Prime numbers of bad reduction for dessins of genus 0

This article expands the special case of the Grothendieck theory to arbitrary fields. A formal definition of Belyi function over an arbitrary field is introduced. It turns out that the properties of Belyi functions over finite fields and the properties of classical Belyi functions are quite different. A definition of the primes of bad reduction is also given, and the primes of bad reduction are calculated for some dessin families. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 25–43, 2005.     

Proprieétés du degré des morphismes de Belyi

Properties of the degree of Belyi functions. A famous theorem of Belyi characterizes the curves defined over a number field by the existence of an element of its function field with certain ramification properties. In this article we are interested in the degree of these functions. We define the Belyi degree of a curve defined over a number field and the Belyi degree of a point on such a curve. We prove finiteness results concerning these invariants. We give an explicit upper bound for the Belyi degree of a point on the projective line, depending on the height and on the degree of its field of definition.     

Poincaré duality algebras mod two

We study Poincaré duality algebras over the field F₂ of two elements. After introducing a connected sum operation for such algebras we compute the corresponding Grothendieck group of surface algebras (i.e., Poincaré algebras of formal dimension 2). We show that the corresponding group for 3-folds (i.e., algebras of formal dimension 3) is not finitely generated, but does have a Krull-Schmidt property.We then examine the isomorphism classes of 3-folds with at most three generators of degree 3, provide a complete classification, settle which such occur as the cohomology of a smooth 3-manifold, and list separating invariants.The closing section and Appendix A provide several different means of constructing connected sum indecomposable 3-folds.     

Proprieétés du degré des morphismes de Belyi

Properties of the degree of Belyi functions. A famous theorem of Belyi characterizes the curves defined over a number field by the existence of an element of its function field with certain ramification properties. In this article we are interested in the degree of these functions. We define the Belyi degree of a curve defined over a number field and the Belyi degree of a point on such a curve. We prove finiteness results concerning these invariants. We give an explicit upper bound for the Belyi degree of a point on the projective line, depending on the height and on the degree of its field of definition.     

On Families of Geometric Parasitic Solutions for Belyi Systems of Genus Zero

This paper is devoted to interrelations between the combinatorial structure of plane graphs and algebraic properties of systems of equations related to Belyi functions for these graphs. The main purpose is to describe some families of plane graphs possessing parasitic solutions and some families of plane graphs such that each solution of the corresponding systems is not parasitic and has multiplicity one.__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 103–111, 2003.     

关于Grothendieck群的几点注记

作者在本文中围绕Ｇｒｏｔｈｅｎｄｉｅｃｋ群对几个问题进行了讨论，主要结果有：１．一切有限生成Ｒ－模的同构类作成一个集合；２．在任意由Ｒ－模作成的集合中稳定同的关系是合同关系；３．Ｇｒｏｔｈｅｎｄｉｅｃｋ群的同构不变性成立。     

Covariant Isotropy of Grothendieck Toposes and Extensive Categories

We provide an explicit characterization of the covariant isotropy group of any Grothendieck topos, i.e. the group of (extended) inner automorphisms of any sheaf over a small site. In order to do so, we first extend previous techniques for computing covariant isotropy from locally finitely presentable categories to locally presentable categories. As a consequence, we also obtain an explicit characterization of the centre of a Grothendieck topos, i.e. the automorphism group of its identity functor. We conclude by providing a more categorical approach to show that these characterizations also extend to any extensive category.

     

Belyi pairs corresponding to dessins d’enfants of genus 3

The paper is focused on a question arising in the dessins d??enfants theory: we are interested in finding the determining equations for the Belyi pair of a given dessin. The low genus (g ?? 2) cases are well-studied, there exist many examples of Belyi pairs of those genera and some efficient methods have been developed. We present a method allowing one to find Belyi pairs of self-dual genus 3 dessins.     

On Real Forms of Belyi Surfaces With Symmetric Groups of Automorphisms

In virtue of the Belyi Theorem an algebraic curve can be defined over the algebraic numbers if and only if the corresponding Riemann surface can be uniformized by a subgroup of a Fuchsian triangle group. Such surfaces are known as Belyi surfaces. Here we study the actions of the symmetric groups S _n on Belyi Riemann surfaces. We show that such surfaces are symmetric and we calculate the number of connected components of the corresponding real forms.     

Grothendieck Inclusion Systems

Inclusion systems have been introduced in algebraic specification theory as a categorical structure supporting the development of a general abstract logic-independent approach to the algebra of specification (or programming) modules. Here we extend the concept of indexed categories and their Grothendieck flattenings to inclusion systems. An important practical significance of the resulting Grothendieck inclusion systems is that they allow the development of module algebras for multi-logic heterogeneous specification frameworks. At another level, we show that several inclusion systems in use in some syntactic (signatures, deductive theories) or semantic contexts (models) appear as Grothendieck inclusion systems too. We also study several general properties of Grothendieck inclusion systems.     

