Homological Algebra

Homological Algebra originated in algebraic topology but quickly became an essential tool in algebraic geometry and modern representation theory. It provides the machinery to measure “how far” a sequence of maps is from being exact, allowing us to compute invariants of complex mathematical objects.

Modules over a Ring

Before discussing homology, we must define the objects we are working with. A Module $M$ over a ring $R$ is a generalization of a vector space. While vector spaces must have a field as their scalars, modules allow for a ring $R$.

• If $R=\mathbb{Z}$, an $R$-module is simply an Abelian group.
• If $R=F$ (a field), an $R$-module is a vector space.

Chain Complexes

A Chain Complex $(C_{\bullet },d_{\bullet })$ is a sequence of modules and homomorphisms: $\cdots \to C_{n+1}\stackrel{d_{n+1}}{\to }{C}_n\stackrel{d_n}{\to }{C}_{n-1}\to \dots$ such that the composition of any two consecutive maps is zero: $d_n\circ d_{n+1}=0$ This property implies that the image of the incoming map is contained within the kernel of the outgoing map: $\text{Im}(d_{n+1})\subseteq \text{Ker}(d_n)$.

Homology Groups

The $n$-th Homology Group $H_n(C)$ is defined as the quotient: $H_n(C)=\frac{\text{Ker}(d_n)}{\text{Im}(d_{n+1})}$

• If $H_n(C)=0$, the sequence is said to be exact at $C_n$.
• Homology measures the failure of a sequence to be exact. In topology, this corresponds to “holes” in a space.

Exact Sequences

An Exact Sequence is a chain complex where all homology groups are zero.

• Short Exact Sequence (SES): A sequence of the form $0\to A\to B\to C\to 0$ where $A$ is a submodule of $B$, and $C\cong B/A$.
• Long Exact Sequence: Given a short exact sequence of complexes, we can derive a long exact sequence connecting their homology groups using the Snake Lemma.

Projective and Injective Resolutions

One of the central techniques in Homological Algebra is “resolving” a module $M$ by mapping simpler modules into it.

• Projective Resolution: A long exact sequence $\cdots \to P_1\to P_0\to M\to 0$ where each $P_i$ is a projective module (generalization of a free module).
• These resolutions allow us to define Derived Functors, such as Ext (measuring extensions) and Tor (measuring torsion).

The Ext and Tor Functors

These are the fundamental “hidden” invariants of modules:

1. ${\text{Tor}}_n^R(A,B)$: Derived from the tensor product functor. It measures “torsion” or dependencies between elements of $A$ and $B$.
2. ${\text{Ext}}_R^n(A,B)$: Derived from the Hom functor. It measures how many ways we can “extend” $B$ by $A$.

Python: Simulating a Chain Complex

We can represent modules as matrices and checks the $d^2=0$ condition using linear algebra (over a field).

import numpy as np

def is_chain_complex(d1, d2):
    """
    Checks if d1 followed by d2 is a zero map.
    d1: C_{n+1} -> C_n
    d2: C_n -> C_{n-1}
    """
    product = np.dot(d2, d1)
    return np.allclose(product, 0)

# Example: Boundary maps in a 2D complex
d1 = np.array([[1], [-1]]) # From an edge to its two vertices
d2 = np.array([1, 1])      # Conceptual d2

print(f"Is chain complex: {is_chain_complex(d1, d2)}")
# Note: In real homology, d2 * d1 must be zero.

Significance

Homological Algebra provides the language for “diagram chasing,” a method of proof where we follow the path of elements through a grid of maps to prove commutativity or exactness. It is the backbone of modern structural mathematics, turning geometric intuition into algebraic calculation.

Exercise