For a given Galois structure on a category C and an eeective descent morphism p : E !B in C we describe the category of so-called weakly split objects over (E; p) in terms of internal actions of the Galois (pre)groupoid of (E; p) with an additional structure. We explain that this generates various known results in categorical Galois theory and in particular two results of M. Barr and R. Diacone...

For us, G-bundles are taken in the fppf topology (or, equivalently, the étale topology). Remember that BunG depends on the choice of X, even though the notation suppresses this! Last time, Francois proved that BunGLn is an algebraic stack: • It satisfies descent in the fppf (equivalently, étale) topology; i.e., it’s a stack; • Its diagonal morphism is representable by an algebraic space (implyi...

In this paper we construct a pushforward-pullback adjoint pair for categories of quasi-coherent sheaves, along a morphism of algebraic stacks, which is represented in algebraic stacks over the site C = Affflat. The construction uses the characterization of algebraic stacks of [H3] and is based on the descent description of the category of quasi-coherent sheaves given in [H2]. We show that an es...

Let R be an integral domain of finite type over Z and let f : X → SpecR be a smooth projective morphism of relative dimension d ≥ 1. We investigate, for a vector bundle E on the total space X , under what arithmetical properties of a sequence (pn, en)n∈N, consisting of closed points pn in SpecR and Frobenius descent data Epn ∼= F n(F) on the closed fibers Xpn , the bundle E0 on the generic fibe...

We generalize the motivic incarnation morphism from the theory of arithmetic integration to the relative case, where we work over a base variety S over a field k of characteristic zero. We develop a theory of constructible effective Chow motives over S, and we show how to associate a motive to any S-variety. We give a geometric proof of relative quantifier elimination for pseudo-finite fields, ...