83 research outputs found
Matlis category equivalences for a ring epimorphism
Under mild assumptions, we construct the two Matlis additive category
equivalences for an associative ring epimorphism . Assuming
that the ring epimorphism is homological of flat/projective dimension , we
discuss the abelian categories of -comodules and -contramodules and
construct the recollement of unbounded derived categories of -modules,
-modules, and complexes of -modules with -co/contramodule cohomology.
Further assumptions allow to describe the third category in the recollement as
the unbounded derived category of the abelian categories of -comodules and
-contramodules. For commutative rings, we also prove that any homological
epimorphism of projective dimension is flat. Injectivity of the map is
not required.Comment: LaTeX 2e with tikz-cd, 30 pages, 6 commutative diagrams. v.1: This is
an improved, expanded version of Sections 16-18 of the long preprint
arXiv:1807.10671v1, which was divided into three parts. v.2: Terminological
change of "u-h-divisible" to "u-divisible"; Remark 1.2(1), Proposition 2.4,
Lemma 3.4, and Remark 3.5 inserted; references added; v.3: Final versio
Smashing localizations of rings of weak global dimension at most one
We show for a ring R of weak global dimension at most one that there is a
bijection between the smashing subcategories of its derived category and the
equivalence classes of homological epimorphisms starting in R. If, moreover, R
is commutative, we prove that the compactly generated localizing subcategories
correspond precisely to flat epimorphisms. We also classify smashing
localizations of the derived category of any valuation domain, and provide an
easy criterion for the Telescope Conjecture (TC) for any commutative ring of
weak global dimension at most one. As a consequence, we show that the TC holds
for any commutative von Neumann regular ring R, and it holds precisely for
those Pr\"ufer domains which are strongly discrete.Comment: 45 pages; version 2: several changes in the presentation (a section
on the homotopy category of dg algebras became an appendix, more explanation
added at various places of the text), main results unchange
Flat Mittag-Leffler modules over countable rings
We show that over any ring, the double Ext-orthogonal class to all flat
Mittag-Leffler modules contains all countable direct limits of flat
Mittag-Leffler modules. If the ring is countable, then the double orthogonal
class consists precisely of all flat modules and we deduce, using a recent
result of \v{S}aroch and Trlifaj, that the class of flat Mittag-Leffler modules
is not precovering in Mod-R unless R is right perfect.Comment: 7 pages; version 2: minor changes, more explanation added in the
proof of Theorem 6 and Lemma 7, references added and update
Cotorsion pairs generated by modules of bounded projective dimension
We apply the theory of cotorsion pairs to study closure properties of classes
of modules with finite projective dimension with respect to direct limit
operations and to filtrations.
We also prove that if the ring is an order in an -noetherian ring Q
of small finitistic dimension 0, then the cotorsion pair generated by the
modules of projective dimension at most one is of finite type if and only if Q
has big finitistic dimension 0. This applies, for example, to semiprime Goldie
rings and Cohen Macaulay noetherian commutative rings.Comment: 30 Page
Recollements from Cotorsion Pairs
Given a complete hereditary cotorsion pair in a
Grothendieck category , the derived category
of the exact category is defined as
the quotient of the category , of unbounded complexes
with terms in , modulo the subcategory
consisting of the acyclic complexes with terms in and cycles in
.
We restrict our attention to the cotorsion pairs such that
coincides with the class of the
acyclic complexes of with terms in . In
this case the derived category fits into a
recollement .
We will explore the conditions under which
and provide many examples.
Symmetrically, we prove analogous results for the exact category
.Comment: Added Lemma 1.2 and fixed statement of Proposition 2.
Covers and direct limits: a contramodule-based approach
We present applications of contramodule techniques to the Enochs conjecture
about covers and direct limits, both in the categorical tilting context and
beyond. In the -tilting-cotilting correspondence situation, if
is a Grothendieck abelian category and the related abelian category
is equivalent to the category of contramodules over a topological ring
belonging to one of certain four classes of topological rings
(e.g., is commutative), then the left tilting class is covering
in if and only if it is closed under direct limits in ,
and if and only if all the discrete quotient rings of the topological ring
are perfect. More generally, if is a module satisfying a
certain telescope Hom exactness condition (e.g., is
-pure--self-orthogonal) and the topological ring
of endomorphisms of belongs to one of certain seven classes
of topological rings, then the class is closed under direct
limits if and only if every countable direct limit of copies of has an
-cover, and if and only if has perfect decomposition. In
full generality, for an additive category with (co)kernels and a
precovering class closed under summands, an object
has an -cover if and only if a certain object
in an abelian category with enough projectives has a
projective cover. The -tilting modules and objects arising from injective
ring epimorphisms of projective dimension form a class of examples which we
discuss.Comment: LaTeX 2e with pb-diagram and xy-pic, 58 pages, 5 commutative
diagrams. v.1: This paper is based on Sections 11-15 and 19 of the long
preprint arXiv:1807.10671v1, which was divided into three parts. v.2: Many
important improvements and additions based on new results in arXiv:1909.12203
and particularly in arXiv:1911.11720; new Section 5 inserted. v.4: Final
versio
-almost perfect commutative rings
Given a multiplicative subset in a commutative ring , we consider
-weakly cotorsion and -strongly flat -modules, and show that all
-modules have -strongly flat covers if and only if all flat -modules
are -strongly flat. These equivalent conditions hold if and only if the
localization is a perfect ring and, for every element , the
quotient ring is a perfect ring, too. The multiplicative subset
is allowed to contain zero-divisors.Comment: 29 pages; v.2: final versio
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