arXiv subjects

Leonid Positselski

Publications and source records attributed to Leonid Positselski.

At least 37 records · Page 2Linked to original sources

arXiv2025

Generalized periodicity theorems

Let $R$ be a ring and $\mathsf S$ be a class of strongly finitely presented (FP${}_\infty$) $R$-modules closed under extensions, direct summands, and syzygies. Let $(\mathsf A,\mathsf B)$ be the (hereditary complete) cotorsion pair generated by $\mathsf S$ in $\textsf{Mod-}R$, and let $(\mathsf C,\mathsf D)$ be the (also hereditary complete) cotorsion pair in which $\mathsf C=\varinjlim\mathsf A=\varinjlim\mathsf S$. We show that any $\mathsf A$-periodic module in $\mathsf C$ belongs to $\mathsf A$, and any $\mathsf D$-periodic module in $\mathsf B$ belongs to $\mathsf D$. Further generalizations of both results are obtained, so that we get a common generalization of the flat/projective and fp-projective periodicity theorems, as well as a common generalization of the fp-injective/injective and cotorsion periodicity theorems. Both are applicable to modules over an arbitrary ring, and in fact, to Grothendieck categories.

math.RA↗

arXiv2024

Compact generators of the contraderived category of contramodules

We consider the contraderived category of left contramodules over a right linear topological ring $\mathfrak R$ with a countable base of neighborhoods of zero. Equivalently, this is the homotopy category of unbounded complexes of projective left $\mathfrak R$-contramodules. Assuming that the abelian category of discrete right $\mathfrak R$-modules is locally coherent, we show that the contraderived category of left $\mathfrak R$-contramodules is compactly generated, and describe its full subcategory of compact objects as the opposite category to the bounded derived category of finitely presentable discrete right $\mathfrak R$-modules. Under the same assumptions, we also prove the flat and projective periodicity theorem for $\mathfrak R$-contramodules.

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arXiv2024

Flat quasi-coherent sheaves as directed colimits, and quasi-coherent cotorsion periodicity

We show that every flat quasi-coherent sheaf on a quasi-compact quasi-separated scheme is a directed colimit of locally countably presentable flat quasi-coherent sheaves. More generally, the same assertion holds for any countably quasi-compact, countably quasi-separated scheme. Moreover, for three categories of complexes of flat quasi-coherent sheaves, we show that all complexes in the category can be obtained as directed colimits of complexes of locally countably presentable flat quasi-coherent sheaves from the same category. In particular, on a quasi-compact semi-separated scheme, every flat quasi-coherent sheaf is a directed colimit of flat quasi-coherent sheaves of finite projective dimension. In the second part of the paper, we discuss cotorsion periodicity in category-theoretic context, generalizing an argument of Bazzoni, Cortes-Izurdiaga, and Estrada. As the main application, we deduce the assertion that any cotorsion-periodic quasi-coherent sheaf on a quasi-compact semi-separated scheme is cotorsion.

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arXiv2024

Flat comodules and contramodules as directed colimits, and cotorsion periodicity

This paper is a follow-up to arXiv:2212.09639. We consider two algebraic settings of comodules over a coring and contramodules over a topological ring with a countable base of two-sided ideals. These correspond to two (noncommutative) algebraic geometry settings of certain kind of stacks and ind-affine ind-schemes. In the context of a coring $\mathcal C$ over a noncommutative ring $A$, we show that all $A$-flat $\mathcal C$-comodules are $\aleph_1$-directed colimits of $A$-countably presentable $A$-flat $\mathcal C$-comodules. In the context of a complete, separated topological ring $\mathfrak R$ with a countable base of neighborhoods of zero consisting of two-sided ideals, we prove that all flat $\mathfrak R$-contramodules are $\aleph_1$-directed colimits of countably presentable flat $\mathfrak R$-contramodules. We also describe arbitrary complexes, short exact sequences, and pure acyclic complexes of $A$-flat $\mathcal C$-comodules and flat $\mathfrak R$-contramodules as $\aleph_1$-directed colimits of similar complexes of countably presentable objects. The arguments are based on a very general category-theoretic technique going back to an unpublished 1977 preprint of Ulmer and rediscovered in arXiv:2310.16773. Applications to cotorsion periodicity and coderived categories of flat objects in the respective settings are discussed. In particular, in any acyclic complex of cotorsion $\mathfrak R$-contramodules, all the contramodules of cocycles are cotorsion.

