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.

We give a combinatorial formula for the weight multiplicities of some infinite-dimensional highest weight ${\mathfrak g}$${\mathfrak l}$ (n)-modules. Our proof, which does not rely on Kazhdan–Lusztig combinatorics, uses a reduction to finite characteristics. The character formula for the corresponding modular representations, which has been computed in a 1997 preprint by the authors, is based on a dual pair which has no obvious counterpart in characteristic zero.

Let Λ be a finite-dimensional hereditary algebra over a finite field k, ${\mathcal H}$(Λ) and ${\mathcal C}$(Λ) be, respectively, the Hall algebra and the composition algebra of Λ, ${\mathcal P}$ be the isomorphism classes of finite dimensional Λ-modules and I the isomorphism classes of simple Λ-modules. We define δ$_α$ and $_α$ δ, α in ${\mathcal P}$, to be the right and left derivations of ${\mathcal H}$(Λ) respectively. By using these derivations and the action of the braid group on the set of exceptional sequences of Λ-mod, we provide an effective algorithm of calculating the root vectors of real Schur roots. This means that we get an inductive method to express u$_λ$ as the combinations of elements u$_i$ in the Hall algebra, where i ∈ I and λ in ${\mathcal P}$ is any exceptional Λ-module. Because of the canonical isomorphism between the Drinfeld–Jimbo quantum group and the generic composition algebra, our algorithm is applicable directly to quantum groups. In particular, all the root vectors are obtained in this way in the finite type cases.

Let k be a perfect field of characteristic p > 0, K$_0$ = Frac(W(k)), π a uniformizer in K$_0$ and π$_n$ ∈ K $_0$ (n∈ N) such that π$_0$ = π and π$^p$$_n$ $_ + 1$ = π$_n$. We write K$_∞$ = ∪$_n$ $_∈N$ K$_0$ (π$_n$), H$_∞$ = Gal (K$_0$/ K$_∞$ and G = Gal(K$_0$/ K$_0$). The main result of this paper is that the functor ’restriction of the Galois action‘ from the category of crystalline representations of G with Hodge–Tate weights in an interval of length [les ] p − 2 to the category of p-adic representations of H$_∞$ is fully faithful and its essential image is stable by sub-object and quotient. The proof uses the comparison between two ways of building mod. p representations of H$_∞$: one thanks to the norm field of K$_∞$, the other thanks to some categories of ’filtered‘ modules with divided powers previously introduced by the author.

We establish sufficient conditions for a biextension algebra of a piecewise hereditary algebra of type Ā$_n$ or D$_n$ by indecomposable modules of derived regular length 2 to be of tame representation type.

We introduce and describe the characteristic class of a difference operator over the difference field (k((t)),τ). Here k is an algebraically closed field of characteristic zero and τ is the k-linear automorphism of k((t)) defined by τ(t)=t/(1+t). The approach is based on the characterization of simple difference operators in terms of their eigenvalues.