arXiv subjects

Leonid Positselski

Publications and source records attributed to Leonid Positselski.

At least 55 records · Page 3Linked to original sources

arXiv2023

Differential graded Koszul duality: an introductory survey

This is an overview on derived nonhomogeneous Koszul duality over a field, mostly based on the author's memoir arXiv:0905.2621. The paper is intended to serve as a pedagogical introduction and a summary of the covariant duality between DG-algebras and curved DG-coalgebras, as well as the triality between DG-modules, CDG-comodules, and CDG-contramodules. Some personal reminiscences are included as a part of historical discussion.

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arXiv2022

Remarks on derived complete modules and complexes

Let $R$ be a commutative ring and $I\subset R$ a finitely generated ideal. We discuss two definitions of derived $I$-adically complete (also derived $I$-torsion) complexes of $R$-modules which appear in the literature: the idealistic and the sequential one. The two definitions are known to be equivalent for a weakly proregular ideal $I$; we show that they are different otherwise. We argue that the sequential approach works well, but the idealistic one needs to be reinterpreted or properly understood. We also consider $I$-adically flat $R$-modules.

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arXiv2022

Abelian right perpendicular subcategories in module categories

We show that an abelian category can be exactly, fully faithfully embedded into a module category as the right perpendicular subcategory to a set of modules or module morphisms if and only if it is a locally presentable abelian category with a projective generator, or in other words, the category of models of an additive algebraic theory of possibly infinite bounded arity. This includes the categories of contramodules over topological rings and other examples. Various versions of the definition of the right perpendicular subcategory are considered, all leading to the same class of abelian categories. We also discuss sufficient conditions under which the natural forgetful functors from the categories of contramodules to the related categories of modules are fully faithful.

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arXiv2022

Countably generated flat modules are quite flat

We prove that if $R$ is a commutative Noetherian ring, then every countably generated flat $R$-module is quite flat, i.e., a direct summand of a transfinite extension of localizations of $R$ in countable multiplicative subsets. We also show that if the spectrum of $R$ is of cardinality less than $κ$, where $κ$ is an uncountable regular cardinal, then every flat $R$-module is a transfinite extension of flat modules with less than $κ$ generators. This provides an alternative proof of the fact that over a commutative Noetherian ring with countable spectrum, all flat modules are quite flat. More generally, we say that a commutative ring is CFQ if every countably presented flat $R$-module is quite flat. We show that all von Neumann regular rings and all $S$-almost perfect rings are CFQ. A zero-dimensional local ring is CFQ if and only if it is perfect. A domain is CFQ if and only if all its proper quotient rings are CFQ. A valuation domain is CFQ if and only if it is strongly discrete.

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arXiv2022

Closure properties of $\varinjlim\mathcal C$

Let $\mathcal C$ be a class of modules and $\mathcal L = \varinjlim \mathcal C$ the class of all direct limits of modules from $\mathcal C$. The class $\mathcal L$ is well understood when $\mathcal C$ consists of finitely presented modules: $\mathcal L$ then enjoys various closure properties. We study the closure properties of $\mathcal L$ in the general case when $\mathcal C \subseteq \mathrm{Mod-}R$ is arbitrary. Then we concentrate on two important particular cases, when $\mathcal C = \operatorname{add} M$ and $\mathcal C = \operatorname{Add} M$, for an arbitrary module $M$. In the first case, we prove that $\varinjlim \operatorname{add} M = \{ N \in \mathrm{Mod-} R \mid \exists F \in \mathcal F_S: N \cong F \otimes_S M \}$ where $S = \operatorname{End} M$, and $\mathcal F_S$ is the class of all flat right $S$-modules. In the second case, $\varinjlim \operatorname{Add} M = \{ \mathfrak F \odot _{\mathfrak S} M \mid \mathfrak F \in \mathcal F_{\mathfrak S} \}$ where $\mathfrak S$ is the endomorphism ring of $M$ endowed with the finite topology, $\mathcal F_{\mathfrak S}$ is the class of all right $\mathfrak S$-contramodules that are direct limits of direct systems of projective right $\mathfrak S$-contramodules, and $\odot_{\mathfrak S}$ denotes the contratensor product. For various classes of modules $\mathcal D$, we show that if $M \in \mathcal D$ then $\varinjlim \operatorname{add} M = \varinjlim \operatorname{Add} M$ (e.g., when $\mathcal D$ consists of pure projective modules), but the equality for an arbitrary module $M$ remains open. Finally, we deal with the question of whether $\varinjlim \operatorname{Add} M = \widetilde{\operatorname{Add} M}$ where $\widetilde{\operatorname{Add} M}$ is the class of all pure epimorphic images of direct sums of copies of a module $M$. We show that the answer is positive in several particular cases, but it is negative in general.

