by Yekutieli, Amnon; Zhang, James J.

In this paper we present a new approach to Grothendieck duality over commutative rings. Our approach is based on the idea of rigid dualizing complexes, which was introduced by Van den Bergh in the context of noncommutative algebraic geometry. The method of rigidity was modified to work over general commutative base rings in our paper (Yekutieli and Zhang, Trans AMS 360:3211–3248, 2008). In the present paper we obtain many of the important local features of Grothendieck duality, yet manage to avoid lengthy and difficult compatibility verifications. Our results apply to essentially finite type algebras over a regular noetherian finite dimensional base ring, and hence are suitable for arithmetic rings. In the sequel paper (Yekutieli, Rigid dualizing complexes on schemes, in preparation) these results will be used to construct and study rigid dualizing complexes on schemes.

DOI: 10.1007/s10468-008-9102-9
Online Date: 5/31/2008
Print publication date: 2/1/2009
View article on SpringerLink

by Gómez Lozano, M. A.; Siles Molina, M.

The aim of this paper is to characterize those elements in a semiprime ring R for which taking local rings at elements and rings of quotients are commuting operations. If Q denotes the maximal ring of left quotients of R, then this happens precisely for those elements if R which are von Neumann regular in Q. An intrinsic characterization of such elements is given. We derive as a consequence that the maximal left quotient ring of a prime ring with a nonzero PI-element is primitive and has nonzero socle. If we change Q to the Martindale symmetric ring of quotients, or to the maximal symmetric ring of quotients of R, we obtain similar results: an element a in R is von Neumann regular if and only if the ring of quotients of the local ring of R at a is isomorphic to the local ring of Q at a.

DOI: 10.1007/s10468-008-9087-4
Online Date: 5/31/2008
Print publication date: 10/1/2008
View article on SpringerLink

by Henke, Anne; Paget, Rowena

Let k be a field of prime characteristic p and E an n-dimensional vector space. We completely describe the tensor space E
^⊗r
viewed as a module for the Brauer algebra B

_k
(r,δ) with parameter δ=2 and n=2. This description shows that while the tensor space still affords Schur–Weyl duality, it typically is not filtered by cell modules, and thus will not be equal to a direct sum of Young modules as defined in Hartmann and Paget (Math Z 254:333–357, 2006). This is very different from the situation for group algebras of symmetric groups. Other results about the representation theory of these Brauer algebras are obtained, including a new description of a certain class of irreducible modules in the case when the characteristic is two.

DOI: 10.1007/s10468-008-9092-7
Online Date: 5/30/2008
Print publication date: 12/1/2008
View article on SpringerLink

by Ardizzoni, A.

The construction of the cotensor coalgebra for an “abelian monoidal” category $\mathcal{M}$ which is also cocomplete, complete and AB5, was performed in Ardizzoni et al. (Comm Algebra 35(1):25–70, 2007). It was also proved that this coalgebra satisfies a meaningful universal property which resembles the classical one. Here the lack of the coradical filtration for a coalgebra E in $\mathcal{M}$ is filled by considering a direct limit $\widetilde{D}$ of a filtration consisting of wedge products of a subcoalgebra D of E. The main aim of this paper is to characterize hereditary coalgebras $\widetilde{D}$, where D is a coseparable coalgebra in $\mathcal{M}$, by means of a cotensor coalgebra: more precisely, we prove that, under suitable assumptions, $\widetilde{D}$ is hereditary if and only if it is formally smooth if and only if it is the cotensor coalgebra $T^c_{D}(D\wedge_E D/D)$ if and only if it is a cotensor coalgebra $T^c_{D}(N)$, where N is a certain D-bicomodule in $\mathcal{M}$. Because of our choice, even when we apply our results in the category of vector spaces, new results are obtained.

DOI: 10.1007/s10468-008-9089-2
Online Date: 5/30/2008
Print publication date: 10/1/2008
View article on SpringerLink

by Yang, Dong

By studying certain centralizer subalgebras of the affine Schur algebra $\widetilde{S}(n,r)$ we show that $\widetilde{S}(n,r)$ is Noetherian and we determine its center. Assuming n ≥ r, we show that $\widetilde{S}(n+1,r)$ is Morita equivalent to $\widetilde{S}(n,r)$, and the Schur functor is an equivalence under certain conditions.

