Absolutely Clean

The notion of absolutely clean N-complexes is studied. We show that an N-complex X is absolutely clean if and only if X is N-exact and Zni(X) is an absolutely clean module for each n∈ℤ and i=1,2,⋯,N. In particular, we prove that a bounded above N-complex X is absolutely clean if and only if Xn is an absolutely clean module for each n∈ℤ. We also show that under certain hypotheses, an N-complex X is Gorenstein AC-injective if and only if Znt(X) is a Gorenstein AC-injective module for each n∈ℤ and t=1,2,⋯,N.