Homological Algebra

Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a powerful tool used in algebraic topology, algebraic geometry, and number theory.

Chain Complexes

A chain complex $(C_\bullet, d_\bullet)$ is a sequence of abelian groups (or modules) and homomorphisms: $\dots \xrightarrow{d_{n+1}} C_n \xrightarrow{d_n} C_{n-1} \xrightarrow{d_{n-1}} \dots$ such that the composition of any two consecutive maps is zero: $d_n \circ d_{n+1} = 0$.

This condition implies that the image of $d_{n+1}$ is contained in the kernel of $d_n$: $im(d_{n+1}) \subseteq ker(d_n)$.

import numpy as np

def is_chain_complex(matrices):
  """Check if a sequence of boundary matrices forms a chain complex."""
  for i in range(len(matrices) - 1):
      # Result of d_n * d_{n+1} should be zero matrix
      prod = np.matmul(matrices[i], matrices[i+1])
      if not np.allclose(prod, 0):
          return False, i
  return True, None

# Example: d1 * d2 = 0
d1 = np.array([[1, -1, 0], [0, 1, -1]]) # Edges to vertices
d2 = np.array([[1], [1], [1]])          # Triangle face to edges

is_valid, index = is_chain_complex([d1, d2])
print(f"Is chain complex: {is_valid}")

Homology Groups

The $n$-th homology group is the quotient group: $H_n(C) = \frac{ker(d_n)}{im(d_{n+1})}$ Elements of $ker(d_n)$ are called cycles, and elements of $im(d_{n+1})$ are called boundaries. Homology measures the extent to which the complex is not “exact.”

Exact Sequences

A sequence is exact if the image of each map is exactly the kernel of the next. A short exact sequence is a sequence of the form: $0 \to A \xrightarrow{f} B \xrightarrow{g} C \to 0$ which implies that $f$ is injective, $g$ is surjective, and $im(f) = ker(g)$.

Diagram Chasing and the Snake Lemma

Diagram chasing is a method of proof used in homological algebra to establish properties of morphisms by tracing elements through commutative diagrams. The Snake Lemma is a fundamental result that relates the kernels and cokernels of two morphisms between short exact sequences.