1,720,985 research outputs found
Matlis category equivalences for a ring epimorphism
Full text linkUnder mild assumptions, we construct the two Matlis additive category equivalences for an associative ring epimorphism u:R⟶U. Assuming that the ring epimorphism is homological of flat/projective dimension 1, we discuss the abelian categories of u-comodules and u-contramodules and construct the recollement of unbounded derived categories of R-modules, U-modules, and complexes of R-modules with u-co/contramodule cohomology. Further assumptions allow to describe the third category in the recollement as the unbounded derived category of the abelian categories of u-comodules and u-contramodules. For commutative rings, we also prove that any homological epimorphism of projective dimension 1 is flat. Injectivity of the map u is not required
Covers and direct limits: a contramodule-based approach
No full textWe present applications of contramodule techniques to the Enochs
conjecture about covers and direct limits, both in the categorical tilting context and
beyond. In the n-tilting-cotilting correspondence situation, if A is a Grothendieck
abelian category and the related abelian category B is equivalent to the category
of contramodules over a topological ring R belonging to one of certain four classes
of topological rings (e. g., R is commutative), then the left tilting class is covering
in A if and only if it is closed under direct limits in A, and if and only if all the
discrete quotient rings of the topological ring R are perfect. More generally, if M is
a module satisfying a certain telescope Hom exactness condition (e. g., M is Σ-pure-Ext -self-orthogonal) and the topological ring R of endomorphisms of M belongs to one of certain seven classes of topological rings, then the class Add(M) is closed under direct limits if and only if every countable direct limit of copies of M has an Add(M)-cover, and if and only if M has perfect decomposition. In full generality, for an additive category A with (co)kernels and a precovering class L ⊂ A closed under summands, an object N ∈ A has an L-cover if and only if a certain object Ψ(N) in an abelian category B with enough projectives has a projective cover. The 1-tilting modules and objects arising from injective ring epimorphisms of projective dimension 1 form a class of examples which we discuss
Definable coaisles over rings of weak global dimension at most one
Full text linkIn the setting of the unbounded derived category of a ring of weak global dimension at most one we consider t-structures with a definable coaisle. The t-structures among these which are stable (that is, the t-structures which consist of a pair of triangulated subcategories) are precisely the ones associated to a smashing localization of the derived category. In this way, our present results generalize those of cite{BS} to the non-stable case. As in the stable case, we confine for the most part to the commutative setting, and give a full classification of definable coaisles in the local case, that is, over valuation domains. It turns out that unlike in the stable case of smashing subcategories, the definable coaisles do not always arise from homological ring epimorphisms. We also consider a non-stable version of the telescope conjecture for t-structures and give a ring-theoretic characterization of the commutative rings of weak global dimension at most one for which it is satisfied
Covering classes and 1-tilting cotorsion pairs over commutative rings
Full text linkWe are interested in characterising the commutative rings for which a 1-tilting cotorsion pair provides for covers, that is when the class A is a covering class. We use Hrbekšfs bijective correspondence between the 1-tilting cotorsion pairs over a commutative ring R and the faithful finitely generated Gabriel topologies on R. Moreover, we use results of Bazzoni-Positselski, in particular a generalisation of Matlis equivalence and their characterisation of covering classes for 1-tilting cotorsion pairs arising from flat injective ring epimorphisms. Explicitly, if is the Gabriel topology associated to the 1-tilting cotorsion pair, and R is the ring of quotients with respect to, we show that if A is covering, then G is a perfect localisation (in Stenstromšfs sense [B. Stenstrom, Rings of Quotients, Springer, New York, 1975]) and the localisation R has projective dimension at most one as an R-module. Moreover, we show that is covering if and only if both the localisation RG and the quotient rings R/J are perfect rings for every J ∈. Rings satisfying the latter two conditions are called G-almost perfect
Derived equivalence induced by infinitely generated n-tilting modules
No full textLet TR be a right n-tilting module over an arbitrary associative ring R. In this paper we prove that there exists an n-tilting module TR′ equiva- lent to TR which induces a derived equivalence between the unbounded derived category D(R) and a triangulated subcategory E⊥ of D(End(T′)) equivalent to the quotient category of D(End(T′)) modulo the kernel of the total left derived functor − ⊗LS′ T ′. If TR is a classical n-tilting module, we have that T = T ′ and the subcategory E⊥ coincides with D(End |(T )), recovering the classical case
S-almost perfect commutative rings
No full textGiven a multiplicative subset S in a commutative ring R, we consider S-weakly cotorsion and S-strongly flat R-modules, and show that all R-modules have S-strongly flat covers if and only if all flat R-modules are S-strongly flat. These equivalent conditions hold if and only if the localization R_S is a perfect ring and, for every element s ∈ S, the quotient ring R/sR is a perfect ring, too. The multiplicative subset S ⊂ R is allowed to contain zero-divisors
Pure projective tilting modules
Full text linkLet TR be a 1-tilting module with tilting torsion pair (Gen T , F ) in Mod-R. The following conditions are proved to be equiv- alent: (1) T is pure projective; (2) Gen T is a definable subcategory of Mod-R with enough pure projectives; (3) both classes GenT and F are finitely axiomatizable; and (4) the heart of the corresponding HRS t-structure (in the derived category Db(Mod-R)) is Grothendieck. This article explores in this context the question raised by Saor ́ın if the Grothendieck condition on the heart of an HRS t-structure implies that it is equivalent to a module category. This amounts to asking if T is tilting equivalent to a finitely presented module. This is re- solved in the positive for a Krull-Schmidt ring, and for a commutative ring, a positive answer follows from a proof that every pure projective 1-tilting module is projective. However, a general criterion is found that yields a negative answer to Saor ́ın’s Question and this criterion is satisfied by the universal enveloping algebra of a semisimple Lie algebra, a left and right noetherian domain
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