Note: Combinatorial Alexander Duality - A Short and Elementary Proof

Let X be a simplicial complex with ground set V . Define its Alexander dual as the simplicial complex X∗ = {σ ⊆ V | V \ σ / ∈ X}. The combinatorial Alexander duality states that the i-th reduced homology group of X is isomorphic to the (|V | − i − 3)-th reduced cohomology group of X∗ (over a given commutative ring R). We give a selfcontained proof from first principles, accessible to the non-expert.