Smoothness in Relative Geometry

In [TVa], Bertrand Toën and Michel Vaquié defined a scheme theory for a closed monoidal category ( ⊗1). In this article, we define a notion of smoothness in this relative (and not necessarily additive) context which generalizes the notion of smoothness in the category of rings. This generalisation consists in replacing homological finiteness conditions by homotopical ones, using the Dold-Kan correspondence. To do this, we provide the category s of simplicial objects in a monoidal category and all the categories sA-mod, sA-alg (A ∈ sComm()) with compatible model structures using the work of Rezk [R]. We then give a general notion of smoothness in sComm(). We prove that this notion is a generalisation of the notion of smooth morphism in the category of rings and is stable under composition and homotopy pushouts. Finally we provide some examples of smooth morphisms, in particular in ℕ-alg and Comm(Set).

Covers inp-adic analytic geometry and log covers II: cospecialization of the (p′)-tempered fundamental group in higher dimensions

The tempered fundamental group of ap-adic variety classifies analytic étale covers that become topological covers for Berkovich topology after pullback by some finite étale cover. This paper constructs cospecialization homomorphisms between the (p′) versions of the tempered fundamental group of the fibers of a smooth morphism with polystable reduction. We study the question for families of curves in another paper. To construct them, we will start by describing the pro-(p′) tempered fundamental group of a smooth and proper variety with polystable reduction in terms of the reduction endowed with its log structure, thus defining tempered fundamental groups for log polystable varieties.

SMOOTH MORPHISMS OF MODULE SCHEMES

Let $\varphi:A\to B$ be a homomorphism of finite-dimensional algebras over an algebraically closed field and $\varphi^{(c)}:{\rm mod}_B^c\to{\rm mod}_A^c$ the induced morphism of the associated module schemes for any integer $c\geq 1$. We prove that if the induced functor ${\rm mod\,} B\to{\rm mod\,} A$ is hom-controlled then the restriction of $\varphi^{(c)}$ to any connected component of ${\rm mod}_B^c$ is a composition of a smooth morphism followed by an immersion. Some new results on the types of singularities in the orbit closures of module schemes are also proved.2000 Mathematical Subject Classification: 14B05, 14L30, 16G10.