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arXiv2022

Topologically semisimple and topologically perfect topological rings

Extending the Wedderburn-Artin theory of (classically) semisimple associative rings to the realm of topological rings with right linear topology, we show that the abelian category of left contramodules over such a ring is split (equivalently, semisimple) if and only if the abelian category of discrete right modules over the same ring is split (equivalently, semisimple). Our results in this direction complement those of Iovanov-Mesyan-Reyes. An extension of the Bass theory of left perfect rings to the topological realm is formulated as a list of conjecturally equivalent conditions, many equivalences and implications between which we prove. In particular, all the conditions are equivalent for topological rings with a countable base of neighborhoods of zero and for topologically right coherent topological rings. Considering the rings of endomorphisms of modules as topological rings with the finite topology, we establish a close connection between the concept of a topologically perfect topological ring and the theory of modules with perfect decomposition. Our results also apply to endomorphism rings and direct sum decompositions of objects in certain additive categories more general than the categories of modules; we call them topologically agreeable categories. We show that any topologically agreeable split abelian category is Grothendieck and semisimple. We also prove that a module $Σ$-coperfect over its endomorphism ring has a perfect decomposition provided that either the endomorphism ring is commutative or the module is countably generated, partially answering a question of Angeleri Hugel and Saorin.

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arXiv2022

Contramodules

Contramodules are module-like algebraic structures endowed with infinite summation (or, occasionally, integration) operations satisfying natural axioms. Introduced originally by Eilenberg and Moore in 1965 in the case of coalgebras over commutative rings, contramodules experience a small renaissance now after being all but forgotten for three decades between 1970-2000. Here we present a review of various definitions and results related to contramodules (drawing mostly from our monographs and preprints arXiv:0708.3398, arXiv:0905.2621, arXiv:1202.2697, arXiv:1209.2995, arXiv:1512.08119, arXiv:1710.02230, arXiv:1705.04960, arXiv:1808.00937) - including contramodules over corings, topological associative rings, topological Lie algebras and topological groups, semicontramodules over semialgebras, and a "contra version" of the Bernstein-Gelfand-Gelfand category O. Several underived manifestations of the comodule-contramodule correspondence phenomenon are discussed.

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arXiv2022

Relative nonhomogeneous Koszul duality

This book contains a detailed exposition of the nonhomogeneous Koszul duality theory in the relative situation over a noncentral, noncommutative, nonsemisimple base ring, as announced in Section 0.4 of arXiv:0708.3398. We prove the Poincare-Birkhoff-Witt theorem in this context and construct the triangulated equivalences of derived Koszul duality. The duality between the ring of differential operators and the de Rham DG-algebra, with the ring of functions as the base ring, is the thematic example. The moderate generality level makes the exposition in this book more accessible than the very heavily technical Chapter 11 of arXiv:0708.3398.

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arXiv2021

Flat commutative ring epimorphisms of almost Krull dimension zero

We consider flat epimorphisms of commutative rings $R\to U$ such that, for every ideal $I\subset R$ for which $IU=U$, the quotient ring $R/I$ is semilocal of Krull dimension zero. Under these assumptions, we show that the projective dimension of the $R$-module $U$ does not exceed $1$. We also describe the Geigle-Lenzing perpendicular subcategory $U^{\perp_{0,1}}$ in $R\mathsf{-Mod}$. Assuming additionally that the ring $U$ and all the rings $R/I$ are perfect, we show that all flat $R$-modules are $U$-strongly flat. Thus we obtain a generalization of some results of the paper arXiv:1801.04820, where the case of the localization $U=S^{-1}R$ of the ring $R$ at a multiplicative subset $S\subset R$ was considered.

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arXiv2021

Contramodules over pro-perfect topological rings

For four wide classes of topological rings $\mathfrak R$, we show that all flat left $\mathfrak R$-contramodules have projective covers if and only if all flat left $\mathfrak R$-contramodules are projective if and only if all left $\mathfrak R$-contramodules have projective covers if and only if all descending chains of cyclic discrete right $\mathfrak R$-modules terminate if and only if all the discrete quotient rings of $\mathfrak R$ are left perfect. Three classes of topological rings for which this holds are the complete, separated topological associative rings with a base of neighborhoods of zero formed by open two-sided ideals such that either the ring is commutative, or it has a countable base of neighborhoods of zero, or it has only a finite number of semisimple discrete quotient rings. The fourth class consists of all the topological rings with a base of neighborhoods of zero formed by open right ideals which have a closed two-sided ideal with certain properties such that the quotient ring is a topological product of rings from the previous three classes. The key technique on which the proofs are based is the contramodule Nakayama lemma for topologically T-nilpotent ideals.

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arXiv2021

Pseudo-dualizing complexes of bicomodules and pairs of t-structures

This paper is a coalgebra version of arXiv:1703.04266 and a sequel to arXiv:1607.03066. We present the definition of a pseudo-dualizing complex of bicomodules over a pair of coassociative coalgebras $\mathcal C$ and $\mathcal D$. For any such complex $\mathcal L^\bullet$, we construct a triangulated category endowed with a pair of (possibly degenerate) t-structures of the derived type, whose hearts are the abelian categories of left $\mathcal C$-comodules and left $\mathcal D$-contramodules. A weak version of pseudo-derived categories arising out of (co)resolving subcategories in abelian/exact categories with enough homotopy adjusted complexes is also considered. Quasi-finiteness conditions for coalgebras, comodules, and contramodules are discussed as a preliminary material.

