by Retert, K.

Noncommutative projective geometry studies noncommutative graded rings by replacing the variety by a suitable Grothendieck category. One way of studying the resulting category is to examine the full subcategories which behave like curves on a commutative variety. Smith and Zhang initiated such a study by considering the subcategory generated by a particular type of module they called a ‘pure curve module in good position’. This paper generalizes their construction by allowing more general modules. The resulting category is shown to be categorically equivalent to a quotient of the category of graded modules over a graded ring. In the course of defining the category equivalence, several dimensions, including projective, injective and Krull dimensions, are calculated. In particular, this extension allows examination of the category created from a line module over more general AS-regular rings than those considered by Smith and Zhang. For instance, suppose that C is a generic line module over R

_d
, Stafford’s Sklyanin-like algebra. Let C denote the category C generates. Then C is equivalent to the category of graded k[x, y]/(x
² − y
²) modules under the Z × Z/2Z-grading where deg (x) = (−1, 0) and deg (y) = (−1,1).

DOI: 10.1023/B:ALGE.0000031205.89426.4f
Print publication date: 8/1/2004
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by Benkart, Georgia; Witherspoon, Sarah

We investigate two-parameter quantum groups corresponding to the general linear and special linear Lie algebras
$${\mathfrak{g}}{\mathfrak{l}}_n $$
and
$${\mathfrak{s}}{\mathfrak{l}}_n $$
. We show that these quantum groups can be realized as Drinfel’d doubles of certain Hopf subalgebras with respect to Hopf pairings. Using the Hopf pairing, we construct a corresponding R-matrix and a quantum Casimir element. We discuss isomorphisms among these quantum groups and connections with multiparameter quantum groups.

DOI: 10.1023/B:ALGE.0000031151.86090.2e
Print publication date: 8/1/2004
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by Künzer, M.; Weber, H.

Let n≥1 and let p be a prime. Expand j∈[0,p

ⁿ
−1]\(p) p-adically as j=∑
_s≥0
a

_s

p

^s
with a

_s
∈[0,p−1]. The #([0,j]\(p))th
_(p)[ζ
_p


ⁿ
]-linear elementary divisor of the cyclotomic Dedekind embedding
$$Z_{(p)} [\zeta _p ^n ] \otimes _{Z_{(p)} } Z_{(p)} [\zeta _p ^n ] \to \prod\limits_{i \in \left( {z/p_{}^n } \right)^* } {Z_{(p)} } [\zeta _p ^n ]$$
has valuation
$$ – 1 + \sum\limits_{s \geqslant 0} {(a_s (s + 1) – a_{s + 1} (s + 2))} p^s $$
at 1−ζ
_p


ⁿ
. There is a similar result for the related cyclic Wedderburn embedding.

DOI: 10.1023/B:ALGE.0000031024.07375.75
Print publication date: 8/1/2004
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