On Belyi pairs over arbitrary fields

The main goal of this article is to extend Grothendieck’s dessins d’enfant theory to arbitrary fields. In this paper, the definitions of a Belyi pair in positive characteristic and primes of bad reduction are given. We consider the graph K _3,3. This abstract graph corresponds to three different dessins. For each dessin we find the Belyi pair and the positive characteristics for which this pair exists. The set of primes of bad reduction is also given. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 3, pp. 3–8, 2006.     

Composite genus one Belyi maps

Motivated by a demand for explicit genus 1 Belyi maps from theoretical physics, we give an efficient method of explicitly computing genus one Belyi maps by (1) composing covering maps from elliptic curves to the Riemann sphere with simpler (univariate) genus zero Belyi maps, as well as by (2) composing further with isogenies between elliptic curves. The computed examples of genus 1 Belyi maps has doubly-periodic dessins d’enfant that are listed in the physics literature as so-called brane-tilings in the context of quiver gauge theories.     

A local homology theory for linearly compact modules

We introduce a local homology theory for linearly compact modules which is in some sense dual to the local cohomology theory of A. Grothendieck [A. Grothendieck, Local Cohomology, Lecture Notes in Math., vol. 20, Springer-Verlag, Berlin/Tokyo/New York, 1967. [10]]. Some basic properties such as the noetherianness, the vanishing and non-vanishing of local homology modules of linearly compact modules are proved. A duality theory between local homology and local cohomology modules of linearly compact modules is developed by using Matlis duality and Macdonald duality. As consequences of the duality theorem we obtain some generalizations of well-known results in the theory of local cohomology for semi-discrete linearly compact modules.     

Geometric Phantom Categories

In this paper we give a construction of phantom categories, i.e. admissible triangulated subcategories in bounded derived categories of coherent sheaves on smooth projective varieties that have trivial Hochschild homology and trivial Grothendieck group. We also prove that these phantom categories are phantoms in a stronger sense, namely, they has trivial K-motives and, hence, all their higher K-groups are trivial too.     

Left-exact localizations of ∞-topoi II: Grothendieck topologies

We revisit the work of Toën–Vezzosi and Lurie on Grothendieck topologies, using the new tools of acyclic classes and congruences. We introduce a notion of extended Grothendieck topology on any ∞-topos, and prove that the poset of extended Grothendieck topologies is isomorphic to that of topological localizations, hypercomplete localizations, Lawvere–Tierney topologies, and covering topologies (a variation on the notion of pretopology). It follows that these posets are small and have the structure of a frame. We revisit also the topological–cotopological factorization by introducing the notion of a cotopological morphism. And we revisit the notions of hypercompletion, hyperdescent, hypercoverings and hypersheaves associated to an extended Grothendieck topology.We also introduce the notion of forcing, which is a tool to compute with localizations of ∞-topoi. We use this in particular to show that the topological part of a left-exact localization of an ∞-topos is universally forcing the generators of this localization to be ∞-connected instead of inverting them.     

Precosheaves of pro-sets and abelian pro-groups are smooth

Let D be the category of pro-sets (or abelian pro-groups). It is proved that for any Grothendieck site X, there exists a reflector from the category of precosheaves on X with values in D to the full subcategory of cosheaves. In the case of precosheaves on topological spaces, it is proved that any precosheaf is smooth, i.e. is locally isomorphic to a cosheaf. Constant cosheaves are constructed, and there are established connections with shape theory.     

Homological Finiteness Conditions for Groups,Monoids, and Algebras

Let ? be an additive category and 𝒞 a full subcategory with split idempotents, and closed under isomorphic images and finite direct sums. We give conditions on ? and 𝒞 implying that ? embeds into an abelian category, so that the objects of 𝒞 turn into injective objects. This construction generalizes the embedding of exactly definable categories into locally coherent categories, while the dual construction generalizes the embedding of finitely accessible categories into Grothendieck categories with a family of finitely generated projective generators. As applications, we characterize exactly definable categories through intrinsic properties and study those locally coherent categories whose fp-injective objects form a Grothendieck category.     

The computation of Belyi pairs of 6-edged dessins d’enfants of genus 3 with automorphism groups of order 12 and 3

In this article, we study dessins d’enfants of genus 3 with six edges and only one vertex, having groups of automorphisms of order at least 3. We find all such dessins and explain the method of the enumeration. For each of them the Belyi pair is computed. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 6, pp. 137–148, 2007.     

On the dual positive Schur property in Banach lattices

The paper contains several characterizations of Banach lattices $E$ with the dual positive Schur property (i.e., $0 \le f_n \xrightarrow {\sigma (E^*,E)} 0$ implies $\Vert f_n\Vert \rightarrow 0$ ) and various examples of spaces having this property. We also investigate relationships between the dual positive Schur property, the positive Schur property, the positive Grothendieck property and the weak Dunford–Pettis property.