Published in Portugaliae Mathematica, Volume 67, Issue 4, January 1, 2010, pages 511-524. Copyright © 2010 European Mathematical Society. The definitive version is available at http://dx.doi.org/10.4171/PM/1875 .
For a general sequence of objects and morphisms, we construct two N-complexes. Then we can define cohomologies (i, k)-type of the N-complexes not only on a diagonal region but also in the triangular region. We obtain an invariant defined on a general sequence of objects and morphisms. For a short exact sequence of N-complexes, we get the associated long exact sequence generalizing the classical long exact sequence. Lastly, several properties of the vanishing cohomologies of N-complexes are given.