Koszul Complex - Theorem

Theorem

Let (R, m) be a Noetherian local ring with maximal ideal m, and let M be a finitely-generated R-module. If x₁, x₂, ..., x_n are elements of the maximal ideal m, then the following are equivalent:

1. The (x_i) form a regular sequence on M,
2. H₁(K_•(x_i)) = 0,
3. H_j(K_•(x_i)) = 0 for all j ≥ 1.