Short exact sequence

Definition

A short exact sequence is a sequence of objects and morphisms in an abelian category that captures the idea of exactness, which means that the image of one morphism equals the kernel of the next. Typically expressed as $A \xrightarrow{f} B \xrightarrow{g} C$, it illustrates how objects relate to each other, where $f$ is injective and $g$ is surjective, highlighting the connections between these objects and the structure of the category. Short exact sequences are essential for understanding many concepts in algebraic topology and homological algebra.

5 Must Know Facts For Your Next Test

1. In a short exact sequence, the first map must be injective, meaning it embeds $A$ into $B$, while the second map must be surjective, meaning it maps onto $C$.
2. Short exact sequences can be used to classify modules over a ring, providing a framework for understanding their relationships.
3. They allow us to derive long exact sequences by applying functors, giving deeper insights into cohomology theories.
4. The splitting property of short exact sequences indicates whether an object can be decomposed into simpler components.
5. Short exact sequences are prevalent in various areas such as group theory, module theory, and sheaf theory, reflecting their broad applicability.

Review Questions

• How does the concept of exactness relate to short exact sequences in an abelian category?
  • Exactness in the context of short exact sequences means that for a sequence $A \xrightarrow{f} B \xrightarrow{g} C$ to be exact, the image of the morphism $f$ must equal the kernel of the morphism $g$. This relationship ensures that all relevant properties and structures between these objects are captured in the sequence. Thus, understanding exactness is crucial to interpreting how these objects interact and how information flows through the sequence.
• Discuss the implications of short exact sequences for module classification and structure in algebra.
  • Short exact sequences have significant implications for classifying modules over rings. By representing modules as extensions of simpler modules through short exact sequences, mathematicians can analyze their properties more thoroughly. When a short exact sequence splits, it indicates that one module can be seen as a direct summand of another, thus simplifying the study of module structures. These insights help to illuminate relationships between modules and provide tools for constructing new modules from known ones.
• Evaluate how short exact sequences facilitate the derivation of long exact sequences in cohomology theories.
  • Short exact sequences play a critical role in deriving long exact sequences in cohomology theories by allowing functors to translate relationships between objects into long sequences that preserve certain properties. When applying a functor to a short exact sequence, one obtains a long exact sequence in cohomology groups. This transformation enables mathematicians to capture deeper relationships between cohomological dimensions and provides a systematic approach to studying topological spaces. The ability to derive long exact sequences highlights not only the importance of short exact sequences but also their central role in connecting various areas of mathematics.

Related terms

Exactness: The property of a sequence of morphisms being exact means that the image of one morphism equals the kernel of the next.

Abelian Category: An abelian category is a category where morphisms can be added, and every morphism has a kernel and cokernel, allowing for the study of exact sequences.

Kernel: The kernel of a morphism is the set of elements that map to the zero object, providing insight into the structure of the morphism.