10.2 Exact sequences and the Snake Lemma
4 min read•august 14, 2024
Exact sequences and the are key tools in homological algebra. They help us understand relationships between different mathematical objects by connecting their . These concepts are crucial for analyzing complex structures and solving problems in algebraic topology.
The Snake Lemma, in particular, allows us to create long exact sequences from short ones. This technique is super useful for computing homology groups and making connections between seemingly unrelated mathematical objects. It's like a secret weapon for tackling tricky algebraic problems!
Exact Sequences
Short and Long Exact Sequences
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- A consists of three objects and two morphisms in an abelian category
- The of the first equals the of the second morphism
- The first morphism is injective while the second morphism is surjective
- A is an infinite of objects and morphisms in an abelian category
- Often arises from short exact sequences of chain complexes
- Connects homology groups of different degrees
- The connecting in a long exact sequence measures the failure of exactness in the corresponding short exact sequence of chain complexes
- Defined using a diagram chase
- Plays a crucial role in relating homology groups of different chain complexes
Splitting Lemmas and Applications
- Splitting lemmas provide conditions under which a short exact sequence splits
- A split exact sequence is isomorphic to a direct sum of the first and third objects
- Splitting occurs when there exists a section (right inverse) or a retraction (left inverse) of one of the morphisms
- The long exact sequence in homology relates the homology groups of a chain complex to those of a subcomplex and the corresponding quotient complex
- Constructed using the Snake Lemma or the Zigzag Lemma
- Allows for computation of homology groups in terms of simpler chain complexes (Mayer-Vietoris sequence)
Snake Lemma for Homology
Statement and Diagram Chase
- The Snake Lemma studies the relationships between homology groups in a of or with exact rows
- Constructs a long exact sequence involving the kernels and cokernels of the vertical maps
- The connecting homomorphism is defined using a diagram chase
- The Snake Lemma is named after the zigzag pattern formed by the kernels, images, and cokernels in the diagram
- The "snake" connects the objects in the top row to those in the bottom row
- Provides a way to relate the homology groups of different chain complexes
Applications to Long Exact Sequences
- The Snake Lemma can be used to derive the long exact sequence in homology associated with a short exact sequence of chain complexes
- The short exact sequence of chain complexes induces short exact sequences of homology groups
- The Snake Lemma connects these short exact sequences into a long exact sequence
- The long exact sequence in homology is a powerful tool for computing homology groups
- Relates the homology of a chain complex to the homology of simpler chain complexes (subcomplex and quotient complex)
- Allows for inductive arguments and the comparison of homology groups
Five Lemma for Exactness
Statement and Commutative Diagrams
- The is a result about commutative diagrams in an abelian category
- If certain maps are isomorphisms and the rows are exact, then the remaining map is also an
- Provides a way to prove that a certain sequence is exact by comparing it to a known exact sequence
- The Five Lemma is often used in conjunction with other diagram lemmas (Snake Lemma, Short Five Lemma)
- Helps to establish the exactness of sequences in homological algebra
- Can be used to prove the functoriality of long exact sequences
Variants and Generalizations
- Variants of the Five Lemma handle similar situations with different numbers of objects and morphisms
- The Four Lemma deals with four objects and three morphisms
- The Six Lemma deals with six objects and five morphisms
- The Five Lemma can be generalized to longer exact sequences and more complex commutative diagrams
- The Nine Lemma and the 3x3 Lemma are examples of such generalizations
- These generalizations are useful in the study of spectral sequences and derived categories
Exact Sequences and Homological Algebra
Central Role of Exact Sequences
- Homological algebra studies sequences of abelian groups or modules connected by homomorphisms
- Focuses on exact sequences and derived functors
- Provides a framework for studying the structure of mathematical objects
- Exact sequences, particularly long exact sequences, encode important structural information about the objects involved
- Relate the homology or groups of different objects
- Allow for the computation of invariants using simpler objects
Tools and Techniques
- The Snake Lemma and the Five Lemma are key tools in homological algebra
- Used to work with exact sequences and prove exactness
- Enable the construction and study of long exact sequences
- Derived functors, such as Ext and Tor, measure the failure of a functor to be exact
- Can be studied using long exact sequences
- Provide additional invariants and structural information
- Spectral sequences are collections of exact sequences related by differentials
- Provide a powerful computational tool in homological algebra
- Enable the calculation of homology and cohomology groups in complex situations (Serre spectral sequence, Adams spectral sequence)
Key Terms to Review (20)
Abelian Groups: An abelian group is a set equipped with an operation that combines any two elements to form a third element, satisfying four key properties: closure, associativity, identity, and invertibility. Additionally, in an abelian group, the operation is commutative, meaning that the order in which two elements are combined does not affect the outcome. This structure is crucial for various mathematical concepts as it lays the foundation for understanding symmetry and other algebraic structures.
Cohomology: Cohomology is a mathematical concept in algebraic topology that assigns algebraic invariants, called cohomology groups, to topological spaces. These groups provide a powerful tool to understand the shape and structure of spaces by capturing their global properties and relationships. Cohomology is closely related to homology, but while homology focuses on 'holes' in a space, cohomology allows for more refined invariants that can be used to define operations such as cup products, making it essential for deeper results in topology and geometry.
Commutative Diagram: A commutative diagram is a visual representation of mathematical structures where the composition of morphisms (arrows) along different paths yields the same result. In algebraic topology, these diagrams help illustrate relationships between spaces and maps, making it easier to understand concepts like homotopy fiber sequences and exact sequences. The beauty of commutative diagrams lies in their ability to convey complex relationships in a clear and concise manner, allowing mathematicians to reason about structures abstractly.
