Glossary

2.4 Exact sequences of chain complexes

4 min read•august 7, 2024

Exact sequences of chain complexes are crucial tools in homological algebra. They help us understand relationships between different chain complexes and their 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 and mapping cones, which are essential for analyzing complex algebraic structures.

Exact Sequences

Short Exact Sequences and Their Properties

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  • A is a sequence of three objects and two morphisms in an where the of the first equals the of the second morphism
  • Short exact sequences have the form 0→A→fB→gC→00 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0 where ff is injective, gg is surjective, and im⁡(f)=ker⁡(g)\operatorname{im}(f) = \ker(g)
  • The injectivity of ff and the surjectivity of gg imply that AA can be viewed as a subobject of BB and CC is isomorphic to the quotient object B/AB/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 is an infinite 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 is a morphism that relates the homology groups of the objects in a long exact sequence
  • Given a short exact sequence 0→A→fB→gC→00 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0, the connecting homomorphism δn:Hn(C)→Hn−1(A)\delta_n: H_n(C) \to H_{n-1}(A) is defined by the composition Hn(C)→Hn(B,A)→Hn−1(A)H_n(C) \to H_n(B,A) \to H_{n-1}(A), where the first map is induced by the quotient map B→CB \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 ff, gg, and hh, the snake lemma provides a long exact sequence connecting the kernels, cokernels, and homology groups of ff, gg, and hh
  • The long exact sequence obtained from the snake lemma has the form 0→ker⁡(f)→ker⁡(g)→ker⁡(h)→δcoker⁡(f)→coker⁡(g)→coker⁡(h)→00 \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 δ\delta is the connecting homomorphism
  • The snake lemma is often used to prove the functoriality of homology and theories and to study the relationships between morphisms in abelian categories

The Five Lemma and the Horseshoe Lemma

  • The 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 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→A∙→B∙→C∙→00 \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∙A_\bullet, B∙B_\bullet, and C∙C_\bullet

Constructions

The Mapping Cone and Its Properties

  • The mapping cone is a construction in homological algebra that associates a to a chain map between two chain complexes
  • Given a chain map f:A∙→B∙f: A_\bullet \to B_\bullet, the mapping cone of ff is the chain complex Cone⁡(f)n=An−1⊕Bn\operatorname{Cone}(f)_n = A_{n-1} \oplus B_n with differential d(a,b)=(−dA(a),f(a)+dB(b))d(a,b) = (-d_A(a), f(a) + d_B(b))
  • The mapping cone fits into a short exact sequence 0→B∙→Cone⁡(f)∙→A∙[−1]→00 \to B_\bullet \to \operatorname{Cone}(f)_\bullet \to A_\bullet[-1] \to 0, where A∙[−1]A_\bullet[-1] denotes the chain complex A∙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 ⋯→Hn(A∙)→f∗Hn(B∙)→Hn(Cone⁡(f)∙)→Hn−1(A∙)→⋯\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

Key Terms to Review (21)

