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We say a lattice L is a subgroup lattice if there exists a group G such that Sub(G)≅L, where Sub(G) is the lattice of subgroups of G, ordered by inclusion. We prove that the lattice of algebraic cl...

Given a prime number p we consider C p the topological completion of the algebraic closure of the field of p-adic numbers with respect to the usual p-adic absolute value. In this paper, we point ou...

Let G be a simply connected, semisimple algebraic group of type B4 or D4 over an algebraically closed field of characteristic p > 0. We determine the characters of certain simple modules for these groups by calculating the composition factors of the Weyl modules.

We study algebraic properties of the Wiener algebra of absolutely convergent power series on the closed unit disc. In particular, we prove a form of Weierstrass preparation for algebraic functions in this algebra.

Let (U; F) be an idempotent algebra. There is an r-ary essentially algebraic operation in F where there is not any (r + 3)-ary algebraic operation depending on at least r + 1 variables. In this paper, we prove that the set of all 4-ary algebraic operations of this algebras forms a finite De Morgan algebra, and then we characterize this De Morgan algebra.

We prove the following theorem: Let 𝔽 be an algebraic closure of a finite field of characteristic p. Let ρ be a continuous homomorphism from the absolute Galois group of ℚ to GL(n, 𝔽) which is isomorphic to a direct sum of one-dimensional representations χ i where p > n + 1 and the product of the conductors of the χ i is squarefree and prime to p. If a certain parity condition holds, then ρ is attached to a Hecke eigenclass in the homology of an arithmetic subgroup Γ of SL(n, ℤ) with coefficients in a module V, where Γ and V are as predicted by Conjecture 2.2 of [5].

Let G be a reductive algebraic group over an algebraically closed field  of characteristic zero and H a closed subgroup of G. Explicit constructions of G-invariant ideals in the algebra [G/H] are given. This allows to obtain an elementary proof of Matsushima's criterion: the homogeneous space G/H is an affine variety if and only if H is reductive.

Let k be a fixed algebraic closure of ℚ and k(t)ac a fixed algebraic closure of k(t). Let S ⊆ k[t]\ {0} be a multiplicative set. Let A = S −1(k[t]) and be the integral closure of A in k(t)ac. We use elliptic curves to develop a necessary condition on S for to be a Bezout domain. We give some examples of S which fail to satisfy this condition. As a consequence, we eliminate some candidates for a good Rumely domain of characteristic 0 with algebraic subring k.

Let ℒ = ⟨F, R, ρ⟩ be a system language. Given a class of ℒ-systems K and an ℒ-algebraic system A = ⟨SEN,⟨N,F⟩⟩, i.e., a functor SEN: Sign → Set, with N a category of natural transformations on SEN, and F:F → N a surjective functor preserving all projections, define the collection K A of A-systems in K as the collection of all members of K of the form 𝔄 = ⟨ SEN,⟨N,F⟩,R 𝔄 ⟩, for some set of relation systems R 𝔄 on SEN. Taking after work of Czelakowski and Elgueta in the context of the model theory of equality-free first-order logic, several relationships between closure properties of the class K, on the one hand, and local properties of K A and global properties connecting K A and K A′, whenever there exists an ℒ-morphism ⟨ F,α⟩ : A → A′, on the other, are investigated. In the main result of the article, it is shown, roughly speaking, that K A is an algebraic closure system, for every ℒ-algebraic system A, provided that K is closed under subsystems and reduced products.

Let R denote a two-dimensional normal standard-graded K-domain over the algebraic closure K of a finite field of characteristic p, and let I ⊂ R denote a homogeneous R +-primary ideal. We prove that the Hilbert–Kunz function of I has the form ϕ (q) = e HK (I)q 2 + γ(q) with rational Hilbert–Kunz multiplicity e HK (I) and an eventually periodic function γ(q).

