by Yoshino, Yuji

We provide a general method to construct the Tate–Vogel homology theory for a general half-exact functor with one variable, aiming at a good generalization of Cohen–Macaulay approximations of modules over commutative Gorenstein rings. For a half exact functor F, using the left and right satellites (S

_n
and S

ⁿ
), we define F
^∨(X)=lim _→
S

_n

S

ⁿ

F(X) and F
^∧(X)=lim _←
S

_n

S

ⁿ

F(X), and call F
^∨ and F
^∧ the Tate–Vogel completions of F. We provide several properties of F
^∨ and F
^∧, and their relations with the G-dimension and the projective dimension of the functor F. A comparison theorem of Tate–Vogel completions with ordinary Tate–Vogel homologies is proved. If F is a half exact functor over the category of R-modules, where R is a commutative Noetherian local ring inspired by Martsinkovsky’s works, we can define the invariants ξ(F) and η(F) of F. If F=Ext
_R


ⁱ
(M, ), then they coincide with Martsinkovsky’s ξ-invariants and Auslander’s delta invariants. Our advantage is that we can consider these invariants for any half exact functors. We also compute these invariants for the local cohomology functors.

DOI: 10.1023/A:1011437901466
Print publication date: 6/1/2001
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by Angeleri Hügel, Lidia; Tonolo, Alberto; Trlifaj, Jan

We relate the theory of envelopes and covers to tilting and cotilting theory, for (infinitely generated) modules over arbitrary rings. Our main result characterizes tilting torsion classes as the pretorsion classes providing special preenvelopes for all modules. A dual characterization is proved for cotilting torsion-free classes using the new notion of a cofinendo module. We also construct unique representing modules for these classes.

DOI: 10.1023/A:1011485800557
Print publication date: 6/1/2001
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by Jespers, Eric; Okniński, Jan

We describe Noetherian semigroup algebras K[S] of submonoids S of polycyclic-by-finite groups over a field K. As an application, we show that these algebras are finitely presented and also that they are Jacobson rings. Next we show that every prime ideal P of K[S] is strongly related to a prime ideal of the group algebra of a subgroup of the quotient group of S via a generalised matrix ring structure on K[S]/P. Applications to the classical Krull dimension, prime spectrum, and irreducible K[S]-modules are given.

DOI: 10.1023/A:1011433816487
Print publication date: 6/1/2001
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by Drabant, Bernhard; Van Daele, Alfons

We define and investigate pairings of multiplier Hopf (*-)algebras which are nonunital generalizations of Hopf algebras. Dual pairs of multiplier Hopf algebras arise naturally from any multiplier Hopf algebra A with integral and its dual Â. Pairings of multiplier Hopf algebras play a basic rôle, e.g., in the study of actions and coactions, and, in particular, in the relation between them. This aspect of the theory is treated elsewhere. In this paper we consider the quantum double construction out of a dual pair of multiplier Hopf algebras. We show that two dually paired regular multiplier Hopf (*-)algebras A and B yield a quantum double which is again a regular multiplier Hopf (*-)algebra. If A and B have integrals, then the quantum double also has an integral. If A and B are Hopf algebras, then the quantum double multiplier Hopf algebra is the usual quantum double. The quantum double construction for dually paired multiplier Hopf (*-)algebras yields new nontrivial examples of multiplier Hopf (*-)algebras.

DOI: 10.1023/A:1011470032416
Print publication date: 6/1/2001
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