Ker-coker exact sequence

Definition

The ker-coker exact sequence is a way to express the relationship between kernels and cokernels in a commutative diagram of morphisms, highlighting their roles in capturing the structure of a sequence of abelian groups or modules. It reveals how the image of one morphism relates to the kernel of another, indicating that the composition of these morphisms leads to exactness, which is essential in understanding properties like isomorphism and homological dimensions.

5 Must Know Facts For Your Next Test

1. In a short exact sequence involving kernel and cokernel, if you have a map from A to B and from B to C, then the sequence is exact if the image of the first map equals the kernel of the second.
2. The kernel measures how many elements are 'lost' when mapping from one object to another, while the cokernel measures how much remains unrepresented in the codomain.
3. The ker-coker sequence plays a crucial role in establishing connections between different homological concepts like Ext and Tor, which are central in derived functor theory.
4. In categorical terms, exactness gives rise to equivalences and limits that help characterize morphisms and objects in a category more precisely.
5. Understanding ker-coker sequences helps in visualizing and working with long exact sequences that arise in various contexts like cohomology and homology theories.

Review Questions

• How does the ker-coker exact sequence illustrate the relationship between kernels and cokernels in a commutative diagram?
  • The ker-coker exact sequence highlights that for any two consecutive morphisms, the image of one must equal the kernel of the next. This illustrates a precise alignment between transformations, emphasizing that what is mapped out (the image) corresponds directly with what is not being captured (the kernel). This relationship is fundamental to maintaining exactness, allowing us to draw conclusions about underlying structures within modules or groups.
• What implications does the concept of exactness have when applied to sequences involving kernels and cokernels?
  • Exactness implies that there is no 'gap' in information between consecutive morphisms, which means each part of the sequence builds on the previous one without loss. This principle allows for effective computations and simplifications in homological algebra. For instance, it allows one to deduce properties about modules from their relationships, leading to results that can be transferred across different settings or applications, such as cohomology groups.
• Evaluate how the ker-coker exact sequence contributes to our understanding of derived functors like Ext and Tor.
  • The ker-coker exact sequence is integral to defining derived functors because it provides a framework for analyzing how kernels and cokernels interact within short exact sequences. By establishing this relationship, we can derive new functors like Ext and Tor that encapsulate important properties of modules beyond direct observations. These functors help uncover deeper connections between algebraic structures, such as resolving modules or computing homological dimensions, thus enriching our overall understanding of their behavior.

Related terms

Exact Sequence: A sequence of morphisms where the image of one morphism equals the kernel of the next, indicating a perfect alignment in their structures.

Kernel: The set of elements that are mapped to zero by a given linear transformation or homomorphism, representing the 'failure' of that mapping.

Cokernel: The quotient of the codomain of a homomorphism by the image of that homomorphism, which captures how much of the codomain is not reached by the mapping.