DOI: 10.1007/s10468-008-9097-2
Online Date: 5/28/2008
Print publication date: 2/1/2009
View article on SpringerLink

by Kuku, Aderemi

Let R be the ring of integers in a number field F, Λ any R-order in a semisimple F-algebra Σ, α an R-automorphism of Λ. Denote the extension of α to Σ also by α. Let Λ
_α
[T] (resp. Σ
_α
[T] be the α-twisted Laurent series ring over Λ (resp. Σ). In this paper we prove that (i) There exist isomorphisms $\mathbb{Q}\otimes K_{n}(\Lambda_{\alpha}[T])\simeq \mathbb{Q}\otimes G_{n}(\Lambda_{\alpha}[T])\simeq \mathbb{Q}\otimes K_{n}(\Sigma_{\alpha}[T])$) for all n ≥ 1. (ii) $G^{\rm pr}_n(\Lambda_{\alpha}[T],\hat{Z}_l)\simeq G_n(\Lambda_{\alpha}[T],\hat{Z}_l)$is an l-complete profinite Abelian group for all n≥2. (iii)${\rm div} G^{\rm pr}_n(\Lambda_{\alpha}[T],\hat{Z}_l)=0$for all n≥2. (iv)$G_n(\Lambda_{\alpha}[T]) \longrightarrow G^{\rm pr}_n(\Lambda_{\alpha}[T],\hat{Z}_l)$is injective with uniquely l-divisible cokernel (for all n≥2). (v) K
_–1(Λ), K
_–1(Λ
_α
[T]) are finitely generated Abelian groups.

DOI: 10.1007/s10468-008-9085-6
Online Date: 5/28/2008
Print publication date: 8/1/2008
View article on SpringerLink

by Miemietz, Vanessa

Ariki’s and Grojnowski’s approach to the representation theory of affine Hecke algebras of type A is applied to type B with unequal parameters to obtain – under certain restrictions on the eigenvalues of the lattice operators – analogous multiplicity-one results and a classification of irreducibles with partial branching rules as in type A.

DOI: 10.1007/s10468-008-9086-5
Online Date: 5/24/2008
Print publication date: 8/1/2008
View article on SpringerLink

by Bavula, V. V.; Hinchcliffe, V.

It is proved that the filter dimension is Morita invariant. A direct consequence of this fact is the Morita invariance of the inequality of Bernstein: if an algebra A is Morita equivalent to the ring ${\cal D} (X)$ of differential operators on a smooth irreducible affine algebraic variety X of dimension n ≥ 1 over a field of characteristic zero then the Gelfand–Kirillov dimension $ {\rm GK} (M)\geq n = \frac{{\rm GK} (A)}{2}$ for all nonzero finitely generated A-modules M. In fact, a stronger result is proved, namely, a Morita invariance of the holonomic number for finitely generated algebra. A direct consequence of this fact is that an analogue of the inequality of Bernstein holds for the (simple) rational Cherednik algebras
H

_c

for integral
c: ${\rm GK} (M)\geq n =\frac{{\rm GK} (H_c)}{2}$
for all nonzero finitely generated H

_c

-modules
M. For these class of algebras, it gives an affirmative answer to a question of Ken Brown about symplectic reflection algebras.

DOI: 10.1007/s10468-008-9091-8
Online Date: 5/23/2008
Print publication date: 10/1/2008
View article on SpringerLink

by Kolb, Stefan

It is shown that central elements in G. Letzter’s quantum group analogs of symmetric pairs lead to solutions of the reflection equation. This clarifies the relation between Letzter’s approach to quantum symmetric pairs and the approach taken by M. Noumi, T. Sugitani, and M. Dijkhuizen. We develop general tools to show that a Noumi-Sugitani-Dijkhuizen type construction of quantum symmetric pairs can be performed preserving spherical representations from the classical situation. These tools apply to the symmetric pair FII and to all symmetric pairs which correspond to an automorphism of the underlying Dynkin diagram. Hence Noumi-Sugitani-Dijkhuizen type constructions with desirable properties are possible for various symmetric pairs for exceptional Lie algebras.

DOI: 10.1007/s10468-008-9093-6
Online Date: 5/21/2008
Print publication date: 12/1/2008
View article on SpringerLink

by Dung, Nguyen Viet; Simson, Daniel

We show how the Gabriel–Roiter measure, introduced by Ringel in (Bull Sci Math 129:726–748, 2005 and Contemp Math 406:105–135, 2006), applies to indecomposable modules of finite length over right pure semisimple rings, and in particular to the study of the open problem whether any right pure semisimple ring is of finite representation type.

DOI: 10.1007/s10468-008-9084-7
Online Date: 5/21/2008
Print publication date: 10/1/2008
View article on SpringerLink