Let v be a henselian valuation of arbitrary rank of a field K and v¯ be its unique prolongation to a fixed algebraic closure K¯ of K. For any α belonging to K¯\K, let Δ K (α) (resp. ω K (α)) denote the invariants defined to be the minimum (resp. maximum) of the set {v¯(α − α’) | α’ ≠ α runs over K-conjugates of α}. In 1998, while correcting a result of James Ax, Khanduja proved that every finite extension of (K, v) contained in K¯ is tame, if and only if to each α ∈ K¯\K, there corresponds a ∈ K such that v¯(α − a) ≥ Δ K (α). It was also shown that the analogue of the above result does not hold in general when K¯ is replaced by a finite extension of K (see J. Alg. 201, 1998, pp. 647–655). In this paper, similar characterizations of finite tame extensions are given, and some of the invariants associated with such an extension are explicitly determined.

For a local or global field F of arbitrary characteristic, we show that every homomorphism from a reductive algebraic group G over F to a commutative algebraic group T over F satisfies the norm principle.

Let k be an algebraically closed field of characteristic ≠2 and let G be a connected, reductive algebraic group with involution θ ∈ Aut(G). A (G, θ)-module is a rational G-module with a compatible action of θ. The purpose of this note is to classify the irreducible (G, θ)-modules.

The families of affine semi-algebraic sets over a real-closed field Kand semi-linear sets over an ordered field enjoy many closure properties with algebraic and geometric significance. This paper studies the natural closure properies of Minkowski sums and scalar dilation. It gives an extension of the underlying vector space structure that enables the study of an arithmetic on the abstract points of their associated spectra. This arithmetic satisfies certain cancellation principles that motivates an investigation into an algebraic object weaker than a group and culminates with a version of the Jordan-Holder theorem. With the subsequent definition of dimension we show that the collection of affine real ultrafilters in K n is n-dimensional over the scalar ultrafilters.

Let p : Y → Xbe a surjection between schemes projective over the algebraic closure of a finite field. Let Lbe a line bundle on Xsuch that p*(L) is globally generated. A natural necessary and sufficient condition is given under which some positive tensor power of Lis globally generated. An application is a sufficient condition for semi-ampleness of nef line bundles on g,n in positive characteristic, which implies the semi-ampleness of λ, ψ i and myriad other nef line bundles. Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.

The aim of this paper is to provide a method for defining Steinberg symbols on a complete algebraic curve over a perfect field k from the commutator of a certain extension of groups. This extension is associated with a group morphism ϕ : k* → G. With this definition the reciprocity law is a consequence of the finiteness of the cohomology groups H 0(C, 𝒪 C ) and H 1(C, 𝒪 C ). Using this method, Hilbert's norm residue symbol on an algebraic curve and the symbol (a, b) v for the field ℚ p (n = 2) can be defined.

Let v be a henselian valuation of any rank of a field K with value group G and be its unique prolongation to a fixed algebraic closure of K. For an element of , which is separable over K, let denote the well known Krasner's constant given by -conjugates of . In 1946, Krasner proved that if belonging to is such that , then . In this paper, we investigate whether is the smallest among all the elements of the divisible closure of G which have the property that whenever , then .

We study two invariants, ν(L/K) and λ(L/K), arising from a field extension L/K. By definition, ν(L/K) is the cardinality of the set of fields F such that K ⊆ F ⊆ L; and λ(L/K) is the supremum of the set of cardinal numbers arising as lengths of chains {Fi } of fields such that K ⊆ Fi ⊆ L. If L can be generated by one element over an infinite field K and 2 ≤ [L : K] = n < ∞, then ν(L/K) ≤ 2 n−2 + 1, with equality if L/K is Galois with the Klein four-group as Galois group. Despite positive results in the case of Abelian Galois groups, λ is not generally additive for finite-dimensional towers; moreover, maximal chains of intermediate fields can exhibit noncatenarian behavior if [L : K] = 12. The theory for infinite-dimensional algebraic extensions is strikingly different. For instance, for each infinite cardinal number N, there exists a field K of cardinality N and a chain of cardinality 2 N consisting of algebraic field extensions of K; in other words, if denotes an algebraic closure of K, then and, a forti...

In this paper we construct the minimal integral model of a finite type of an algebraic torus defined over a complete non-Archimedean extension of an algebraic number field. The main problem is to study this model in the case of the ramified splitting field. We describe the models of quasi-split tori, norm tori and tori of low dimension. We also calculate the reduction of these models.