Abelian Category: An abelian category is a type of category in mathematics where morphisms can be added together, and every morphism has a kernel and a cokernel, allowing for the construction of exact sequences. This structure provides a framework to discuss concepts like exactness, kernels, cokernels, and homological algebra more generally, making it crucial for understanding how to work with chain complexes and derived functors.
Chain Complex: A chain complex is a sequence of abelian groups or modules connected by homomorphisms, where the composition of any two consecutive homomorphisms is zero. This structure allows for the study of algebraic properties and relationships in various mathematical contexts, particularly in the fields of topology and algebra. Chain complexes are fundamental in defining homology and cohomology theories, which provide powerful tools for analyzing mathematical objects.
Cohomology: Cohomology is a mathematical concept that assigns algebraic invariants to topological spaces and chain complexes, capturing information about their structure and relationships. It provides a dual perspective to homology, focusing on the study of cochains and cocycles, which can reveal properties of spaces that homology alone might miss. This tool is essential in various areas of mathematics, connecting geometry, algebra, and topology.
Connecting Homomorphism: A connecting homomorphism is a morphism that arises in the context of exact sequences, specifically serving as a bridge between different chain complexes. It captures the relationship between the homology of the chain complexes and helps facilitate the transfer of algebraic information across these complexes. This concept plays a crucial role in linking different algebraic structures, especially when analyzing how they interact under sequences and derived functors.
Derived Functors: Derived functors are a way to extend the concept of functors in category theory to measure how much a given functor fails to be exact. They provide a systematic way to derive additional information from a functor by analyzing its relationship with exact sequences and chain complexes. Derived functors are particularly useful in homological algebra as they connect various algebraic structures, allowing us to study properties like the existence of certain modules and their relationships.
Exact Sequence: 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.
Exactness: Exactness in homological algebra refers to a property of sequences of objects and morphisms that captures the idea of preserving structure in a way that connects the input and output accurately. It ensures that the image of one morphism equals the kernel of the next, providing a precise relationship among the objects involved and facilitating the understanding of how algebraic structures behave under various operations.
Five Lemma: The Five Lemma is a key result in homological algebra that provides a method for proving the isomorphism of homology groups in the context of a commutative diagram of chain complexes. It connects the properties of morphisms in exact sequences, enabling one to deduce information about the mapping of objects based on the behavior of their images and kernels. This lemma is essential for understanding how exact sequences function and is closely linked to various other concepts, including exactness and the structure of derived functors.
Henri Cartan: Henri Cartan was a prominent French mathematician known for his significant contributions to algebraic topology and homological algebra, particularly in developing the theory of sheaves and derived functors. His work laid essential groundwork for later developments in category theory and cohomology, impacting various mathematical areas including group cohomology and spectral sequences.
Homology: Homology is a mathematical concept that associates a sequence of algebraic objects, typically abelian groups or vector spaces, to a topological space or a chain complex, providing a way to classify and measure the 'holes' or 'voids' within that space. It connects deeply with various structures in mathematics, revealing relationships between algebra and topology through its formulation and applications.
Horseshoe Lemma: The Horseshoe Lemma is a result in homological algebra that describes the relationship between two exact sequences of chain complexes. It specifically provides a method to construct a commutative diagram involving homology groups, allowing the transfer of information between different algebraic structures. This lemma is crucial for understanding how exact sequences behave and interact, especially when dealing with derived functors like Tor and Ext.
Image: In mathematics, particularly in the study of algebraic structures and homomorphisms, the image of a function is the set of all outputs it can produce. This concept is fundamental for understanding how elements from one set map to another, revealing properties of homomorphisms and chain complexes, as well as providing insight into exact sequences and homology groups.
Jean-Pierre Serre: Jean-Pierre Serre is a prominent French mathematician known for his work in algebraic geometry, topology, and number theory. His contributions have had a significant impact on various areas of mathematics, particularly through the development of cohomological methods and the theory of derived categories.
Kernel: The kernel of a homomorphism is the set of elements in the domain that map to the identity element of the codomain. It acts as a measure of how much a homomorphism fails to be injective and plays a crucial role in understanding algebraic structures and their relationships through exact sequences, chain complexes, and homology groups.
Long exact sequence: A long exact sequence is a sequence of abelian groups or modules connected by homomorphisms, which satisfies exactness at every point, indicating that the image of each homomorphism equals the kernel of the next. This concept is crucial in understanding the behavior of homology and cohomology theories, allowing one to relate different algebraic structures through their exact sequences and facilitating computations in various contexts.
Morphism: A morphism is a structure-preserving map between two objects in a category, serving as a central concept in category theory. Morphisms can represent functions, transformations, or relationships and play a crucial role in defining how objects interact within mathematical structures. They help to create frameworks for understanding mathematical concepts like exact sequences, functors, and transformations between categories.
Projective Resolution: A projective resolution is a specific type of exact sequence that helps to approximate modules using projective modules. It consists of a chain of projective modules connected by homomorphisms that leads to a given module, allowing the study of properties like homological dimensions and derived functors, which are essential in understanding the structure and classification of modules.
Short Exact Sequence: A short exact sequence is a sequence of algebraic structures and homomorphisms where the image of one morphism equals the kernel of the next, typically represented as 0 → A → B → C → 0. This definition serves as a foundation for understanding various concepts in algebra, particularly in relation to how structures relate to one another through homomorphisms and their exactness properties.
Snake Lemma: The Snake Lemma is a fundamental result in homological algebra that relates the exactness of sequences of homomorphisms through a commutative diagram. It provides a way to construct long exact sequences from short exact sequences and is pivotal in understanding the behavior of exact sequences in various contexts, including category theory and chain complexes.
Split Exact Sequence: A split exact sequence is a specific type of exact sequence in which a short exact sequence of modules or chain complexes splits, meaning that there is a way to express the middle object as a direct sum of the other two objects. This concept is crucial because it implies that the sequence can be broken down into simpler components, making it easier to analyze and understand. The existence of a section (a right inverse) for the inclusion map provides an intuitive way to visualize how the modules relate to each other.
Tor Functors: Tor functors are derived functors that measure the failure of a functor to be exact, specifically in the context of modules over a ring. They arise from taking the left derived functors of the tensor product, providing important information about the homological properties of modules, particularly in situations involving projective resolutions and exact sequences of chain complexes. Understanding Tor functors is crucial for analyzing relationships between modules and their structure through derived functors.
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