scholarly journals Almost Homogeneous Varieties of Albanese Codimension One

2020 ◽  
Author(s):  
Bruno Laurent
Keyword(s):  
Group Scheme ◽  
Arbitrary Field ◽  
Neutral Component ◽  
Perfect Field ◽  
Codimension One ◽  
Normal Varieties ◽  
Characteristic 2 ◽  
Homogeneous Varieties

Abstract We classify almost homogeneous normal varieties of Albanese codimension $1$, defined over an arbitrary field. We prove that such a variety has a unique normal equivariant completion. Over a perfect field, the group scheme of automorphisms of this completion is smooth, except in one case in characteristic $2$, and we determine its (reduced) neutral component.

On the structure of flat families of complete homogeneous varieties

2017 ◽  
Vol 28 (11) ◽  
pp. 1750079
Author(s):  
Preena Samuel
Keyword(s):  
Automorphism Group ◽  
Homogeneous Space ◽  
Group Scheme ◽  
Base Change ◽  
Neutral Component ◽  
Natural Action ◽  
Abelian Scheme ◽  
Closed Fields ◽  
Algebraically Closed Fields ◽  
Homogeneous Varieties

In this paper, we investigate flat families of complete homogeneous varieties. Over a reduced, noetherian base of characteristic 0, such a family turns out to be a homogeneous space under the natural action of the neutral component of its automorphism group scheme; further, after an étale base change, such a family can be expressed as a product of an abelian scheme and a Borel scheme. The structure of the neutral component of the automorphism group scheme of such a family is also obtained. These results extend already known structure results for complete homogeneous varieties over algebraically closed fields.

scholarly journals Finite symmetric tensor categories with the Chevalley property in characteristic 2

2020 ◽  
pp. 2140010
Author(s):  
Pavel Etingof ◽  
Shlomo Gelaki
Keyword(s):  
Tensor Category ◽  
Group Scheme ◽  
Symmetric Tensor ◽  
Function Algebra ◽  
Equivalence Classes ◽  
Closed Field ◽  
Arbitrary Field ◽  
Tensor Categories ◽  
Characteristic 2 ◽  
Chevalley Property

We prove an analog of Deligne’s theorem for finite symmetric tensor categories [Formula: see text] with the Chevalley property over an algebraically closed field [Formula: see text] of characteristic [Formula: see text]. Namely, we prove that every such category [Formula: see text] admits a symmetric fiber functor to the symmetric tensor category [Formula: see text] of representations of the triangular Hopf algebra [Formula: see text]. Equivalently, we prove that there exists a unique finite group scheme [Formula: see text] in [Formula: see text] such that [Formula: see text] is symmetric tensor equivalent to [Formula: see text]. Finally, we compute the group [Formula: see text] of equivalence classes of twists for the group algebra [Formula: see text] of a finite abelian [Formula: see text]-group [Formula: see text] over an arbitrary field [Formula: see text] of characteristic [Formula: see text], and the Sweedler cohomology groups [Formula: see text], [Formula: see text], of the function algebra [Formula: see text] of [Formula: see text].

A Further Extension of Cayley's Parameterization

1959 ◽  
Vol 11 ◽  
pp. 48-50 ◽  
Author(s):  
Martin Pearl
Keyword(s):  
Symmetric Matrix ◽  
Arbitrary Field ◽  
Complex Field ◽  
Result Theorem ◽  
Characteristic 2 ◽  
Image Position ◽  
Row Space ◽  
Real Matrices ◽  
Skew Symmetric Matrix

In a recent paper (3)* the following theorem was proved for real matrices.Theorem 1. If A is a symmetric matrix and Q is a skew-symmetric matrix such that A + Q is non-singular, then1is a cogredient automorph (c.a.) of A whose determinant is + 1 and having theproperty that A and I + P span the same row space.Conversely, if P is a c.a. of A whose determinant is + 1 and if P has theproperty that I + P and A span the same row space, then there exists a skew symmetricmatrix Q such that P is given by equation (1).Theorem 1 reduces to the well-known Cayley parameterization in the case where A is non-singular. A similar and somewhat simpler result (Theorem 4) was given for the case when the underlying field is the complex field. It was also shown that the second part of the theorem (in either form) is false when the characteristic of the underlying field is 2. The purpose of this paper is to simplify the proof of Theorem 1 and at the same time, to extend these results to matrices over an arbitrary field of characteristic ≠ 2.

An automorphism of the symplectic group Sp4(2n)

1968 ◽  
Vol 64 (1) ◽  
pp. 5-9 ◽  
Author(s):  
Anne Duncan
Keyword(s):  
Symplectic Group ◽  
Explicit Description ◽  
Linear Groups ◽  
Perfect Field ◽  
Characteristic 2

In his discussion (3) of automorphisms of finite simple linear groups, Steinberg shows that the group denoted by B2(K) admits graph automorphisms when K is a perfect field of characteristic 2. The group B2(K) is identified in (2) as the symplectic group Sp4(K) when K is finite and of characteristic 2, and it is the object of this note to give, by means of this identification, an explicit description of the graph automorphisms of B2(2n).

scholarly journals A solution of a problem of Plotkin and Vovsi and an application to varieties of groups

1999 ◽  
Vol 67 (3) ◽  
pp. 329-355 ◽  
Author(s):  
C. K. Gupta ◽  
A. N. Krasil'nikov
Keyword(s):  
Free Group ◽  
Invariant Subgroup ◽  
Arbitrary Field ◽  
Finitely Based ◽  
Maximal Condition ◽  
Infinite Rank ◽  
Characteristic 2 ◽  
Countably Infinite ◽  
Fully Invariant Subgroups ◽  
Relatively Free Group

