In a cochain complex of modules
the module of -coboundaries is the image of . It is a submodule of and is contained in the module of -cocycles .
The cochain complex is called exact at if .
In the right complex of -modules
for all , the th module is , and the th coboundary operator maps every element of to the residue class of in . The module of -coboundaries is the set of the residue classes of 0 and in , and the module of -cocycles the set of the residues classes of all even numbers .