Local cohomology

Definition

Local cohomology is a tool in algebraic geometry and commutative algebra that helps to study the properties of sheaves and modules at a particular prime ideal or over a local ring. It captures information about the support of a module and provides insights into its local behavior, particularly in relation to depth, regular sequences, and Cohen-Macaulay rings. This concept bridges the gap between global properties of schemes and local phenomena.

5 Must Know Facts For Your Next Test

1. Local cohomology modules are denoted as $H^i_I(M)$ for a module $M$ and an ideal $I$, capturing information about sections supported on the set where $I$ vanishes.
2. The local cohomology groups can provide insight into whether or not a module is Cohen-Macaulay by examining the vanishing of these groups at certain levels.
3. One important result is that if $M$ is finitely generated over a Cohen-Macaulay ring, then all local cohomology modules are finitely generated as well.
4. Local cohomology is particularly useful in dimension theory because it connects local properties with global invariants of schemes or varieties.
5. The support of the local cohomology module $H^i_I(M)$ corresponds precisely to the variety defined by the ideal $I$, thus linking algebraic geometry with commutative algebra.

Review Questions

• How does local cohomology relate to the concept of depth in modules over rings?
  • Local cohomology relates closely to the concept of depth because it provides insight into how well a module behaves with respect to regular sequences. Specifically, when analyzing local cohomology modules, one can determine whether they vanish at certain levels, which gives information about the depth of the module. If the depth equals the Krull dimension, it often indicates that the module exhibits good behavior concerning regular sequences.
• In what ways does understanding local cohomology enhance our knowledge of Cohen-Macaulay rings?
  • Understanding local cohomology enhances knowledge of Cohen-Macaulay rings by revealing how these rings behave under localization. The vanishing of certain local cohomology modules can serve as a criterion for Cohen-Macaulayness. Additionally, examining local cohomological dimensions helps identify whether a ring maintains its desirable properties across various modules, thereby deepening our understanding of its structure.
• Evaluate how local cohomology can be utilized to address questions about regular sequences and their implications for module theory.
  • Local cohomology can be utilized to address questions about regular sequences by examining their relationships with depth and support in modules. Regular sequences are integral for defining depth, which in turn influences local cohomological behavior. By analyzing local cohomology groups associated with modules, one can determine if certain sequences are indeed regular, allowing for deeper insights into module theory and how these concepts interplay within algebraic structures.

Related terms

Cohen-Macaulay ring: A Cohen-Macaulay ring is a type of ring that has 'nice' properties, meaning that its depth equals its Krull dimension. This property is important in the study of local cohomology as it affects the vanishing of local cohomology modules.

Depth: Depth is a measure of how many elements from a ring can be taken to form a regular sequence. It relates to the structure of the module and plays a crucial role in determining properties such as Cohen-Macaulayness and the behavior of local cohomology.

Regular sequence: A regular sequence is a sequence of elements in a ring such that each element is not a zero divisor on the quotient by the ideal generated by the previous elements. This concept is key for understanding depth and local cohomology.