An exact sequence is a sequence of algebraic objects and morphisms between them, where the image of one morphism is equal to the kernel of the next. This concept is central in homological algebra as it allows us to study the relationships between different algebraic structures and provides insight into their properties through the notions of homology and cohomology.
congrats on reading the definition of Exact Sequence. now let's actually learn it.
Exact sequences can be classified as short exact sequences, long exact sequences, or more generally as sequences involving various algebraic structures.
In a short exact sequence of the form $$0 \to A \to B \to C \to 0$$, the injectivity of the first map and the surjectivity of the second map guarantee that A embeds into B and B maps onto C without any 'gaps'.
Exactness is preserved under various operations, such as taking products or coproducts, allowing for flexibility in manipulation within category theory.
Exact sequences can be used to derive important invariants like homology groups, which play a crucial role in algebraic topology and other areas.
The snake lemma is an important tool in deriving long exact sequences from short exact sequences, linking different homology groups together.
Review Questions
How do exact sequences relate to kernels and cokernels in algebraic structures?
Exact sequences highlight the relationship between kernels and cokernels by defining conditions where the image of one morphism equals the kernel of the next. For instance, in an exact sequence, if you have a morphism from A to B, then the image of this morphism must correspond precisely to the kernel of the subsequent morphism from B to C. This establishes a tight connection between these concepts, allowing for analysis of how elements are transformed and what structures emerge from these transformations.
Discuss how short exact sequences can be used to derive information about long exact sequences.
Short exact sequences provide a foundational tool for deriving long exact sequences through techniques like the snake lemma. When you have a short exact sequence like $$0 \to A \to B \to C \to 0$$, it sets up relationships between various objects that can extend into longer sequences involving additional modules or spaces. These longer sequences then reveal further properties about homology or cohomology groups by connecting them through multiple layers of mappings.
Evaluate the importance of exact sequences in understanding derived functors and their computations.
Exact sequences play a vital role in the computation and understanding of derived functors by establishing clear pathways between different algebraic structures. Derived functors often arise from taking left or right derived functors of certain functors, which are defined using resolutions that may include exact sequences. By studying these exact sequences, one can derive key properties and invariants about modules or spaces that are essential for deeper exploration in categories like homological algebra and algebraic topology.
The kernel of a homomorphism is the set of elements that map to the zero element in the target structure, essentially measuring how much 'information' is lost when applying the homomorphism.
The cokernel of a homomorphism is the quotient of the target structure by the image of the homomorphism, providing a way to analyze how much 'information' is added or how structures differ.
A chain complex is a sequence of abelian groups (or modules) connected by homomorphisms, where the composition of any two consecutive homomorphisms is zero, allowing for the study of homology.