Diagram chasing: Diagram chasing is a method used in category theory and algebraic topology to derive relationships between objects and morphisms in a commutative diagram. This technique allows one to navigate through the diagram's structure to deduce properties or prove results about the sequences of maps and their interactions, especially within exact sequences. By systematically following arrows in the diagrams, one can identify how elements relate to one another and establish connections between different algebraic structures.
Exact Sequence: An exact sequence is a sequence of algebraic structures and morphisms between them such that the image of one morphism equals the kernel of the next. This concept is essential in various areas of mathematics, as it captures the idea of how structures are connected and allows for the analysis of their properties through homological methods.
Five Lemma: The Five Lemma is a result in homological algebra that relates the long exact sequences in homology or cohomology associated with two short exact sequences. It provides conditions under which a morphism between two objects can be determined by the morphisms of their corresponding parts, establishing an important connection in the study of exact sequences and their properties.
Henri Poincaré: Henri Poincaré was a French mathematician, theoretical physicist, and philosopher of science, often regarded as one of the founders of algebraic topology. His work laid the groundwork for many areas of modern mathematics and theoretical physics, particularly through his contributions to topology and the concepts of homology and fundamental groups.
Homology Groups: Homology groups are algebraic structures that associate a sequence of abelian groups or modules to a topological space, providing a way to measure the 'holes' in that space. They capture important topological features, such as connectedness and the presence of cycles, and are essential in various computations and theorems in algebraic topology.
Homomorphism: A homomorphism is a structure-preserving map between two algebraic structures, such as groups, rings, or vector spaces, that respects the operations defined on those structures. It allows for the comparison and analysis of different mathematical objects by providing a way to translate properties and operations from one structure to another, making it essential in understanding concepts like the fundamental group, vector bundles, and exact sequences.
Image: In algebraic topology, the image refers to the set of outputs of a function or a mapping, particularly in the context of homology and sequences. It represents the elements that can be reached or represented by applying the function to all elements in the domain. Understanding images is crucial for computing properties of topological spaces, especially when analyzing how spaces can be decomposed and connected through continuous maps.
Injectivity: Injectivity is a property of a function where each element of the domain maps to a unique element in the codomain, ensuring that no two distinct inputs produce the same output. This concept is crucial for understanding the behavior of functions, particularly in the context of mathematical structures and their relationships. An injective function preserves the distinctness of elements, making it vital in various applications, including exact sequences and the Snake Lemma.
Isomorphism: Isomorphism is a mathematical concept indicating a structure-preserving correspondence between two objects, meaning they can be transformed into each other without losing their essential properties. This concept is vital in various branches of mathematics as it helps to classify objects by their structures rather than their appearances, revealing deeper connections between seemingly different entities.
Kernel: The kernel of a linear map or homomorphism is the set of elements that are mapped to the zero element of the codomain. In algebraic topology, understanding the kernel is crucial for exploring properties of spaces through tools like cellular homology and exact sequences, as it helps identify which elements or cycles do not contribute to the overall structure of a space.
Long exact sequence: A long exact sequence is a sequence of groups or modules connected by homomorphisms such that the image of one homomorphism equals the kernel of the next. This concept plays a crucial role in connecting different cohomological and homological properties, enabling us to derive important relationships between various algebraic structures and topological spaces.
Modules: Modules are algebraic structures that generalize vector spaces by allowing scalars to come from a ring instead of a field. This means that in modules, you can still perform addition and scalar multiplication, but the scalars are elements of a ring, which might not have multiplicative inverses for every element. This extension is crucial in understanding concepts such as exact sequences and the Snake Lemma, which rely on how modules behave under homomorphisms and their relationships in sequences.
Morphism: A morphism is a structure-preserving map between two mathematical objects, such as sets, groups, or topological spaces. It encapsulates the idea of a 'transformation' that maintains the intrinsic properties of the objects involved. In the context of algebraic topology, morphisms play a crucial role in defining relationships between different structures and facilitate the study of their properties through concepts like exact sequences and the Snake Lemma.
Samuel Eilenberg: Samuel Eilenberg was a prominent mathematician known for his foundational contributions to algebraic topology, particularly through the development of Eilenberg-MacLane spaces and homology theories. His work has significantly influenced the understanding of topological spaces and their algebraic invariants, connecting various concepts in topology and category theory.
Short Exact Sequence: A short exact sequence is a sequence of algebraic objects and morphisms between them that captures essential relationships, where the image of one morphism equals the kernel of the next. It usually takes the form of a diagram with three groups and two homomorphisms, showing how these groups are connected. This concept is vital for understanding how chain complexes relate to homology and how to work with exact sequences in various mathematical contexts.
Snake Lemma: The Snake Lemma is a fundamental result in homological algebra that describes a relationship between two exact sequences of abelian groups and provides a way to construct a connecting homomorphism between their kernels and cokernels. This lemma is crucial for understanding the behavior of chain complexes and is often used to derive long exact sequences in homology, establishing connections between different algebraic structures.
Surjectivity: Surjectivity is a property of a function where every element in the codomain has at least one pre-image in the domain. This means that the function 'covers' its codomain completely, ensuring that no part of the codomain is left unmapped. In the context of exact sequences and the Snake Lemma, understanding surjectivity helps in analyzing how mappings between algebraic structures interact and preserves certain properties across sequences.