Ext Functor

Definition

The Ext functor is a tool in homological algebra that measures the extent to which a module fails to be projective, essentially quantifying extensions of modules. It can be thought of as a way to study the relationships between modules by capturing how they can be extended by other modules, providing insights into their structure. Specifically, when examining the depth of modules and rings, the Ext functor reveals crucial information about the syzygies and resolutions of modules, shedding light on their homological properties.

5 Must Know Facts For Your Next Test

1. The Ext functor, denoted as $$\text{Ext}^n(M,N)$$, provides a way to measure how many distinct extensions exist between two modules M and N.
2. For modules over a Noetherian ring, Ext groups capture important information about the depth of modules, as higher Ext groups often indicate lower depth.
3. The first Ext group $$\text{Ext}^1(M,N)$$ classifies extensions of the module N by the module M, essentially describing how M can fit into N.
4. Ext functors can be computed using projective resolutions, allowing for practical techniques to find extensions in various algebraic contexts.
5. If $$\text{Ext}^1(M,N) = 0$$, it indicates that every short exact sequence of the form $$0 \to N \to E \to M \to 0$$ splits, meaning M is projective with respect to N.

Review Questions

• How does the Ext functor relate to the concept of projective and injective modules?
  • The Ext functor plays a key role in understanding whether a module is projective or injective. If $$\text{Ext}^1(M,N) = 0$$ for some module N, this indicates that M is projective because it shows that all extensions of N by M split. Conversely, if we consider an injective module I and look at $$\text{Hom}(M,I)$$, we see that injectivity is closely tied to resolving modules and building up structures with the help of Ext.
• Discuss the importance of computing higher Ext groups in relation to the depth of modules.
  • Computing higher Ext groups like $$\text{Ext}^2(M,N)$$ or more can give significant insight into the depth of modules. For instance, if these groups are non-zero, they indicate that there are obstructions to lifting certain maps and that the module M might not have sufficient projective resolutions. This connection between depth and Ext groups helps understand not just individual modules but also their interactions within a larger category of modules.
• Evaluate how the Ext functor can be used to analyze syzygies and resolutions of modules in commutative algebra.
  • The Ext functor is essential in studying syzygies and resolutions because it helps in constructing long exact sequences in homological algebra. By looking at projective resolutions of a given module M, we can compute various Ext groups which reveal information about its syzygies—essentially capturing how far M is from being free or projective. Analyzing these groups allows mathematicians to build a comprehensive picture of the module's structure and its interaction with other modules, providing deep insights into both algebraic properties and geometric interpretations.

Related terms

Projective Module: A module that has the property that every surjective homomorphism onto it splits, meaning it can be lifted along any surjection.

Injective Module: A module where every injective homomorphism from it to another module can be extended to any larger module, allowing for the construction of certain exact sequences.

Homological Dimension: The minimum number of steps required to resolve a module using projective or injective modules, indicating its complexity in terms of extension problems.