Definition. Chain complex [O7WN]

2026-10-04 · Lîm Tsú-thuàn

An algebraic chain complex (C∙,∂∙)(C_\bullet, \partial_\bullet) is a sequence of RR-modules over a commutative ring RR

⋯→Cn+1→∂n+1Cn→∂nCn−1→⋯\cdots \to C_{n+1} \xrightarrow{\partial_{n+1}} C_n \xrightarrow{\partial_n} C_{n-1} \to \cdots

indexed over n∈Zn \in \mathbb{Z} and connected by homomorphisms ∂n\partial_n satisfying the condition ∂n∘∂n+1=0\partial_n \circ \partial_{n+1} = 0Every nn-boundary is an nn-cycle.