An étale tate twist with finite coefficients and duality in mixed characteristics

is an isomorphism (the relative Poincaré duality, [SGA4], XVII, XVIII). In this paper, we will give a generalization of this duality theory to the case where B is the spectrum of a Dedekind ring of mixed characteristic (e.g., Spec Z), and where n is not invertible on either B or X. More precisely, we will not need the smoothness of f but assume only that X is regular and semistable over B around fibers where n is not invertible. Contrary to the case where n is invertible on X, the usual Tate twist (i.e., the étale sheaf μ n,X ) does not work well in our situation. Therefore we will need to discuss, first of all, a correct object in D(Xét, Z/nZ) playing the role of a Tate twist.