Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Abstract Algebra → Modules ↓


Exact Sequences


Introduction

In the field of abstract algebra, particularly when studying modules over rings, exact sequences provide a structured way of understanding the relationships between different modules and homomorphisms. Exact sequences are fundamental in various areas of mathematics because they encapsulate the concept of "exactness" or exactness of mappings in algebraic structures. This is important for understanding extensions, complexes, and maps within the algebraic hierarchy.

What is the exact sequence?

First, let's define what an exact sequence is. An exact sequence is a sequence of modules and module homomorphisms between them, such that the image of one homomorphism is the kernel of the next homomorphism. Let's express this with some modules and homomorphisms:

        M₁ → F M₂ → G M₃

The sequence is exact on M₂ if Im(f) = Ker(g), which means that the image of f is exactly the kernel of g. The sequence can be extended further:

        M₀ → H M₁ → F M₂ → G M₃ → I M₄

Here, exactness on M₁ means Im(h) = Ker(f), and so on. If the sequence is such that the maps are alternately injective and surjective when viewed consecutively, then it is exact.

Visual representation

To see this visually, think of the mapping as arrows and the modules as nodes:

M₁ M₂ M₃

In this diagram, the arrows represent homomorphisms, and the spaces between the module nodes represent images and kernels that represent accuracy.

Simple examples of exact sequences

It is important to understand exact sequences through examples. Consider the simplest exact sequence of a vector space:

        0 → F v → G v → 0

Here, 0 denotes the zero module, V is a vector space, and f and g are the inclusion and identity maps. The sequence is exact because:

  • Im(f) = Ker(g), since f is injective.
  • g is an identity and so V covers completely.

This trivial sequence can be simplified, but it is a basic foundation for understanding more complex sequences.

Exact sequences in group theory

Exact sequences apply not only to modules and vector spaces, but also to group homomorphisms. Consider a group homomorphism sequence:

        G₁ → F G₂ → G G₃

The sequence is exact if Im(f) = Ker(g). For example, suppose f: G₁ → G₂ is inclusion, and g: G₂ → G₃ is quotienting the kernel of some other homomorphism. Then, exactness on G₂ ensures the effectiveness of these transformations in capturing group properties.

Long exact sequences

Long exact sequences provide more in-depth information. They often appear in complex algebraic topology, such as the long exact sequence of homotopy or cohomology groups. Consider this sequence:

        0 → α A → β B → γ C → δ D → 0

Accuracy here means:

  • α is injective since Im(0)=0 collines with Ker(α).
  • Im(α) = Ker(β), ensuring that β captures the necessary transformations.
  • Passing through D, δ must be oscillating since the sequence terminates at zero.

Exact sequences in cohomology

Cohomology theory, especially in topology and geometry, uses exact sequences to classify algebraic invariants. Consider the short exact sequence of coefficient groups:

        0 → A → B → C → 0

This yields an exact sequence in cohomology as the functions move around in these groups, which is analyzed to extract useful topological features. This application is important in showing the robustness of exact sequences in theoretical applications.

Connecting homeomorphisms

Cycles are generated in specific steps, often in a sequence. For example, the exact sequences are given:

        0 → A → B → C'

And:

        0 → C' → D → E

A connecting homomorphism connects C' to D, preserving exactness in distinct sections. Such operations are important in preserving fundamental exactness properties in module sets in sequence algebra.

Conclusion

Exact sequences in modules introduce important concepts of injectivity, surjectivity, homomorphism and kernel cohesion in algebra. They facilitate complex algebraic structure analysis and provide important insights into mathematical foundations and applications. With these theories, exact sequences become tools for further exploration and development in abstract algebra and its many branches.

Understanding and effectively applying exact sequences is part of the important groundwork for advanced mathematical education, leading to more profound theoretical and practical breakthroughs across a broad academic field.


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