Local cohomology

Definition

Local cohomology is a powerful tool in algebraic geometry that captures the behavior of sheaves on a space near a specified subspace, allowing for the study of local properties of varieties. It provides insight into how sections of sheaves behave in the vicinity of this subspace and helps in understanding the global properties by examining the local structure. This concept is essential for computations in sheaf cohomology, particularly when dealing with support conditions.

5 Must Know Facts For Your Next Test

1. Local cohomology is denoted by $H^i_I(X)$, where $I$ is an ideal defining a closed subset and $X$ is the space under consideration.
2. One of the key properties of local cohomology is its relationship with support; it can capture information about sheaves localized at specific points or subsets.
3. Local cohomology groups are left exact, which means they preserve finite limits and provide useful information about the dimensions of certain sheaf cohomology groups.
4. The Artin-Rees lemma plays an important role in local cohomology, allowing for control over sections in relation to ideals and their powers.
5. Local cohomology can be computed using spectral sequences and can often yield deeper insights into the global structure of a variety.

Review Questions

• How does local cohomology relate to the concepts of support and sheaf cohomology?
  • Local cohomology is fundamentally linked to support since it specifically studies sections of sheaves near a given subspace or ideal. It provides insights into how these sections behave in relation to their support, capturing essential local information that may not be evident from global cohomology alone. By analyzing this relationship, one can gain a deeper understanding of how properties like vanishing or non-vanishing occur locally.
• Discuss the significance of the Artin-Rees lemma in computing local cohomology and its implications on sheaf theory.
  • The Artin-Rees lemma is crucial for working with local cohomology as it establishes conditions under which sections can be controlled by powers of ideals. This lemma allows mathematicians to relate sections over a closed subset with sections defined globally via localization. Its implications extend into various areas of sheaf theory, impacting how we approach problems involving vanishing theorems and providing insight into depth and regularity conditions.
• Evaluate how local cohomology contributes to understanding the geometric and algebraic properties of varieties through computational methods.
  • Local cohomology significantly enhances our understanding of both geometric and algebraic properties of varieties by providing a framework to study localized behaviors in relation to specific subspaces. Using computational methods like spectral sequences allows researchers to extract valuable information about singularities, intersection theory, and other intricate structures within varieties. This approach facilitates deeper explorations into how local behaviors influence global characteristics, ultimately bridging algebraic geometry with computational techniques for insightful analysis.

Related terms

Sheaf Cohomology:

A mathematical framework that generalizes the notion of cohomology to sheaves, providing tools to analyze global sections and their properties.

Support: The support of a sheaf or a function refers to the closure of the set of points where it does not vanish, which is critical in defining local cohomology.

Cohomological Dimension: The maximum length of a resolution by projective modules or free sheaves, which can provide insights into the properties of local cohomology.