2.1 Definition and properties of chain complexes

Chain complexes are sequences of abelian groups connected by boundary maps. They're crucial in homological algebra, helping us study algebraic structures and topological spaces. The key property is that composing two consecutive boundary maps gives zero.

This section dives into the definition and properties of chain complexes. We'll explore cycles, boundaries, and homology groups, which measure "holes" in algebraic objects. Understanding these concepts is essential for grasping the broader ideas in homological algebra.

Chain Complexes and Graded Modules

Definition and Structure

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• A chain complex is a sequence of abelian groups or modules $C_n$ connected by homomorphisms $d_n: C_n \rightarrow C_{n-1}$ called boundary maps or differentials
• The boundary maps satisfy the condition $d_{n-1} \circ d_n = 0$ for all $n$, meaning the composition of any two consecutive boundary maps is the zero map
• A graded module is a module that can be expressed as a direct sum of submodules indexed by the integers, i.e., $M = \bigoplus_{n \in \mathbb{Z}} M_n$
• The elements of $M_n$ are said to have degree $n$, and the degree of a homogeneous element $x \in M_n$ is denoted by $|x| = n$

Boundary Operators and Differentials

• The boundary operator or differential $d_n: C_n \rightarrow C_{n-1}$ lowers the degree of the elements by 1
• The condition $d_{n-1} \circ d_n = 0$ implies that the image of $d_n$ is contained in the kernel of $d_{n-1}$, i.e., $\text{im } d_n \subseteq \ker d_{n-1}$
• This condition is necessary for the homology groups of the chain complex to be well-defined
• The differential can be thought of as a generalization of the boundary operator in simplicial or cellular homology, which maps a chain to its boundary

Exactness and Homology

Exact Sequences

• A sequence of abelian groups or modules and homomorphisms between them is called exact if the image of each homomorphism is equal to the kernel of the next homomorphism
• In the context of chain complexes, a short exact sequence is a sequence of the form $0 \rightarrow A \xrightarrow{f} B \xrightarrow{g} C \rightarrow 0$, where $f$ is injective, $g$ is surjective, and $\text{im } f = \ker g$
• Exact sequences are important tools in homological algebra, as they allow for the study of relationships between different chain complexes and their homology groups

Cycles, Boundaries, and Homology

• A cycle in a chain complex $(C_n, d_n)$ is an element $x \in C_n$ such that $d_n(x) = 0$, i.e., an element in the kernel of the boundary map
• A boundary in a chain complex $(C_n, d_n)$ is an element $x \in C_n$ such that there exists a $y \in C_{n+1}$ with $d_{n+1}(y) = x$, i.e., an element in the image of the boundary map
• The $n$-th homology group of a chain complex $(C_n, d_n)$ is defined as the quotient group $H_n(C) = \ker d_n / \text{im } d_{n+1}$, i.e., the cycles modulo the boundaries
• Homology groups measure the "holes" or "voids" in a topological space or algebraic object, with different dimensions of holes corresponding to different homology groups

Chain Homotopy

Definition and Properties

• A chain homotopy between two chain maps $f, g: C \rightarrow D$ is a sequence of homomorphisms $h_n: C_n \rightarrow D_{n+1}$ such that $f_n - g_n = d^D_{n+1} \circ h_n + h_{n-1} \circ d^C_n$ for all $n$
• If there exists a chain homotopy between $f$ and $g$, then $f$ and $g$ are said to be chain homotopic, denoted by $f \simeq g$
• Chain homotopy is an equivalence relation on the set of chain maps between two chain complexes
• Chain homotopic maps induce the same homomorphisms on homology, i.e., if $f \simeq g$, then $f_* = g_*: H_n(C) \rightarrow H_n(D)$ for all $n$, where $f_*$ and $g_*$ are the induced maps on homology
• The concept of chain homotopy is analogous to the notion of homotopy between continuous maps in topology, but adapted to the algebraic setting of chain complexes