Algebraic geometry over algebraic structures. II. Foundations

In this paper, we introduce elements of algebraic geometry over an arbitrary algebraic structure. We prove so-called unification theorems that describe coordinate algebras of algebraic sets in several different ways.     

Stein domains in Banach algebraic geometry

In this article we give a homological characterization of the topology of Stein spaces over any valued base field. In particular, when working over the field of complex numbers, we obtain a characterization of the usual Euclidean (transcendental) topology of complex analytic spaces. For non-Archimedean base fields the topology we characterize coincides with the topology of the Berkovich analytic space associated to a non-Archimedean Stein algebra. Because the characterization we used is borrowed from a definition in derived geometry, this work should be read as a derived perspective on analytic geometry.     

Algebraic patching over complete domains

We extend the method of algebraic patching due to Haran-Jarden-Völklein from complete absolute valued fields to complete domains. We apply the extended method to reprove a result of Lefcourt obtained by formal patching — every finite group is regularly realizable over the quotient field of a complete domain.     

Algebraic complexities and algebraic curves over finite fields

Algebraic schemes of computation of bilinear forms over various rings of scalars are examined. The problem of minimal complexity of these schemes is considered for computation of polynomial multiplication and multiplication in commutative algebras, and finite extensions of fields. While for infinite fields minimal complexities are known (Winograd, Fiduccia, Strassen), for finite fields precise minimal complexities are not yet determined. We prove lower and upper bounds on minimal complexities. Both are linear in the number of inputs. These results are obtained using the relationship with linear coding theory and the theory of algebraic curves over finite fields.     

Equations and algebraic geometry over profinite groups

The notion of an equation over a profinite group is defined, as well as the concepts of an algebraic set and of a coordinate group. We show how to represent the coordinate group as a projective limit of coordinate groups of finite groups. It is proved that if the set π(G) of prime divisors of the profinite period of a group G is infinite, then such a group is not Noetherian, even with respect to one-variable equations. For the case of Abelian groups, the finiteness of a set π(G) gives rise to equational Noetherianness. The concept of a standard linear pro-p-group is introduced, and we prove that such is always equationally Noetherian. As a consequence, it is stated that free nilpotent pro-p-groups and free metabelian pro-p-groups are equationally Noetherian. In addition, two examples of equationally non-Noetherian pro-p-groups are constructed. The concepts of a universal formula and of a universal theory over a profinite group are defined. For equationally Noetherian profinite groups, coordinate groups of irreducible algebraic sets are described using the language of universal theories and the notion of discriminability.     

Algebraic leaves of algebraic foliations over number fields

Summary — We prove an algebraicity criterion for leaves of algebraic foliations defined over number fields. Namely, consider a number field K embedded in C, a smooth algebraic variety X over K, equipped with a K-rational point P, and F an algebraic subbundle of the its tangent bundle T_X, defined over K. Assume moreover that the vector bundle F is involutive, i.e., closed unter Lie bracket. Then it defines an holomorphic foliation of the analytic mainfold X(C), and one may consider its leaf ℱ through P. We prove that ℱ is algebraic if the following local conditions are satisfied: i) For almost every prime ideal p of the ring of integers 𝒪_K of the number field K, the p-curvature of the reduction modulo p of the involutive bundle F vanishes at P (where p denotes the characteristic of the residue field 𝒪_K / p ). ii) The analytic manifold ℱ satisfies the Liouville property; this arises, in particular, if ℱ is the image by some holomorphic map of the complement in a complex algebraic variety of a closed analytic subset. This algebraicity criterion unifies and extends various results of D. V. and G. V. Chudnovsky, André, and Graftieaux, and also admits new consequences. For instance, applied to an algebraic group G over K, it shows that a K-Lie subalgebra h of Lie G is algebraic if and only if for almost every non-zero prime ideal p of 𝒪_K , of residue characteristic p, the reduction modulo p of h is a restricted Lie subalgebra of the reduction modulo p of Lie G (i.e., is stable under p-th powers). This solves a conjecture of Ekedahl and Shepherd-Barron. The algebraicity criterion above follows from a more basic algebraicity criterion concerning smooth formal germs in algebraic varieties over number fields. The proof of the latter relies on “transcendence techniques”, recast in a modern geometric version involving elementary concepts of Arakelov geometry, and on some analytic estimates, related to the First Main Theorem of higher-dimensional Nevanlinna theory.

