Cokernel

Definition

The cokernel is a concept in mathematics that refers to a specific construction associated with a morphism in an abelian category. It represents the quotient of the codomain of a morphism by the image of that morphism, effectively capturing the idea of what remains after accounting for the transformation applied by the morphism. This construction allows for a deeper understanding of the relationships between objects and morphisms within the framework of homological algebra.

5 Must Know Facts For Your Next Test

1. The cokernel of a morphism \( f: A \to B \) is denoted as \( \text{coker}(f) = B / \text{Im}(f) \), where \( \text{Im}(f) \) is the image of \( f \).
2. Cokernels can be thought of as measuring the 'failure' of a morphism to be surjective, with the cokernel providing information about what is not captured in the image.
3. In an abelian category, every morphism has a cokernel, which allows for a well-defined notion of quotient objects.
4. Cokernels play a crucial role in the study of homological algebra, particularly in understanding derived functors and chain complexes.
5. The cokernel can be represented diagrammatically in commutative diagrams, showcasing its relationship with kernels and exact sequences.

Review Questions

• How does the cokernel relate to the concept of surjectivity in morphisms?
  • The cokernel directly relates to surjectivity by providing insight into how much of the codomain is actually covered by the image of a given morphism. If a morphism is surjective, its cokernel will be trivial, meaning that there are no elements left over after mapping. Conversely, if there are elements in the codomain not accounted for by the image, the cokernel captures those leftover elements, illustrating that the morphism does not cover its entire codomain.
• Describe how cokernels interact with kernels in the context of exact sequences.
  • In an exact sequence, cokernels and kernels work together to reveal structural properties about objects and morphisms. Specifically, an exact sequence highlights that at any point where a morphism maps into another object, its cokernel must equal the kernel of the subsequent morphism. This interconnection allows for a coherent understanding of how images and preimages relate across various transformations within an abelian category, ultimately illustrating important relationships between different algebraic structures.
• Evaluate the importance of cokernels in homological algebra and their impact on understanding derived functors.
  • Cokernels are essential in homological algebra because they help define derived functors, which measure how well certain functors behave with respect to exactness. Understanding cokernels allows mathematicians to analyze complex structures by breaking them down into simpler components through quotients. This ability to decompose and reconstruct objects leads to deeper insights into properties such as projective and injective modules, ultimately aiding in resolving problems regarding extensions and cohomology theories across various fields within mathematics.

Related terms

Kernel: The kernel is the set of elements in the domain of a morphism that are mapped to zero in the codomain, representing an important aspect of how morphisms behave.

Exact Sequence: An exact sequence is a sequence of objects and morphisms such that the image of one morphism equals the kernel of the next, illustrating the interplay between cokernels and kernels.

Isomorphism: An isomorphism is a morphism that establishes a bijective correspondence between two objects, signifying that they are structurally identical.