134 research outputs found
On -Dominant Dimension
Let and be artin algebras and a
faithfully balanced selforthogonal bimodule. We show that the -dominant
dimensions of and are identical. As applications to
the results obtained, we give some characterizations of double dual functors
(with respect to ) preserving monomorphisms and being
left exact respectively.Comment: 13 page
Proper Resolutions and Gorenstein Categories
Let be an abelian category and an additive full
subcategory of . We provide a method to construct a proper
-resolution (resp. coproper -coresolution) of one
term in a short exact sequence in from that of the other two
terms. By using these constructions, we answer affirmatively an open question
on the stability of the Gorenstein category posed by
Sather-Wagstaff, Sharif and White; and also prove that
is closed under direct summands. In addition, we
obtain some criteria for computing the -dimension and the
-dimension of an object in .Comment: 35 pages. arXiv admin note: substantial text overlap with
arXiv:1012.170
On the grade of modules over Noetherian rings
Let be a left and right noetherian ring and the
category of finitely generated left -modules. In this paper we show
the following results: (1) For a positive integer , the condition that the
subcategory of consisting of -torsionfree modules coincides
with the subcategory of consisting of -syzygy modules for any
is left-right symmetric. (2) If is an Auslander ring
and is in with \grade N=k<\infty, then is pure
of grade if and only if can be embedded into a finite direct sum of
copies of the st term in a minimal injective resolution of as
a right -module. (3) Assume that both the left and right
self-injective dimensions of are . If \grade {\rm
Ext}_{\Lambda}^k(M, \Lambda)\geq k for any and \grade {\rm
Ext}_{\Lambda}^i(N, \Lambda)\geq i for any and , then the socle of the last term in a minimal injective resolution
of as a right -module is non-zero.Comment: 17 pages. To appear in Communications in Algebr
Duality of Preenvelopes and Pure Injective Modules
Let be an arbitrary ring and (-)^+=\Hom_{\mathbb{Z}}(-,
\mathbb{Q}/\mathbb{Z}) where is the ring of integers and
is the ring of rational numbers, and let be a
subcategory of left -modules and a subcategory of right
-modules such that for any and all
modules in are pure injective. Then a homomorphism of
left -modules with is a -(pre)envelope of
provided is a -(pre)cover of . Some
applications of this result are given.Comment: 9 pages, to appear in Canadian Mathematical Bulleti
Generalized tilting modules with finite injective dimension
Let be a left noetherian ring, a right noetherian ring and a
generalized tilting module with . The injective dimensions of
and are identical provided both of them are finite. Under the
assumption that the injective dimensions of and are finite, we
describe when the subcategory is a finitely generated
right -module is closed under submodules. As a consequence, we obtain a
negative answer to a question posed by Auslander in 1969. Finally, some partial
answers to Wakamatsu Tilting Conjecture are given.Comment: 18 pages. This is the final version. To appear in Journal of Algebr
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