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arXiv2021

Projective covers of flat contramodules

We show that a direct limit of projective contramodules (over a right linear topological ring) is projective if it has a projective cover. A similar result is obtained for $\infty$-strictly flat contramodules of projective dimension not exceeding $1$, using an argument based on the notion of the topological Jacobson radical. Covers and precovers of direct limits of more general classes of objects, both in abelian categories with exact and with nonexact direct limits, are also discussed, with an eye towards the Enochs conjecture about covers and direct limits, using locally split (mono)morphisms as the main technique. In particular, we offer a simple elementary proof of the Enochs conjecture for the left class of an $n$-tilting cotorsion pair in an abelian category with exact direct limits.

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arXiv2021

Flat ring epimorphisms of countable type

Let $R\to U$ be an associative ring epimorphism such that $U$ is a flat left $R$-module. Assume that the related Gabriel topology $\mathbb G$ of right ideals in $R$ has a countable base. Then we show that the left $R$-module $U$ has projective dimension at most $1$. Furthermore, the abelian category of left contramodules over the completion of $R$ at $\mathbb G$ fully faithfully embeds into the Geigle-Lenzing right perpendicular subcategory to $U$ in the category of left $R$-modules, and every object of the latter abelian category is an extension of two objects of the former one. We discuss conditions under which the two abelian categories are equivalent. Given a right linear topology on an assocative ring $R$, we consider the induced topology on every left $R$-module, and for a perfect Gabriel topology $\mathbb G$ compare the completion of a module with an appropriate Ext module. Finally, we characterize the $U$-strongly flat left $R$-modules by the two conditions of left positive-degree Ext-orthogonality to all left $U$-modules and all $\mathbb G$-separated $\mathbb G$-complete left $R$-modules.

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arXiv2021

Derived, coderived, and contraderived categories of locally presentable abelian categories

For a locally presentable abelian category $\mathsf B$ with a projective generator, we construct the projective derived and contraderived model structures on the category of complexes, proving in particular the existence of enough homotopy projective complexes of projective objects. We also show that the derived category $\mathsf D(\mathsf B)$ is generated, as a triangulated category with coproducts, by the projective generator of $\mathsf B$. For a Grothendieck abelian category $\mathsf A$, we construct the injective derived and coderived model structures on complexes. Assuming Vopenka's principle, we prove that the derived category $\mathsf D(\mathsf A)$ is generated, as a triangulated category with products, by the injective cogenerator of $\mathsf A$. More generally, we define the notion of an exact category with an object size function and prove that the derived category of any such exact category with exact $κ$-directed colimits of chains of admissible monomorphisms has Hom sets. In particular, the derived category of any locally presentable abelian category has Hom sets.

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arXiv2021

Unbounded derived categories of small and big modules: Is the natural functor fully faithful?

Consider the obvious functor from the unbounded derived category of all finitely generated modules over a left noetherian ring $R$ to the unbounded derived category of all modules. We answer the natural question whether this functor defines an equivalence onto the full subcategory of complexes with finitely generated cohomology modules in two special cases. If $R$ is a quasi-Frobenius ring of infinite global dimension, then this functor is not full. If $R$ has finite left global dimension, this functor is an equivalence. We also prove variants of the latter assertion for left coherent rings, for noetherian schemes and for locally noetherian Grothendieck categories.

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arXiv2021

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 $n$-tilting-cotilting correspondence situation, if $\mathsf A$ is a Grothendieck abelian category and the related abelian category $\mathsf B$ is equivalent to the category of contramodules over a topological ring $\mathfrak R$ belonging to one of certain four classes of topological rings (e.g., $\mathfrak R$ is commutative), then the left tilting class is covering in $\mathsf A$ if and only if it is closed under direct limits in $\mathsf A$, and if and only if all the discrete quotient rings of the topological ring $\mathfrak R$ are perfect. More generally, if $M$ is a module satisfying a certain telescope Hom exactness condition (e.g., $M$ is $Σ$-pure-$\operatorname{Ext}^1$-self-orthogonal) and the topological ring $\mathfrak R$ of endomorphisms of $M$ belongs to one of certain seven classes of topological rings, then the class $\mathsf{Add}(M)$ is closed under direct limits if and only if every countable direct limit of copies of $M$ has an $\mathsf{Add}(M)$-cover, and if and only if $M$ has perfect decomposition. In full generality, for an additive category $\mathsf A$ with (co)kernels and a precovering class $\mathsf L\subset\mathsf A$ closed under summands, an object $N\in\mathsf A$ has an $\mathsf L$-cover if and only if a certain object $Ψ(N)$ in an abelian category $\mathsf B$ 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.

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arXiv2020

Matlis category equivalences for a ring epimorphism

Under mild assumptions, we construct the two Matlis additive category equivalences for an associative ring epimorphism $u\colon R\to U$. 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.

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arXiv2020

Cofibrant generation of pure monomorphisms

We show that pure monomorphisms are cofibrantly generated---generated from a set of morphisms by pushouts, transfinite composition, and retracts---in any locally finitely presentable additive category. In particular, this is true in any category of $R$-modules.

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