AbstractLet K be an arbitrary field of characteristic 2, F a free group of countably infinite rank. We construct a finitely generated fully invariant subgroup U in F such that the relatively free group F/U satisfies the maximal condition on fully invariant subgroups but the group algebra K (F/U) does not satisfy the maximal condition on fully invariant ideals. This solves a problem posed by Plotkin and Vovsi. Using the developed techniques we also construct the first example of a non-finitely based (nilpotent of class 2)-by-(nilpotent of class 2) variety whose Abelian-by-(nilpotent of class at most 2) groups form a hereditarily finitely based subvariety.

Field-theoretic and linear-algebraic invariants

2015 ◽  
Author(s):  
Brian Conrad ◽  
Gopal Prasad
Keyword(s):  
Simple Group ◽  
Root System ◽  
Reductive Group ◽  
Simple Groups ◽  
Arbitrary Field ◽  
Isomorphism Classes ◽  
Algebraic Invariants ◽  
Characteristic 2 ◽  
Minimal Type

This chapter deals with field-theoretic and linear-algebraic invariants. It first presents a construction of non-standard pseudo-split absolutely pseudosimple k-groups with root system A1 over any imperfect field k of characteristic 2. It then considers an absolutely pseudo-simple group over a field k, along with a pseudo-split pseudo-reductive group over an arbitrary field k. It also establishes the equality over k of minimal fields of definition for projection onto maximal geometric adjoint semisimple quotients. This is followed by two examples that illustrate the root field in A1-cases. The chapter concludes with a discussion of a classification of the isomorphism classes of pseudo-split pseudo-simple groups G over an imperfect field k of characteristic p subject to the hypothesis that G is of minimal type. The associated irreducible root datum, which is sufficient to classify isomorphism classes in the semisimple case, is supplemented with additional field-theoretic and linear-algebraic data.

scholarly journals On the Theory of Differential Forms on Algebraic Varieties

1957 ◽  
Vol 11 ◽  
pp. 13-39
Author(s):  
Yûsaku Kawahara
Keyword(s):  
Function Field ◽  
Projective Variety ◽  
Hyperplane Section ◽  
Differential Forms ◽  
Classical Case ◽  
Algebraic Varieties ◽  
Simple Point ◽  
Perfect Field ◽  
Linearly Independent ◽  
Normal Varieties

Let K be a function field of one variable over a perfect field k and let v be a valuation of K over k. Then is the different-divisor (Verzweigungsdivisor) of K/k(x), and (x)∞ is the denominator-divisor (Nennerdivisor) of x. In §1 we consider a generalization of this theorem in the function fields of many variables under some conditions. In §2 and §3 we consider the differential forms of the first kind on algebraic varieties, or the differential forms which are finite at every simple point of normal varieties and subadjoint hypersurfaces which are developed by Clebsch and Picard in the classical case. In §4 we give a proof of the following theorem. Let Vr be a normal projective variety defined over a field k of characteristic 0, and let ω1, …, ωs be linearly independent simple closed differential forms which are finite at every simple point of Vr. Then the induced forms on a generic hyperplane section are also linearly independent.

scholarly journals Group schemes and local densities of ramified hermitian lattices in residue characteristic 2. Part II

2018 ◽  
Vol 30 (6) ◽  
pp. 1487-1520 ◽  
Author(s):  
Sungmun Cho
Keyword(s):  
Mass Formula ◽  
Local Density ◽  
Group Scheme ◽  
Field Extension ◽  
Integral Group ◽  
Group Schemes ◽  
Quadratic Extensions ◽  
Local Densities ◽  
Characteristic 2 ◽  
Residue Characteristic

Abstract This paper is the complementary work of [S. Cho, Group schemes and local densities of ramified hermitian lattices in residue characteristic 2: Part I, Algebra Number Theory 10 2016, 3, 451–532]. Ramified quadratic extensions {E/F} , where F is a finite unramified field extension of {\mathbb{Q}_{2}} , fall into two cases that we call Case 1 and Case 2. In our previous work, we obtained the local density formula for a ramified hermitian lattice in Case 1. In this paper, we obtain the local density formula for the remaining Case 2, by constructing a smooth integral group scheme model for an appropriate unitary group. Consequently, this paper, combined with [W. T. Gan and J.-K. Yu, Group schemes and local densities, Duke Math. J. 105 2000, 3, 497–524] and our previous work, allows the computation of the mass formula for any hermitian lattice {(L,H)} , when a base field is unramified over {\mathbb{Q}} at a prime {(2)} .

Fields countably generated over a proper subfield

1986 ◽  
Vol 33 (2) ◽  
pp. 219-226
Author(s):  
James K. Deveney ◽  
Joe Yanik
Keyword(s):  
Affirmative Answer ◽  
Arbitrary Field ◽  
Arbitrary Characteristic ◽  
Characteristic 2

For an arbitrary field K there are two related questions that can be asked:(1) Is there a proper subfield, L, of K such that K is countably generated over L?(2) Given a proper subfield M of K is there a proper subfield, L, of K containing M such that K is countably generated over L?We give an affirmative answer to (1) in characteristic p ≠ 0 and provide counterexamples to (2) for arbitrary characteristic ≠ 2.

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