Local cohomology

Definition

Local cohomology is a derived functor that measures the 'local' properties of sheaves, particularly how they behave in the vicinity of a specified support. This concept is crucial in understanding sheaf cohomology because it provides a way to analyze the cohomological behavior of sheaves concentrated around certain subsets of the space, allowing for insights into their global properties. Local cohomology can highlight the relationships between local and global sections, which is fundamental in various contexts in algebraic geometry and commutative algebra.

5 Must Know Facts For Your Next Test

1. Local cohomology is defined with respect to a specified open subset of a topological space, focusing on sections supported within that subset.
2. It is typically denoted by $H^i_I(M)$ where $I$ is an ideal sheaf and $M$ is a module over a ring.
3. Local cohomology can be computed using Čech cohomology or through spectral sequences, linking it with other cohomological methods.
4. The functors associated with local cohomology are left exact, meaning they preserve finite limits but may not preserve colimits.
5. Local cohomology provides crucial information in algebraic geometry, particularly in understanding the vanishing of sections and how they relate to varieties.

Review Questions

• How does local cohomology function as a derived functor, and what implications does this have for understanding sheaf behavior?
  • Local cohomology acts as a derived functor by measuring how sheaves behave locally in relation to certain supports. It captures information that global sections might overlook, allowing us to gain insights into local properties around specified subsets. This helps in understanding the interplay between local and global sections, which is essential for applications in algebraic geometry.
• Discuss the significance of local cohomology in relation to the support of sheaves and its impact on their global properties.
  • The significance of local cohomology lies in its ability to focus on sections of sheaves that are supported in specific open subsets. This localized viewpoint allows mathematicians to determine when sections vanish or exhibit certain behaviors near those subsets. Consequently, this has a profound impact on understanding the global properties of sheaves since it informs us about how local conditions can influence overarching structures.
• Evaluate the role of local cohomology in algebraic geometry and its connections to other mathematical concepts like derived functors and cohomology.
  • Local cohomology plays a pivotal role in algebraic geometry by providing tools to analyze sheaves concentrated around subvarieties or closed sets. Its connection with derived functors allows it to extend our understanding of classical cohomology, offering insights into more intricate relationships between local and global sections. By linking local cohomology with concepts like support and the behavior of ideals, it deepens our comprehension of the geometric structure underlying algebraic varieties.

Related terms

Cohomology: A mathematical tool that studies topological spaces by associating algebraic invariants, often through the use of sheaves.

Derived Functors: Functors that provide a way to extend the concept of homology and cohomology, capturing deeper information about objects and their relationships.

Support: The set of points in a space where a sheaf or a function does not vanish, providing essential information about its local properties.