Artin-Rees Lemma

Definition

The Artin-Rees Lemma is a crucial result in commutative algebra that provides conditions under which an ideal's power intersects with a submodule of a module. It essentially states that for an ideal and a finitely generated module, there exists some power of the ideal such that its intersection with any submodule can be controlled by a power of the ideal, linking it to properties of local cohomology and modules. This lemma is particularly useful when dealing with the Tor functor and sheaf cohomology, as it helps manage the behavior of ideals in these contexts.

5 Must Know Facts For Your Next Test

1. The Artin-Rees Lemma is often used to show that certain submodules can be approximated by intersections with powers of ideals.
2. It provides an important tool for proving results related to the regularity of modules and their local properties.
3. In the context of the Tor functor, it helps understand how torsion elements behave under the interaction with ideals.
4. The lemma is also fundamental when discussing vanishing conditions for local cohomology, providing necessary control over elements in modules.
5. The Artin-Rees Lemma ensures that given an ideal and a submodule, one can always find a sufficiently large power of the ideal to work with.

Review Questions

• How does the Artin-Rees Lemma facilitate the understanding of the relationship between an ideal and a submodule in the context of finitely generated modules?
  • The Artin-Rees Lemma provides a powerful way to analyze how an ideal interacts with a submodule by ensuring that there exists a power of the ideal such that its intersection with the submodule is manageable. This allows mathematicians to deduce important properties about the structure of modules, particularly when considering their generation and the presence of torsion elements. The control offered by this lemma is essential in understanding how these interactions play out in various algebraic settings.
• In what ways does the Artin-Rees Lemma contribute to results regarding local cohomology?
  • The Artin-Rees Lemma significantly contributes to local cohomology by providing conditions under which vanishing results can be obtained for local cohomology modules. It allows for the approximation of submodules using powers of ideals, which is crucial in establishing when certain local cohomology groups vanish. By controlling how ideals intersect with modules, it supports deeper insights into the structure and behavior of sheaves over spaces.
• Evaluate the implications of the Artin-Rees Lemma on the study of the Tor functor and its applications in homological algebra.
  • The implications of the Artin-Rees Lemma on the study of the Tor functor are profound, as it helps in analyzing how torsion elements interact with ideals within modules. By providing a way to manage intersections between ideals and submodules, it lays the groundwork for understanding derived functors in homological algebra. Moreover, it enables mathematicians to establish connections between local properties of modules and their global behavior, facilitating broader applications across various algebraic contexts.

Related terms

Tor Functor: A functor that measures the failure of flatness of modules and is used to study derived functors in homological algebra.

Local Cohomology: A type of cohomology that captures information about the behavior of sheaves over a space, particularly around certain subvarieties or closed subsets.

Finitely Generated Module: A module that can be generated by a finite number of elements, playing a key role in various results in commutative algebra.