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Résumé — Nous établissons un critère d'algébricité concernant les feuilles des feuilletages algébriques définis sur un corps de nombres. Soit en effet K un corps de nombres plongé dans C, X une variété algébrique lisse sur K, munie d'un point K-rationnel P, et F un sous-fibré du fibré tangent T_X, défini sur K. Supposons de plus que le fibré vectoriel F soit involutif, i.e.., stable par crochet de Lie. Il définit alors un feuilletage holomorphe de la variété analytique X(C) et l'on peut considérer la feuille ℱ de ce feuilletage passant par P. Nous montrons que ℱ est algébrique lorque les conditions locales suivantes son satisfaites: i) Pour presque tout idéal premier p de l'annneau des entiers 𝒪_K de K, la réduction modulo p du fibré F est stablé par l'opération de puissance p-ième (où p désigne la caractéristique du corps résiduel 𝒪_K / p ). ii) La variété analytique ℱ satisfait à la propriété de Liouville; cela a lieu, par exemple, lorsque ℱ est l'image par une application holomorphe du complémentaire d'un sous-ensemble analytique fermé dans une variété algébrique. Ce critère d'algébricité unifie et généralise divers résultats de D. V. and G. V. Chudnovsky, André et Graftieaux. Il conduit aussi à de nouvelles conséquences. Par exemple, appliqué à un groupe algébrique G sur K, il montre qu'une sous-algèbre de Lie h de Lie G, définie sur K, est algébrique si et seulement si, pour presque tout idéal premier p de 𝒪_K , de caractéristique résiduelle p, la réduction modulo p de h est une sous-p-algèbre de Lie de la réduction modulo p de Lie G (i.e., est stable par puissance p-ième). Cet énoncé résout une conjecture d'Ekedahl et Shepherd-Barron. Le critère d'algébricité ci-dessus découle d'un critère d'algébricité plus général, concernant les germes de sous-variétés formelles des variétés sur les corps de nombres. La démonstration de ce dernier repose sur des “techniques de transcendance”, reformulées dans une version géométrique utilisant diverses notions élémentaires de géométrie d'Arakelov, et sur des estimations analytiques reliées au premier théorème fondamental de la théorie de Nevanlinna en dimension supérieure.

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Manucsrit re?u le 27 septembre 2000.     

Moduli of curves and spin structures via algebraic geometry

Here we investigate some birational properties of two collections of moduli spaces, namely moduli spaces of (pointed) stable curves and of (pointed) spin curves. In particular, we focus on vanishings of Hodge numbers of type and on computations of Kodaira dimension. Our methods are purely algebro-geometric and rely on an induction argument on the number of marked points and the genus of the curves.

     

Algebraic structures determined by 3 by 3 matrix geometry

Using a ``3 by 3 matrix trick' we show that multiplication (an algebraic structure) in a *-algebra is determined by the geometry of the *-algebra of the 3 by 3 matrices with entries from , . This is an example of an algebra-geometry duality which, we claim, has applications.

     

Algebraic geometry over free metabelian lie algebras. I. U-algebras and universal classes

This paper is the first in a series of three, the object of which is to lay the foundations of algebraic geometry over the free metabelian Lie algebra F. In the current paper, we introduce the notion of a metabelian U-Lie algebra and establish connections between metabelian U-Lie algebras and special matrix Lie algebras. We define the Δ-localization of a metabelian U-Lie algebra A and the direct module extension of the Fitting radical of A and show that these algebras lie in the universal closure of A. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 37–63, 2003.     

Arrow's theorem: unusual domains and extended codomains

In this paper we establish Arrow's theorem in a general ordinal case. When some configurations are allowed in the domain and if this domain is included in the codomain, the only social functions satisfying the independence condition and the weak Pareto Principle are the absolute dictatorships or the absolute oligarchies.     

Algebraic geometry over the residue field of the infinite place

Nikolai Durov introduced the theory of generalized rings and schemes to study Arakelov geometry in an alternative algebraic framework, and introduced the residue field at the infinite place, 𝔽_∞. We show an elementary algebraic approach to modules and algebras over this object, define prime congruences, show that the polynomial ring of n variables is of Krull dimension n, and derive a prime decomposition theorem for these primes.     

Algebraic geometry and soliton dynamics

Algebro-geometric sectors of solutions of the KP hierarchy are described in terms of τ-functions and vertex operators. Some useful identities involving theta functions and prime forms on Riemann surfaces are provided which are applied to obtain explicit solutions in the bilinear formalism. By using a dressing method for τ-functions the soliton dynamics against the background of quasiperiodic solutions is characterized. Furthermore, a formula for the soliton shifts in terms of prime forms on Riemann surfaces is obtained.     

Categorical Abstract Algebraic Logic: Ordered Equational Logic and Algebraizable PoVarieties

A syntactic apparatus is introduced for the study of the algebraic properties of classes of partially ordered algebraic systems (a.k.a. partially ordered functors (pofunctors)). A Birkhoff-style order HSP theorem and a Mal’cev-style order SLP theorem are proved for partially ordered varieties and partially ordered quasivarieties, respectively, of partially ordered algebraic systems based on this syntactic apparatus. Finally, the notion of a finitely algebraizable partially-ordered quasi-variety, in the spirit of Pałasińska and Pigozzi, is introduced and some of the properties of these quasi-povarieties are explored in the categorical framework.