scholarly journals Cancellation problem for projective modules over affine algebras

2008 ◽  
Vol 3 (3) ◽  
pp. 561-581 ◽  
Author(s):  
Manoj Kumar Keshari
Keyword(s):  
Monic Polynomial ◽  
Regular Sequence ◽  
Closed Field ◽  
Projective Modules ◽  
Affine Algebra ◽  
Algebraically Closed Field ◽  
Cancellation Problem ◽  
Affine Algebras ◽  
Open Question

AbstractLet A be an affine algebra of dimension n over an algebraically closed field k with 1/n! ∈ k. Let P be a projective A-module of rank n − 1. Then, it is an open question due to N. Mohan Kumar, whether P is cancellative. We prove the following results:(i) If A = R[T,T−1], then P is cancellative.(ii) If A = R[T,1/f] or A = R[T,f1/f,…,fr/f], where f(T) is a monic polynomial and f,f1,…,fr is R[T]-regular sequence, then An−1 is cancellative. Further, if k = p, then P is cancellative.

On bad groups, bad fields, and pseudoplanes

1991 ◽  
Vol 56 (3) ◽  
pp. 915-931 ◽  
Author(s):  
Ali Nesin
Keyword(s):  
Projective Planes ◽  
Closed Field ◽  
Morley Rank ◽  
Algebraically Closed Field ◽  
Desarguesian Projective Planes ◽  
Natural Geometry ◽  
Special Cases ◽  
Finite Morley Rank ◽  
Rank 2 ◽  
Open Question

Cherlin introduced the concept of bad groups (of finite Morley rank) in [Ch1]. The existence of such groups is an open question. If they exist, they will contradict the Cherlin-Zil'ber conjecture that states that an infinite simple group of finite Morley rank is a Chevalley group over an algebraically closed field. In this paper, we prove that bad groups of finite Morley rank 3 act on a natural geometry Γ (namely on a special pseudoplane; see Corollary 20) sharply flag-transitively.We show that Γ is not very far from being a projective plane and when it is so rk(Γ) = 2 and Γ is not Desarguesian (Theorem 2). Baldwin [Ba] recently discovered non-Desarguesian projective planes of Morley rank 2. This discovery, together with this paper, makes the existence of bad groups (also of bad fields) more plausible. A bad field is a pair (K, A) of finite Morley rank, where K is an algebraically closed field, A <≠K* and A is infinite. There existence is also unknown.In this paper, we define the concept of a sharp-field as a pair (K, A), where K is a field, A < K*and1. K = A − A,2. If a + b − 1 ∈ A, a ∈ A, b ∈ A, then either a = 1 or b = 1.If K is finite this is equivalent to 1 and2.′ ∣K∣ = ∣A∣2 ∣A∣ + 1.Finite sharp-fields are special cases of difference sets [De]

Representations of 𝔰𝔩3 in characteristic 3

2017 ◽  
Vol 16 (01) ◽  
pp. 1750012
Author(s):  
Xin Wen
Keyword(s):  
Irreducible Representations ◽  
Closed Field ◽  
Projective Modules ◽  
Verma Modules ◽  
Algebraically Closed Field ◽  
Simple Modules ◽  
Rank 2 ◽  
Characteristic 3 ◽  
Cartan Invariants

Let [Formula: see text] be the special linear Lie algebra [Formula: see text] of rank 2 over an algebraically closed field [Formula: see text] of characteristic 3. In this paper, we classify all irreducible representations of [Formula: see text], which completes the classification of the irreducible representations of [Formula: see text] over an algebraically closed field of arbitrary characteristic. Moreover, the multiplicities of baby Verma modules in projective modules and simple modules in baby Verma modules are given. Thus we get the character formulae for simple modules and the Cartan invariants of [Formula: see text].

Cartan invariants for the Witt superalgebra W(2) in positive characteristic

2016 ◽  
Vol 15 (03) ◽  
pp. 1650039 ◽  
Author(s):  
Feifei Duan ◽  
Fang Li
Keyword(s):  
Positive Characteristic ◽  
Lie Superalgebra ◽  
Character Formula ◽  
Closed Field ◽  
Projective Modules ◽  
Algebraically Closed Field ◽  
Grassmann Superalgebra ◽  
Cartan Invariants

The aim of this paper is to study the projective modules of the [Formula: see text]-reduced enveloping superalgebra [Formula: see text], where [Formula: see text] is the Lie superalgebra of superderivations on the Grassmann superalgebra [Formula: see text], over an algebraically closed field k of characteristic [Formula: see text]. Mainly, the Cartan invariants and the dimensions of indecomposable projective modules of [Formula: see text] are determined for any [Formula: see text]-character [Formula: see text] up to isomorphism.