INTRODUCTION In this paper we investigate representations of simple algebraic groups over an algebraically closed field of characteristic 2 and of their Lie algebras. For the groups of rank 4 or less, we shall determine all of the extensions of simple modules. The central theme will be the study of some intimate connections among the groups of types Bl, C l and Dl (and F 4 when l = 4). We also give calculations for those other groups of rank 4 or less which have not already been treated elsewhere ([1], [18]), but this is primarily for the sake of completeness.

This paper is devoted to studying nilpotent algebraic groups (by means of the Putcha-Renner theory of algebraic monoids) and related solvable algebraic monoids defined over an algebraically closed field. The main results are as follow:. 1). If G is a connected algebraic group of dimension ¦ 2, then G is nilpotent iff all the connected closed subgroups of codimension 1 are so. 2). Let G be a connected solvable algebraic group with dim G > 2. If dim G u = 2 and C(G) c ≅G a , then G is nonnilpotent and G u ≅ (G a )2. 3). Let M be a three-dimensional regular nonnilpotent solvable algebraic monoid of rank 1 and M ¦ G(M) = G. If G u ≇ (G a ) 2, then M = G∪E(M) and E(M) \ {1} is an (irreducible closed two-dimensional) subsemigroup, which is either a left zero semigroup or a right zero semigroup.

This paper gives examples of zero characteristic holomorphy rings of algebraic functions, where infinitely many primes are allowed to appear as poles and where Diophantine problem has no solution. The methodology used is applicable to any function field whose field of constants is not algebraically closed.1.

Let G be a connected simply-connected simple algebraic group of type G2 over an algebraically closed field K of characteristic p ≥ 13,and G1 the kernel of the Frobenius morphism F on G. In the present paper we show how one can obtain the extensions of any two simple modules for G by using the same method as [7] and [9]. Note that there are 6 positive roots in the root system of G, and 12 alcove types corresponding to the 12 alcoves in the restricted box, hence there are 12 “generic decomposition patterns” which indicate composition factor multiplicities in the Weyl modules of G. In particular, all patterns involve the same number of alcoves and the same distribution of multiplicities, and the total number of composition factors is 119. Moreover, composition factor multiplicities are from 1 to 4. All these facts make the calculation quite complicated. We shall omit almost all details of proofs which are not essentially defferent from those in [7] in order to make the size of this paper not too large. The paper is organized as follows. In Section 1 we introduce our basic notations and preliminaries on the extensions of simple modules for G and G1. In Section 2 and 3 we determine the extensions of any two simple Gmodules with “small” highest weights, and the extensions of any two simple G1-modules. Section 4 is devoted to determining the G-socle of the tensor product of two simple G-modules. Finally in Section 5 we obtain our main results on ExtG. The authors wish to thank the referee for his helpful comments, especially for his pointing out both mistakes of (8) and the definition of B(λ) in the original version. (The same mistakes also occur in [7].) ——————————∗ Current Address: Department of Mathematics, Xinyang Teachers College, Henan 464000. ∗∗ Supported by the National Natural Science Foundation of China.

In [5] it was shown that for a polynomial P of precise degree n with coefficients in an arbitrary m-ary algebra of dimension d as a vector space over an algebraically closed fields, the zeros of P together with the homogeneous zeros of the dominant part of P form a set of cardinality nd or the cardinality of the base field. We investigate polynomials with coefficients in a d dimensional algebra A without assuming the base field k to be algebraically closed. Separable polynomials are defined to be those which have exactly nd distinct zeros in [Ktilde] ⨷k A [Ktilde] where [Ktilde] denotes an algebraic closure of k. The main result states that given a separable polynomial of degree n, the field extension L of minimal degree over k for which L ⨷k A contains all nd zeros is finite Galois over k. It is shown that there is a non empty Zariski open subset in the affine space of all d-dimensional k algebras whose elements A have the following property: In the affine space of polynomials of precise degree n with co...

(1981). Abstract homomorphisms of algebraic groups: problems and bibliography. Communications in Algebra: Vol. 9, No. 1, pp. 95-114.