FP-injective objects in the category of N-complexes

We show that an N-complex of modules C is FP-injective if and only if C is N-exact and $$\textrm{Z}_n^i(C)$$ Z n i ( C ) is an FP-injective module for each $$i=1,2,\cdots ,N$$ i = 1 , 2 , ⋯ , N and each $$n\in \mathbb {Z}$$ n ∈ Z by virtue of Gaussian binomial coefficients. Applications of this result go in three directions. Firstly, over a coherent ring, we prove that a bounded above N-complex C is FP-injective if and only if C is N-exact and $$C_n$$ C n is an FP-injective module for each $$n\in \mathbb {Z}$$ n ∈ Z . Secondly, we obtain some examples of FP-injective N-complexes for some fixed integer N. Finally, we give a characterization of coherent rings.