Theorem of M. A. Auslander

Definition

The theorem of M. A. Auslander is a significant result in homological algebra that provides a way to compute the functors Tor and Ext, which are fundamental in understanding the properties of modules over a ring. This theorem connects the derived functors with projective and injective resolutions, allowing for more efficient calculations and deeper insights into the structure of modules. It highlights the relationship between finitely generated modules and their behavior under these functors.

5 Must Know Facts For Your Next Test

1. Theorem of M. A. Auslander shows that for certain classes of modules, the computation of Tor and Ext can be reduced to simpler cases, particularly when dealing with finitely generated modules over Noetherian rings.
2. One of the key implications of this theorem is the identification of projective modules as being particularly useful in the computation of Ext groups.
3. This theorem establishes criteria under which the Tor functor can vanish, offering insight into when certain extensions exist or are trivial.
4. Auslander's work emphasizes the importance of resolution techniques in homological algebra, especially how projective and injective resolutions relate to derived functors.
5. The theorem has applications in various areas, including representation theory and algebraic geometry, where understanding module structures is essential.

Review Questions

• How does the theorem of M. A. Auslander impact the computation techniques for the functors Tor and Ext?
  • The theorem of M. A. Auslander significantly impacts computation techniques by simplifying how we approach the calculation of Tor and Ext. It provides conditions under which these functors can be computed more easily, particularly for finitely generated modules over Noetherian rings. By linking derived functors with projective and injective resolutions, it allows us to focus on simpler cases or specific properties of modules, enhancing our overall understanding of their behavior.
• In what ways does Auslander's theorem help identify when Tor or Ext vanish for specific classes of modules?
  • Auslander's theorem offers criteria that indicate when Tor or Ext functors vanish, which is crucial for understanding module extensions. For example, if a module is projective or flat, then certain values of Tor can be shown to equal zero, suggesting that extensions may not exist in those cases. This insight allows mathematicians to better classify modules based on their properties and understand the implications on their relationships with other modules.
• Evaluate the broader implications of Auslander's theorem in the context of algebraic geometry and representation theory.
  • Evaluating Auslander's theorem reveals its extensive implications in fields like algebraic geometry and representation theory. In algebraic geometry, understanding the homological properties of sheaves relates closely to module theory, thus impacting how we classify coherent sheaves on algebraic varieties. In representation theory, identifying projective modules through this theorem helps analyze representations of algebras, leading to better insights into their structure and classification. Overall, Auslander's results provide powerful tools that bridge these areas with fundamental concepts in homological algebra.

Related terms

Tor Functor: The Tor functor measures the failure of flatness and provides a way to compute homological dimensions of modules.

Ext Functor: The Ext functor is used to classify extensions of modules and to study projective and injective resolutions.

Projective Resolution: A projective resolution is an exact sequence of module homomorphisms where each module is projective, providing a way to compute Ext groups.