328 research outputs found
Canonical Filtrations of Gorenstein Injective Modules
The principle "Every result in classical homological algebra should have a
counterpart in Gorenstein homological algebra" is given in [3]. There is a
remarkable body of evidence supporting this claim (cf. [2] and [3]). Perhaps
one of the most glaring exceptions is provided by the fact that tensor products
of Gorenstein projective modules need not be Gorenstein projective, even over
Gorenstein rings. So perhaps it is surprising that tensor products of
Gorenstein injective modules over Gorenstein rings of finite Krull dimension
are Gorenstein injective.
Our main result is in support of the principle. Over commutative, noetherian
rings injective modules have direct sum decompositions into indecomposable
modules. We will show that Gorenstein injective modules over Gorenstein rings
of finite Krull dimension have filtrations analogous to those provided by these
decompositions. This result will then provide us with the tools to prove that
all tensor products of Gorenstein injective modules over these rings are
Gorenstein injective.Comment: 9 pages; It has been accepted for publication in Proceedings of the
American Mathematical Societ
Transfinite tree quivers and their representations
The idea of "vertex at the infinity" naturally appears when studying
indecomposable injective representations of tree quivers. In this paper we
formalize this behavior and find the structure of all the indecomposable
injective representations of a tree quiver of size an arbitrary cardinal
. As a consequence the structure of injective representations of
noetherian -trees is completely determined. In the second part we will
consider the problem whether arbitrary trees are source injective
representation quivers or not.Comment: to appear in Mathematica Scandinavic
Deconstructibility and the Hill lemma in Grothendieck categories
A full subcategory of a Grothendieck category is called deconstructible if it
consists of all transfinite extensions of some set of objects. This concept
provides a handy framework for structure theory and construction of
approximations for subcategories of Grothendieck categories. It also allows to
construct model structures and t-structures on categories of complexes over a
Grothendieck category. In this paper we aim to establish fundamental results on
deconstructible classes and outline how to apply these in the areas mentioned
above. This is related to recent work of Gillespie, Enochs, Estrada, Guil
Asensio, Murfet, Neeman, Prest, Trlifaj and others.Comment: 20 pages; version 2: minor changes, misprints corrected, references
update
Locally projective monoidal model structure for complexes of quasi-coherent sheaves on P^1(k)
We will generalize the projective model structure in the category of
unbounded complexes of modules over a commutative ring to the category of
unbounded complexes of quasi-coherent sheaves over the projective line.
Concretely we will define a locally projective model structure in the category
of complexes of quasi-coherent sheaves on the projective line. In this model
structure the cofibrant objects are the dg-locally projective complexes. We
also describe the fibrations of this model structure and show that the model
structure is monoidal. We point out that this model structure is necessarily
different from other known model structures such as the injective model
structure and the locally free model structure
Balance with Unbounded Complexes
Given a double complex there are spectral sequences with the terms
being either H (H or HH. But if
both spectral sequences have all their terms 0. This can
happen even though there is nonzero (co)homology of interest associated with
. This is frequently the case when dealing with Tate (co)homology. So in
this situation the spectral sequences may not give any information about the
(co)homology of interest. In this article we give a different way of
constructing homology groups of when HH. With this
result we give a new and elementary proof of balance of Tate homology and
cohomology
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