Let X be an arbitrary variety over a finite field k and p=char k,n∈ N. We will construct a complex of étale sheaves on X together with trace isomorphism from the highest étale cohomology group of this complex onto Z/p$^n$Z such that for every constructible Z/p$^n$Z-sheaf on X the Yoneda pairing is a nondegenerate pairing of finite groups. If X is smooth, this complex is the Gersten resolution of the logarithmic de Rham–Witt sheaf introduced by Gros and Suwa. The proof is based on the special case proven by Milne when the sheaf is constant and X is smooth, as well as on a purity theorem which in turn follows from a theorem about the cohomological dimension of C$_i$-fields due to Kato and Kuzumaki. If the existence of the Lichtenbaum complex is proven, the theorem will be the p-part of a general duality theorem for varieties over finite fields.