3.2 The snake lemma and its applications
2 min read•august 7, 2024
The is a powerful tool in homological algebra. It connects and in commutative diagrams with exact rows, creating a . This lemma is crucial for understanding how different parts of algebraic structures relate to each other.
The snake lemma has wide-ranging applications in algebra. It's used to prove other important results like the and can be extended to more complex situations. Understanding this lemma is key to grasping how short and long exact sequences work together.
The Snake Lemma and Connecting Homomorphisms
Snake Lemma and its Components
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- States that given a with exact rows, there exists a long connecting the kernels and cokernels of the vertical maps
- Involves the , which maps the cokernel of one vertical map to the kernel of the next vertical map
- Utilizes the , which is a long exact sequence that alternates between kernels and cokernels of the vertical maps in the commutative diagram
- Applies the , which states that if a composition of two maps is zero, then the image of the first map is contained in the kernel of the second map
Applications and Extensions
- Used to prove the five lemma, which states that if the first two and last two vertical maps are isomorphisms in a commutative diagram with exact rows, then the middle vertical map is also an
- Can be generalized to the 3x3 lemma, which involves a 3x3 commutative diagram with exact rows and columns
- Applies to various algebraic structures, such as modules over a ring, abelian groups, and chain complexes
- Helps in understanding the relationship between short exact sequences and long exact sequences in homological algebra
Homological Algebra Fundamentals
Commutative Diagrams and Chain Complexes
- A commutative diagram is a diagram of objects and morphisms where all directed paths between two objects lead to the same result
- Commutativity is a crucial property in homological algebra, as it allows for the study of functorial properties and the construction of exact sequences
- A is a sequence of abelian groups or modules connected by homomorphisms (differentials) such that the composition of any two consecutive homomorphisms is zero
- Chain complexes are used to define , which measure the "holes" or "cycles modulo boundaries" in the complex
Homology and its Properties
- Homology is a functor that assigns to each chain complex a sequence of abelian groups (homology groups), which are the quotients of the kernel of one differential by the image of the previous differential
- Homology groups are invariants of the chain complex and provide information about its algebraic structure
- The homology functor is a covariant functor from the category of chain complexes to the category of graded abelian groups
- Homology satisfies the Eilenberg-Steenrod axioms, which characterize homology theories in algebraic topology and allow for the comparison of different homology theories
Key Terms to Review (21)
C, d, e: In the context of the snake lemma, c, d, and e refer to specific morphisms in the commutative diagram that arises when dealing with exact sequences. These morphisms play a crucial role in establishing the relationships between objects and their mappings in a way that helps to deduce properties of exactness and isomorphism in homological algebra.
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.
Cokernels: Cokernels are an important concept in category theory and homological algebra, representing a certain type of quotient in the context of morphisms between objects. Specifically, given a morphism $f: A \to B$, the cokernel of $f$ is defined as the coimage of $f$, which captures the 'difference' between $B$ and the image of $f$. Cokernels allow us to study how objects relate through morphisms and form a crucial part of understanding exact sequences and the snake lemma.
Commutative Diagram: A commutative diagram is a visual representation in category theory where objects are represented as points and morphisms as arrows, illustrating relationships between objects in such a way that all paths with the same start and endpoints yield the same result when composed. This concept is essential for understanding how different mathematical structures interact and is crucial for analyzing concepts like exact sequences and functors.
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.
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.
F, g, h: In the context of the snake lemma, 'f', 'g', and 'h' typically refer to morphisms (or maps) between objects in a category, which are crucial in establishing the long exact sequence that arises from a commutative diagram. These morphisms play a vital role in connecting different objects and help illustrate the relationships between them, especially when dealing with exact sequences. Understanding how these morphisms interact is essential for applying the snake lemma effectively.
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 groups: Homology groups are algebraic structures that arise from chain complexes and serve to classify topological spaces by measuring their 'holes' in various dimensions. They provide crucial insights into the properties of spaces and are integral to understanding concepts in algebra, geometry, and topology.
Injectivity: Injectivity is a property of functions that indicates a one-to-one correspondence between elements of the domain and the codomain, meaning that different elements in the domain map to different elements in the codomain. This concept plays a crucial role in various mathematical contexts, particularly in determining whether morphisms in categories preserve structure and enable certain constructions. In homological algebra, understanding injectivity is vital for applying the snake lemma, as it helps clarify how certain sequences can be analyzed and manipulated.
Isomorphism: An isomorphism is a mapping between two mathematical structures that establishes a one-to-one correspondence, preserving the operations and relations of the structures. It shows that two structures are fundamentally the same in terms of their properties, even if they may appear different at first glance. This concept connects deeply with how we understand relationships in category theory and other mathematical frameworks.
Ker-coker exact sequence: The ker-coker exact sequence is a way to express the relationship between kernels and cokernels in a commutative diagram of morphisms, highlighting their roles in capturing the structure of a sequence of abelian groups or modules. It reveals how the image of one morphism relates to the kernel of another, indicating that the composition of these morphisms leads to exactness, which is essential in understanding properties like isomorphism and homological dimensions.
Kernels: In algebra, a kernel is the set of elements that are mapped to zero by a given linear transformation or homomorphism. This concept is crucial in understanding the structure of algebraic systems and plays a significant role in establishing properties like exactness and injectivity in sequences and diagrams.
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.
Samuel Eilenberg: Samuel Eilenberg was a prominent mathematician known for his groundbreaking contributions to the field of homological algebra, particularly in the development of derived functors and the Ext functor. His work laid the foundation for many essential concepts in modern algebra, influencing various mathematical areas including topology and category theory.
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.
Surjectivity: Surjectivity is a property of a function or map where every element in the target set has at least one pre-image in the domain. This concept is crucial because it ensures that the mapping covers the entire target space, making it possible to solve equations and transfer structures effectively. In the context of exact sequences and the snake lemma, understanding surjectivity helps clarify when certain elements can be constructed from others, linking different mathematical objects meaningfully.
Three x Three Lemma: The three x three lemma is a result in homological algebra that describes conditions under which a commutative diagram of modules and morphisms can be used to derive certain exact sequences. This lemma is particularly important in the context of the snake lemma, as it helps to establish connections between kernels and cokernels in the diagrams. By providing a systematic approach to handling short exact sequences, this lemma facilitates the analysis of homological properties of modules.
Zig-zag lemma: The zig-zag lemma is a result in homological algebra that describes how to construct a long exact sequence of homology or cohomology groups from a zig-zag of short exact sequences. This lemma plays a crucial role in connecting various sequences of objects and morphisms, allowing us to deduce important properties of functors and derived categories.