2.4 Exact sequences of chain complexes

Exact sequences of chain complexes are crucial tools in homological algebra. They help us understand relationships between different chain complexes and their homology groups. These sequences form the backbone of many important results in the field.

This topic builds on our understanding of chain complexes and introduces key concepts like short and long exact sequences. We'll explore powerful tools like the snake lemma and mapping cones, which are essential for analyzing complex algebraic structures.

Exact Sequences

Short Exact Sequences and Their Properties

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• A short exact sequence is a sequence of three objects and two morphisms in an abelian category where the image of the first morphism equals the kernel of the second morphism
• Short exact sequences have the form $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ where $f$ is injective, $g$ is surjective, and $\operatorname{im}(f) = \ker(g)$
• The injectivity of $f$ and the surjectivity of $g$ imply that $A$ can be viewed as a subobject of $B$ and $C$ is isomorphic to the quotient object $B/A$
• Short exact sequences are used to study the relationships between objects in an abelian category and can provide information about the structure of the objects involved

Long Exact Sequences and Connecting Homomorphisms

• A long exact sequence is an infinite exact sequence of objects and morphisms in an abelian category
• Long exact sequences are often obtained by applying a covariant or contravariant functor to a short exact sequence
• The connecting homomorphism is a morphism that relates the homology groups of the objects in a long exact sequence
• Given a short exact sequence $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$, the connecting homomorphism $\delta_n: H_n(C) \to H_{n-1}(A)$ is defined by the composition $H_n(C) \to H_n(B,A) \to H_{n-1}(A)$, where the first map is induced by the quotient map $B \to C$ and the second map is the boundary map in the long exact sequence of relative homology
• The connecting homomorphism provides a way to relate the homology groups of the objects in a short exact sequence and is a crucial component in the construction of long exact sequences

Lemmas

The Snake Lemma and Its Applications

• The snake lemma is a powerful tool in homological algebra that relates the kernels, cokernels, and homology groups of morphisms between short exact sequences
• Given a commutative diagram of short exact sequences with vertical morphisms $f$, $g$, and $h$, the snake lemma provides a long exact sequence connecting the kernels, cokernels, and homology groups of $f$, $g$, and $h$
• The long exact sequence obtained from the snake lemma has the form $0 \to \ker(f) \to \ker(g) \to \ker(h) \xrightarrow{\delta} \operatorname{coker}(f) \to \operatorname{coker}(g) \to \operatorname{coker}(h) \to 0$, where $\delta$ is the connecting homomorphism
• The snake lemma is often used to prove the functoriality of homology and cohomology theories and to study the relationships between morphisms in abelian categories

The Five Lemma and the Horseshoe Lemma

• The five lemma states that if a commutative diagram of exact sequences has isomorphisms for the first, second, fourth, and fifth vertical arrows, then the middle vertical arrow is also an isomorphism
• The five lemma is useful for proving that certain morphisms are isomorphisms by verifying that the surrounding morphisms in a commutative diagram are isomorphisms
• The horseshoe lemma is a result in homological algebra that relates the homology of a complex to the homology of its subcomplexes and quotient complexes
• Given a short exact sequence of chain complexes $0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0$, the horseshoe lemma provides a long exact sequence connecting the homology groups of $A_\bullet$, $B_\bullet$, and $C_\bullet$

Constructions

The Mapping Cone and Its Properties

• The mapping cone is a construction in homological algebra that associates a chain complex to a chain map between two chain complexes
• Given a chain map $f: A_\bullet \to B_\bullet$, the mapping cone of $f$ is the chain complex $\operatorname{Cone}(f)_n = A_{n-1} \oplus B_n$ with differential $d(a,b) = (-d_A(a), f(a) + d_B(b))$
• The mapping cone fits into a short exact sequence $0 \to B_\bullet \to \operatorname{Cone}(f)_\bullet \to A_\bullet[-1] \to 0$, where $A_\bullet[-1]$ denotes the chain complex $A_\bullet$ shifted by one degree
• The homology of the mapping cone is related to the homology of the source and target complexes by a long exact sequence $\cdots \to H_n(A_\bullet) \xrightarrow{f_*} H_n(B_\bullet) \to H_n(\operatorname{Cone}(f)_\bullet) \to H_{n-1}(A_\bullet) \to \cdots$
• The mapping cone construction is used to study the relationships between chain maps and their induced morphisms on homology groups