3.1 Short and long exact sequences

Exact sequences are powerful tools in algebra, connecting objects through precise relationships. They reveal structural connections between groups, rings, and modules, helping us understand complex mathematical structures.

Short exact sequences link three objects, while long exact sequences extend infinitely. Both types use homomorphisms to preserve structure and provide insights into algebraic relationships. They're essential for computing invariants and analyzing object properties.

Exact Sequences and Homomorphisms

Short Exact Sequences

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• A short exact sequence is a sequence of homomorphisms between three objects $A \xrightarrow{f} B \xrightarrow{g} C$ such that $f$ is injective, $g$ is surjective, and $\text{im } f = \text{ker } g$
• The injectivity of $f$ implies that $A$ can be viewed as a subobject of $B$, while the surjectivity of $g$ implies that $C$ is a quotient object of $B$
• Short exact sequences provide a way to relate three objects in a category, often revealing important structural relationships
• Example: In the category of abelian groups, the sequence $0 \to \mathbb{Z} \xrightarrow{\times 2} \mathbb{Z} \xrightarrow{\text{mod } 2} \mathbb{Z}/2\mathbb{Z} \to 0$ is a short exact sequence

Long Exact Sequences

• A long exact sequence is an infinite sequence of homomorphisms between objects $\cdots \to A_{n-1} \xrightarrow{f_{n-1}} A_n \xrightarrow{f_n} A_{n+1} \to \cdots$ such that the composition of any two consecutive homomorphisms is zero and the kernel of each homomorphism is equal to the image of the previous one
• Long exact sequences often arise from short exact sequences through the application of functors, such as the homology or cohomology functors
• They provide a powerful tool for computing invariants of objects by relating them to the invariants of other objects in the sequence
• Example: The long exact sequence in homology associated to a pair of topological spaces $(X, A)$ is given by $\cdots \to H_n(A) \to H_n(X) \to H_n(X, A) \xrightarrow{\partial} H_{n-1}(A) \to \cdots$

Exactness and Homomorphisms

• A sequence of homomorphisms is called exact if the image of each homomorphism is equal to the kernel of the next homomorphism
• Exactness is a crucial property in the study of exact sequences, as it ensures that the objects and homomorphisms fit together in a precise way
• Homomorphisms are structure-preserving maps between algebraic objects, such as groups, rings, or modules
• In the context of exact sequences, homomorphisms play a central role in relating the objects and their invariants
• Example: In the short exact sequence $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$, the homomorphisms $f$ and $g$ preserve the algebraic structure of the objects $A$, $B$, and $C$

Kernels, Images, and Cokernels

Kernels

• The kernel of a homomorphism $f: A \to B$ is the subobject of $A$ consisting of all elements that are mapped to the zero element in $B$
• Kernels capture the notion of the "zero-set" of a homomorphism and play a crucial role in the construction of exact sequences
• The kernel of a homomorphism is always a normal subobject, which allows for the formation of quotient objects
• Example: In the category of groups, the kernel of a group homomorphism $f: G \to H$ is the normal subgroup $\text{ker } f = \{g \in G \mid f(g) = e_H\}$

Images

• The image of a homomorphism $f: A \to B$ is the subobject of $B$ consisting of all elements that are the result of applying $f$ to some element in $A$
• Images capture the notion of the "range" of a homomorphism and are essential in the study of surjectivity
• In an exact sequence, the image of each homomorphism is equal to the kernel of the next homomorphism
• Example: In the category of vector spaces, the image of a linear map $f: V \to W$ is the subspace $\text{im } f = \{f(v) \mid v \in V\}$

Cokernels

• The cokernel of a homomorphism $f: A \to B$ is the quotient object of $B$ by the image of $f$
• Cokernels capture the notion of the "coimage" of a homomorphism and are dual to the concept of kernels
• In an exact sequence, the cokernel of a homomorphism can be identified with the kernel of the next homomorphism
• Example: In the category of abelian groups, the cokernel of a group homomorphism $f: A \to B$ is the quotient group $\text{coker } f = B/\text{im } f$

Splitting Lemma

Splitting Exact Sequences

• The splitting lemma states that a short exact sequence $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ splits if and only if there exists a homomorphism $h: C \to B$ such that $g \circ h = \text{id}_C$
• A split exact sequence implies that $B$ is isomorphic to the direct sum of $A$ and $C$, i.e., $B \cong A \oplus C$
• The splitting lemma provides a way to decompose an object in an exact sequence into simpler components
• Example: In the category of vector spaces, a short exact sequence $0 \to U \to V \to W \to 0$ splits if and only if there exists a linear map $s: W \to V$ such that the composition $W \to V \to W$ is the identity on $W$

Applications of the Splitting Lemma

• The splitting lemma is a powerful tool for understanding the structure of objects in an exact sequence
• It allows for the classification of extensions of objects, which has applications in various areas of algebra and topology
• The splitting lemma is often used in conjunction with other techniques, such as the snake lemma, to compute invariants and study the properties of algebraic objects
• Example: In the study of group extensions, the splitting lemma can be used to determine whether an extension is trivial or not, providing insights into the structure of the groups involved