological properties and the behavior of sections over different open sets. It provides insight into how a sheaf interacts with the underlying space, which is crucial for deeper discussions on cohomology.

5 Must Know Facts For Your Next Test

1. The support of a sheaf is a closed subset of the underlying topological space, often denoted as \(\text{Supp}(\mathcal{F})\) for a sheaf \(\mathcal{F}\).
2. A sheaf is zero outside its support, meaning that the sections over open sets outside the support do not provide any information.
3. The support of a sheaf can change when considering different sheaves on the same space, reflecting varying data associated with the space.
4. In cohomology, understanding the support can lead to results about vanishing and non-vanishing of cohomology groups for specific sheaves.
5. When studying morphisms of sheaves, the supports must be compatible; this means that morphisms must respect the structure given by their respective supports.

Review Questions

• How does the support of a sheaf relate to its sections over various open sets?
  • The support of a sheaf defines where its sections can be non-zero. If you have an open set that intersects with the support, you may find non-trivial sections there. In contrast, if an open set lies entirely outside the support, all sections defined over that set will be zero. This relationship highlights how crucial it is to identify the support when working with sheaves in cohomological contexts.
• Discuss how understanding the support of a sheaf can impact the study of cohomological properties.
  • Understanding the support of a sheaf is vital because it can influence which cohomology groups are non-zero. For instance, if a sheaf has a compact support, it often leads to finite-dimensional cohomology groups. Additionally, knowing where a sheaf is supported can inform whether certain spectral sequences or exact sequences might collapse or yield significant information about global sections.
• Evaluate the role of supports in analyzing morphisms between sheaves and their implications for cohomological studies.
  • Supports play an essential role in understanding morphisms between sheaves because they dictate how these morphisms behave regarding locality. When analyzing morphisms, one must ensure that supports align; if not, sections may vanish unexpectedly or fail to preserve essential properties. This alignment is crucial in cohomology since understanding which sections survive under morphisms can directly affect derived functors and ultimately lead to significant insights into the structure and properties of various spaces.

Related terms

Sheaf: A mathematical object that associates data to open sets of a topological space, satisfying specific gluing and restriction properties.

Cohomology:

A tool in algebraic topology that studies the global properties of topological spaces using cochains, cocycles, and their relationships.

Global sections: The sections of a sheaf that are defined over the entire space rather than just local open sets, providing a way to capture the overall structure.