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...

(ii) Denote this map by φ, which is induced by the compositions with εi : Xi ↪→ ⊔ j Xj , i ∈ I. If φ(fi) = φ(gj) for some fi ∈ Hom(Y,Xi) and gj ∈ Hom(Y,Xj), then we must have i = j because φ(fi) and φ(gj) have the same image. Since φ(fi) = εi ◦ fi, φ(gi) = εi ◦ gi and εi is an injective map, we deduce that fi(y) = gi(y), ∀y ∈ Y , i.e. fi = gi. (iii) This map is induced by the composition with t...

We give a criterion for an HNN extension of a finite p-group to be residually p. 1. Statement of the Main Results By an HNN pair we mean a pair (G,φ) where G is a group and φ : A → B is an isomorphism between subgroups A and B of G. Given such an HNN pair (G,φ) we consider the corresponding HNN extension G∗ = 〈G, t | t−1at = φ(a), a ∈ A〉 of G, which we denote, by slight abuse of notation, as G∗...

Let R be a regular local ring, K its field of fractions and (V, φ) a quadratic space over R. In the case of R containing a field of characteristic zero we show that if (V, φ)⊗R K is isotropic over K, then (V, φ) is isotropic over R. 1 Characteristic zero case 1.0.1 Theorem (Main). Let R be a regular local ring, K its field of fractions and (V, φ) a quadratic space over R. Suppose R contains a f...

Let φ : V → V be a self-morphism of a quasiprojective variety defined over a number field K and let P ∈ V (K) be a point with infinite orbit under iteration of φ. For each prime p of good reduction, let mp(φ, P ) be the size of the φ-orbit of the reduction of P modulo p. Fix any ǫ > 0. We show that for almost all primes p in the sense of analytic density, the orbit size mp(φ, P ) is larger than...

The harmonic morphisms φ : M → Nn are studied using the methods of the moving frame and exterior differential systems and three main results are achieved. The first result is a local structure theorem for such maps in the case that φ is a submersion, in particular, a normal form is found for all such φ once the metric on the target manifold N is specified. The second result is a finiteness theo...

Definition 1.1. If X is a topological space with a sheaf of ring OX , an OX-module is a sheaf F of abelian groups on X, together with the structure of an OX(U)-module on each F (U), which is compatible with restriction, in the sense that if V ⊆ U , for any s ∈ F (U) and f ∈ OX(U) we have ρUV (fs) = ρUV (f) · ρUV (s) in F (V ). A morphism between OX -modules F and G is a morphism φ of the underl...

For any integer k > 2 , the infinite -bonacci word W ( ) on alphabet is defined as fixed point of morphism φ : N ⁎ → starting with 0, where i + j = { 1 if 0 … − . We obtain structure all square factors occurring in Moreover, we prove that critical exponent 3 Finally, provide

A run in a string is a periodic substring which is extendable neither to the left nor to the right with the same period. Strings containing many runs are of interest. In this paper, we focus on the series of strings {ψ(φ(a))}i≥0 generated by two kinds of morphisms, φ : {a, b, c} → {a, b, c}∗ and ψ : {a, b, c} → {0, 1}∗. We reveal a simple morphism φr plays a critical role to generate run-rich s...

One of the main notions in category theory is the notion of limit. Similarly, one of the most commonly used techniques in homotopy theory is the notion of “homotopy limit” commonly called “holim” for short. The purpose of the this paper is to begin to develop the notion of limit for n-categories, which should be a bridge between the categorical notion of limit and the homotopical notion of holi...