In a chain complex of modules ...->C_(i + 1) ->^(d_(i + 1)) C_i ->^(d_i) C_(i - 1)->..., the module B_i of i-boundaries is the image of d_(i + 1). It is a submodule of C_i and is contained in the module of i-cycles Z_i, which is the kernel of d_i. The complex is called exact at C_i if B_i = Z_i. In the chain complex ...->Z_8 ->^(·4) Z_8 ->^(·4) Z_8->... where all boundary operators are the multiplication by 4, for all i the module of i-boundaries is B_i = {0^_, 4^_}, whereas the module of i-cycles is Z_i = {0^_, 2^_, 4^_, 6^_}.

boundary operator | chain complex | coboundary | homology