数学上，同调代数领域中的一个链复形是一个交换群或者模的串行A₀, A₁, A₂... 通过一系列同态d_n : A_n→A_n-1相连，使得每两个连接的映射的复合为零：d_n o d_n+1 = 0对于所有n。它们常常写作如下形式：

定义链复形的同调群为 。当所有同调群为零时，此链复形为正合的。

链复形概念的一个变种是上链复形。一个上链复形是一个交换群或者模的串行A₀, A₁, A₂...由一系列同态d_n : A_n→A_n+1相连，使得任何两个接连的映射的复合为零：d_n+1 o d_n = 0 对于所有的n:

In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of a topological space. More generally, homological algebra includes the study of chain complexes in the abstract, without any reference to an underlying space. In this case, chain complexes are studied axiomatically as algebraic structures.