Mapping cone

Definition

A mapping cone is a construction in homological algebra that provides a way to associate a new complex to a chain map between two complexes. It captures the idea of 'extending' the original complex by adding the codomain and shifting the grading, allowing for an analysis of the properties and behavior of the original map. The mapping cone is essential in understanding derived functors, cohomology, and the relationship between chain maps and induced homomorphisms.

5 Must Know Facts For Your Next Test

1. The mapping cone of a chain map `f: A -> B` is constructed as a new chain complex `C(f)`, which consists of `B` in degree 0 and `A` in degree 1, with the differential defined using `f`.
2. The mapping cone is useful for studying the properties of morphisms in homological algebra, particularly in establishing relationships between different complexes.
3. When considering the long exact sequence in homology arising from a short exact sequence of complexes, the mapping cone plays a key role.
4. Mapping cones can be used to define derived functors like Ext and Tor, which are fundamental tools for studying modules over a ring.
5. The mapping cone construction helps visualize how chain maps affect homological properties, making it easier to understand induced homomorphisms between homology groups.

Review Questions

• How does the construction of a mapping cone relate to the study of chain maps?
  • The mapping cone is directly constructed from a chain map and serves to extend the original complex by adding its codomain shifted by one degree. This construction allows us to analyze how the chain map influences homological properties by providing a new complex that encapsulates this relationship. By looking at the mapping cone, we can better understand how elements in the domain map into the codomain and how these interactions shape the overall structure of homology.
• What is the significance of the mapping cone in deriving long exact sequences in homology?
  • The mapping cone is crucial for establishing long exact sequences when dealing with short exact sequences of chain complexes. By applying the mapping cone to these sequences, we can derive relationships between homology groups that reveal important information about the properties of these complexes. The long exact sequence allows us to track how elements are transformed through the morphisms, leading to insights about kernels and images across different degrees.
• Evaluate how understanding mapping cones can enhance your comprehension of derived functors like Ext and Tor.
  • Understanding mapping cones significantly enriches your comprehension of derived functors such as Ext and Tor by illustrating how these concepts relate to chain maps and their induced homomorphisms. Mapping cones provide concrete examples of how one can derive new complexes from existing ones, allowing you to see how Ext measures extensions of modules and Tor measures tensor products. By linking these derived functors back to mapping cones, you gain insight into their definitions and can visualize their roles within broader categorical frameworks, strengthening your grasp on their applications in homological algebra.

Related terms

Chain Complex:

A sequence of abelian groups or modules connected by homomorphisms, where the composition of two consecutive homomorphisms is zero.

Exact Sequence:

A sequence of abelian groups or modules and homomorphisms between them such that the image of one homomorphism equals the kernel of the next.

Homotopy:

A concept that describes when two continuous functions are equivalent if one can be transformed into the other through a continuous deformation.