Makar-Limanov's conjecture on free subalgebras

It is proved that over every countable field K there is a nil algebra R such that the algebra obtained from R by extending the field K contains noncommutative free subalgebras of arbitrarily high rank.It is also shown that over every countable field K there is an algebra R without noncommutative free subalgebras of rank two such that the algebra obtained from R by extending the field K contains a noncommutative free subalgebra of rank two. This answers a question of Makar-Limanov [Lenny Makar-Limanov, private communication, Beijing, June 2007].     

The Stanley Depth in the Upper Half of the Koszul Complex

Let R = K[X₁, ?c, X_n] be a polynomial ring over some field K. In this article, we prove that the kth syzygy module of the residue class field K of R has Stanley depth n ? 1 for ?n/2? ≤k < n, as it had been conjectured by Bruns et al. in 2010. In particular, this gives the Stanley depth for a whole family of modules whose graded components have dimension greater than 1. So far, the Stanley depth is known only for a few examples of this type. Our proof consists in a close analysis of a matching in the Boolean algebra.     

Local automorphisms of nilpotent algebras of matrices of small orders

Let K be an associative commutative ring with identity, and let R be the algebra of lower niltriangular n × n matrices over K. For n = 3, we prove that local automorphisms and Lie automorphisms of the algebra R generate all its local Lie automorphisms. For the case when K is a field and n = 4, we describe local automorphisms, local derivations, and local Lie automorphisms of the algebra R.     

Discrete Valuations Extend to Certain Algebras of Quantum Type

So-called “quantized” algebras are popular objects of study in non-commutative algebra. Usually such algebras are either positively graded Ore domains R with R ₀ = K a field and R = K[R ₁], R ₁ being a finite dimensional K vectorspace, or else filtered rings having a ring of forementioned type for its associated graded ring. We show that every discrete valuation of K extends to a valuation, in the sence of O. Schilling (cf. [S]), of the skewfield of fractions, Δ = Q_cl (R), of the Ore domain R (Proposition 2.3. and Corollary 2.8.). Such extension property has long been known to fail for finite dimensional skewfields over K; its validity in the case of several quantized algebras may be viewed as a consequence of the rigidity of their defining relations. Our result opens the door for a more arithmetical study of Δ e.g. in case K is a numberfield or an algebraic function field of a curve; for an application in this direction we refer to a first version of some divisor calculus started in [VW].     

Group algebras and enveloping algebras with nonmatrix and semigroup identities

Let Kbe a field of characteristic p> 0. Denote by ω(R) the augmentation ideal of either a group algebra (R) = K[G] or a restricted enveloping algebra R= u(L) over K. We first characterize those Rfor which ω(R) satisfies a polynomial identity not satisfied by the algebra of all 2 × 2 matrices over K. Then, we examine those Rfor which U J(R) satisfies a semigroup identity (that is, a polynomial identity which can be written as the difference of two monomials).     

Galois theory over complete local domains

Complete local domains play an important role in commutative algebra and algebraic geometry, and their algebraic properties were already described by Cohen’s structure theorem in 1946. However, the Galois theoretic properties of their quotient fields only recently began to unfold. In 2005 Harbater and Stevenson considered the two dimensional case. They proved that the absolute Galois group of the field K((X, Y)) (where K is an arbitrary field) is semi-free. In this work we settle the general case, and prove that if R is a complete local domain of dimension exceeding 1, then the quotient field of R has a semi-free absolute Galois group.     

Anneaux de polynomes de ore sur des anneaux artiniens

Let Ra commutative noetherian domainKits quotient field with caracteristic O. A R-Lie algebra is a free R-modul of finish type, with a Lie product [,]. The envelopping algebra U(g)of gis studied under any conditions on gand R.     

Milnor K-theory of rings,higher Chow groups and applications

If R is a smooth semi-local algebra of geometric type over an infinite field, we prove that the Milnor K-group K ^M _n (R) surjects onto the higher Chow group CH ⁿ (R , n) for all n≥0. Our proof shows moreover that there is an algorithmic way to represent any admissible cycle in CH ⁿ (R , n) modulo equivalence as a linear combination of “symbolic elements” defined as graphs of units in R. As a byproduct we get a new and entirely geometric proof of results of Gabber, Kato and Rost, related to the Gersten resolution for the Milnor K-sheaf. Furthermore it is also shown that in the semi-local PID case we have, under some mild assumptions, an isomorphism. Some applications are also given. Oblatum 17-XII-1998 & 1-X-2001?Published online: 18 January 2002     

On Duality in the Homology Algebra of a Koszul Complex

The homology algebra of the Koszul complex K(x ₁, ..., x _n ; R) of a Gorenstein local ring R has Poincare duality if the ideal I = (x ₁, ..., x _n ) of R is strongly Cohen-Macaulay (i.e., all homology modules of the Koszul complex are Cohen-Macaulay) and under the assumption that dim R - grade I ⩽ 4 the converse is also true.__________Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 1, pp. 77–81, 2003.     

Stable range one for rings with central units

Abstract

The purpose of this paper is to give a partial positive answer to a question raised by Khurana et al. as to whether a ring R with stable range one and central units is commutative. We show that this is the case under any of the following additional conditions: R is semiprime or R is one-sided Noetherian or R has unit-stable range 1 or R has classical Krull dimension 0 or R is an algebra over a field K such that K is uncountable and R has only countably many primitive ideals or R is affine and either K has characteristic 0 or has infinite transcendental degree over its prime subfield or is algebraically closed. However, the general question remains open.     