Exactness of Complexes of Modules over Schur Superalgebras

2006 ◽  
Vol 13 (01) ◽  
pp. 99-110 ◽  
Author(s):  
Alexandr N. Grishkov ◽  
František Marko ◽  
Alexandr N. Zubkov
Keyword(s):  
Tensor Products ◽  
Closed Field ◽  
Projective Modules ◽  
Characteristic P ◽  
Algebraically Closed Field ◽  
Injective Modules

In this paper, we study tensor products of injective modules and exactness of certain naturally appearing complexes of projective and injective modules over Schur superalgebras over an algebraically closed field K of characteristic p > 2. In particular, for the complex [Formula: see text] of projective modules P(i) = Sξ 1i(m+1)r-i given by multiplication by ui = ξ 1i(m+1)r-i, 1i+1 (m+1)r-i-1, we determine all of the terms where it is exact.

scholarly journals The strong simple connectedness of tame algebras with separating almost cyclic coherent Auslander–Reiten components

2021 ◽  
Author(s):  
Piotr Malicki
Keyword(s):  
Closed Field ◽  
Algebraically Closed Field ◽  
Main Application ◽  
Finite Dimensional ◽  
Simple Connectedness ◽  
Tame Algebras

AbstractWe study the strong simple connectedness of finite-dimensional tame algebras over an algebraically closed field, for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. As the main application we describe all analytically rigid algebras in this class.

scholarly journals On Unramified Separable Abelian p-Extensions of Function Fields I

1959 ◽  
Vol 14 ◽  
pp. 223-234 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Galois Group ◽  
Function Field ◽  
Function Fields ◽  
Abelian Extension ◽  
Closed Field ◽  
Jacobian Variety ◽  
Algebraically Closed Field

Let k be an algebraically closed field of characteristic p>0. Let K/k be a function field of one variable and L/K be an unramified separable abelian extension of degree pr over K. The galois automorphisms ε1, …, εpr of L/K are naturally extended to automorphisms η(ε1), … , η(εpr) of the jacobian variety JL of L/k. If we take a svstem of p-adic coordinates on JL, we get a representation {Mp(η(εv))} of the galois group G(L/K) of L/K over p-adic integers.

scholarly journals CHARACTER CLUSTERS FOR LIE ALGEBRA MODULES OVER A FIELD OF NONZERO CHARACTERISTIC

2013 ◽  
Vol 89 (2) ◽  
pp. 234-242 ◽  
Author(s):  
DONALD W. BARNES
Keyword(s):  
Lie Algebra ◽  
Saturated Formation ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Direct Sum ◽  
Finite Dimensional ◽  
Nonzero Characteristic ◽  
Composition Factors

AbstractFor a Lie algebra $L$ over an algebraically closed field $F$ of nonzero characteristic, every finite dimensional $L$-module can be decomposed into a direct sum of submodules such that all composition factors of a summand have the same character. Using the concept of a character cluster, this result is generalised to fields which are not algebraically closed. Also, it is shown that if the soluble Lie algebra $L$ is in the saturated formation $\mathfrak{F}$ and if $V, W$ are irreducible $L$-modules with the same cluster and the $p$-operation vanishes on the centre of the $p$-envelope used, then $V, W$ are either both $\mathfrak{F}$-central or both $\mathfrak{F}$-eccentric. Clusters are used to generalise the construction of induced modules.

scholarly journals Classification of one-dimensional superattracting germs in positive characteristic

2014 ◽  
Vol 35 (7) ◽  
pp. 2242-2268 ◽  
Author(s):  
MATTEO RUGGIERO
Keyword(s):  
Positive Characteristic ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Dimensional Version

We give a classification of superattracting germs in dimension $1$ over a complete normed algebraically closed field $\mathbb{K}$ of positive characteristic up to conjugacy. In particular, we show that formal and analytic classifications coincide for these germs. We also give a higher-dimensional version of some of these results.

scholarly journals Geometric Weil representation in characteristic two

2011 ◽  
Vol 11 (2) ◽  
pp. 221-271 ◽  
Author(s):  
Alain Genestier ◽  
Sergey Lysenko
Keyword(s):  
Weil Representation ◽  
Closed Field ◽  
Triangulated Category ◽  
Algebraically Closed Field ◽  
Witt Vectors ◽  
Suitable Sense ◽  
Characteristic Two ◽  
Greenberg Realization ◽  
Geometric Analogue

AbstractLet k be an algebraically closed field of characteristic two. Let R be the ring of Witt vectors of length two over k. We construct a group stack Ĝ over k, the metaplectic extension of the Greenberg realization of $\operatorname{\mathbb{S}p}_{2n}(R)$. We also construct a geometric analogue of the Weil representation of Ĝ, this is a triangulated category on which Ĝ acts by functors. This triangulated category and the action are geometric in a suitable sense.

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