A word is cubefree if it contains no non-empty subword of the form xxx. A morphism h : Σ * → Σ * is k-uniform if h(a) has length k for all a ∈ Σ. A morphism is cubefree if it maps cubefree words to cubefree words. We show that for all k ≥ 0 there exists a k-uniform cubefree binary morphism.

Proposition 1.1. (a) Any open immersion is étale. (b) The composite of two étale morphisms is étale. (c) Any base change of an étale morphism is étale. (d) If φ ◦ ψ and φ are étale, then so is ψ. Proposition 1.2. Let f : X → Y be an étale morphism. (a) For all x ∈ X, OX,x and OY,f(x) have the same Krull dimension. (b) The morphism f is quasi-finite. (c) The morphism f is open. (d) If Y is reduc...

a toroidalization of a dominant morphism $varphi: xto y$ of algebraic varieties over a field of characteristic zero is a toroidal lifting of $varphi$ obtained by performing sequences of blow ups of nonsingular subvarieties above $x$ and $y$. we give a proof of toroidalization of locally toroidal morphisms of 3-folds.

In this paper C denotes a category and o1, o2, o3 denote objects of C. Let C be a non empty category structure with units and let o be an object of C. Observe that 〈o, o〉 is non empty. The following propositions are true: (1) Let v be a morphism from o1 to o2, u be a morphism from o1 to o3, and f be a morphism from o2 to o3. If u = f · v and f −1 · f = id(o2) and 〈o1, o2〉 6= ∅ and 〈o2, o3〉 6= ∅...

Abstract. We introduce the concept of a morphism between coloured nets. Our definition generalizes Petris definition for ordinary nets. A morphism of coloured nets maps the topological space of the underlying undirected net as well as the kernel and cokernel of the incidence map. The kernel are flows along the transitionbordered fibres of the morphism, the cokernel are classes of markings of th...

0.1. The problem. Roughly speaking, the semistable reduction problem we address here asks for the following: Let X → B be a surjective morphism of complex projective varieties with geometrically integral generic fiber. Find a generically finite proper surjective morphism (that is, an alteration) B1 → B, and a proper birational morphism (that is, a modification) Y → X ×B B1, such that the morphi...

We continue our study on infinitesimal lifting properties of maps between locally noetherian formal schemes started in [3]. In this paper, we focus on some properties which arise specifically in the formal context. In this vein, we make a detailed study of the relationship between the infinitesimal lifting properties of a morphism of formal schemes and those of the corresponding maps of usual s...