Ideals and simplicity of unitary Lie algebras

Double graded ideals and simplicity of elementary unitary Lie algebra eu _n (R,⁻, γ) and Steinberg unitary Lie algebra stu _n (R,⁻, γ) are characterized, where R is a unital involutory associative algebra over a field F of characteristic zero, n ⩾ 5.     

Automorphisms of the symplectic group over a local ring

Let R denote a commutative local ring with maximal ideal m and residue field K = R/m. Let V be a symplectic space over R. In this paper we determine the group automorphisms of the symplectic group Sp_n(V) when n 6, the characteristic of k is not 2, and k is not the finite field of three elements.     

Coinvariants for a coadjoint action of quantum matrices

Let K be a (algebraically closed ) field. A morphism A ⟼ g ⁻¹ Ag, where A ∈ M(n) and g ∈ GL(n), defines an action of a general linear group GL(n) on an n × n-matrix space M(n), referred to as an adjoint action. In correspondence with the adjoint action is the coaction α: K[M(n)] → K[M(n)] ⊗ K[GL(n)] of a Hopf algebra K[GL(n)] on a coordinate algebra K[M(n)] of an n × n-matrix space, dual to the conjugation morphism. Such is called an adjoint coaction. We give coinvariants of an adjoint coaction for the case where K is a field of arbitrary characteristic and one of the following conditions is satisfied: (1) q is not a root of unity; (2) char K = 0 and q = ±1; (3) q is a primitive root of unity of odd degree. Also it is shown that under the conditions specified, the category of rational GL _q × GL _q -modules is a highest weight category.     

Graded central polynomials for the algebra <Emphasis Type="Italic">M</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(<Emphasis Type="Italic">K</Emphasis>)

Let M _n (K) be the algebra of all n × n matrices over an infinite field K. This algebra has a natural ℤ _n -grading and a natural ℤ-grading. Finite bases for its ℤ _n -graded identities and for its ℤ-graded identities are known. In this paper we describe finite generating sets for the ℤ _n -graded and for the ℤ-graded central polynomials for M _n (K) Partially supported by CNPq 620025/2006-9     

Modular Adjacency Algebras of Hamming Schemes

To each association scheme G and to each field R, there is associated naturally an associative algebra, the so-called adjacency algebra RG of G over R. It is well-known that RG is semisimple if R has characteristic 0. However, little is known if R has positive characteristic. In the present paper, we focus on this case. We describe the algebra RG if G is a Hamming scheme (and R a field of positive characteristic). In particular, we show that, in this case, RG is a factor algebra of a polynomial ring by a monomial ideal.     

Characterization off-vectors of families of convex sets inR d Part I: Necessity of Eckhoff’s conditions

LetK=K ₁,...,K_n be a family ofn convex sets inR ^d . For 0≦i<n denote byf _i the number of subfamilies ofK of sizei+1 with non-empty intersection. The vectorf(K) is called thef-vectors ofK. In 1973 Eckhoff proposed a characterization of the set off-vectors of finite families of convex sets inR ^d by a system of inequalities. Here we prove the necessity of Eckhoff's inequalities. The proof uses exterior algebra techniques. We introduce a notion of generalized homology groups for simplicial complexes. These groups play a crucial role in the proof, and may be of some independent interest.     

Explicit linear minimal free resolutions over a natural class of Rees algebras

Homological properties of the Rees algebra R of a Koszul K-algebra A over a field K, with respect to the maximal homogeneous ideal, are studied. In particular, for a finitely generated graded A-module N with linear minimal free R-resolution over A, the minimal free resolution of is explicitly constructed. This resolution is again linear.Received: 23 October 2000     

Formally Real Involutions on Central Simple Algebras

An involution # on an associative ring R is formally real if a sum of nonzero elements of the form r ^# r where r ? R is nonzero. Suppose that R is a central simple algebra (i.e., R = M _n (D) for some integer n and central division algebra D) and # is an involution on R of the form r ^# = a ^?1 r? a, where ? is some transpose involution on R and a is an invertible matrix such that a? = ±a. In Section 1 we characterize formal reality of # in terms of a and ?| _D . In later sections we apply this result to the study of formal reality of involutions on crossed product division algebras. We can characterize involutions on D = (K/F, Φ) that extend to a formally real involution on the split algebra D ? _F K ? M _n (K). Every such involution is formally real but we show that there exist formally real involutions on D which are not of this form. In particular, there exists a formally real involution # for which the hermitian trace form x ? tr(x ^# x) is not positive semidefinite.     

The Birman–Murakami–Wenzl Algebras of Type D n

The Birman–Murakami–Wenzl algebra (BMW algebra) of type D _n is shown to be semisimple and free of rank (2 ⁿ  + 1)n!! ? (2 ^n?1 + 1)n! over a specified commutative ring R, where n!! =1·3…(2n ? 1). We also show it is a cellular algebra over suitable ring extensions of R. The Brauer algebra of type D _n is the image of an R-equivariant homomorphism and is also semisimple and free of the same rank, but over the ring ?[δ^±1]. A rewrite system for the Brauer algebra is used in bounding the rank of the BMW algebra above. As a consequence of our results, the generalized Temperley–Lieb algebra of type D _n is a subalgebra of the BMW algebra of the same type.