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arXiv2024

Coderived and contraderived categories of locally presentable abelian DG-categories

The concept of an abelian DG-category, introduced by the first-named author in arXiv:2110.08237, unites the notions of abelian categories and (curved) DG-modules in a common framework. In this paper we consider coderived and contraderived categories in the sense of Becker. Generalizing some constructions and results from the preceding papers by Becker arXiv:1205.4473 and by the present authors arXiv:2101.10797, we define the contraderived category of a locally presentable abelian DG-category $\mathbf B$ with enough projective objects and the coderived category of a Grothendieck abelian DG-category $\mathbf A$. We construct the related abelian model category structures and show that the resulting exotic derived categories are well-generated. Then we specialize to the case of a locally coherent Grothendieck abelian DG-category $\mathbf A$, and prove that its coderived category is compactly generated by the absolute derived category of finitely presentable objects of $\mathbf A$, thus generalizing a result from the second-named author's preprint arXiv:1412.1615. In particular, the homotopy category of graded-injective left DG-modules over a DG-ring with a left coherent underlying graded ring is compactly generated by the absolute derived category of DG-modules with finitely presentable underlying graded modules. We also describe compact generators of the coderived categories of quasi-coherent matrix factorizations over coherent schemes.

math.CT↗

arXiv2024

Locally coherent exact categories

A locally coherent exact category is a finitely accessible additive category endowed with an exact structure in which the admissible short exact sequences are the directed colimits of admissible short exact sequences of finitely presentable objects. We show that any exact structure on a small idempotent-complete additive category extends uniquely to a locally coherent exact structure on the category of ind-objects; in particular, any finitely accessible category has the unique maximal and the unique minimal locally coherent exact category structures. All locally coherent exact categories are of Grothendieck type in the sense of Stovicek. We also discuss the canonical embedding of a small exact category into the abelian category of additive sheaves in connection with the locally coherent exact structure on the ind-objects, and deduce two periodicity theorems as applications.

math.CT↗

arXiv2024

Exact categories of topological vector spaces with linear topology

We explain why the naive definition of a natural exact category structure on complete, separated topological vector spaces with linear topology fails. In particular, contrary to arXiv:0711.2527, the category of such topological vector spaces is not quasi-abelian. We present a corrected definition of exact category structure which works OK. Then we explain that the corrected definition still has a shortcoming in that a natural tensor product functor is not exact in it, and discuss ways to refine the exact category structure so as to make the tensor product functors exact.

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arXiv2024

Notes on limits of accessible categories

Let $κ$ be a regular cardinal, $λ<κ$ be a smaller infinite cardinal, and $\mathsf K$ be a $κ$-accessible category where colimits of $λ$-indexed chains exist. We show that various category-theoretic constructions applied to $\mathsf K$, such as the inserter and the equifier, produce $κ$-accessible categories $\mathsf E$ again, and the most obvious expected description of the full subcategory of $κ$-presentable objects in $\mathsf E$ in terms of $κ$-presentable objects in $\mathsf K$ holds true. In particular, if $\mathsf C$ is a $κ$-small category, then the category of functors $\mathsf C\rightarrow\mathsf K$ is $κ$-accessible, and its $κ$-presentable objects are precisely all the functors from $\mathsf C$ to the $κ$-presentable objects of $\mathsf K$. We proceed to discuss the preservation of $κ$-accessibility by conical pseudolimits, lax and oplax limits, and weighted pseudolimits. The results of this paper go back to an unpublished 1977 preprint of Ulmer. Our motivation comes from the theory of flat modules and flat quasi-coherent sheaves.

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arXiv2024

Philosophy of contraherent cosheaves

Contraherent cosheaves are module objects over algebraic varieties defined by gluing using the colocalization functors. Contraherent cosheaves are designed to be used for globalizing contramodules and contraderived categories for the purposes of Koszul duality and semi-infinite algebraic geometry. One major technical problem associated with contraherent cosheaves is that the colocalization functors, unlike the localizations, are not exact. The reason is that, given a commutative ring homomorphism $R\rightarrow S$ arising in connection with a typical covering in algebraic geometry, the ring $S$ is usually a flat, but not a projective $R$-module. We argue that the relevant difference between projective and flat modules, from the standpoint of homological algebra, is not that big, as manifested by the flat/projective and cotorsion periodicity theorems. The difference becomes even smaller if one observes that the ring $S$ is often a very flat $R$-module.

math.AG↗

arXiv2024

A bounded below, noncontractible, acyclic complex of projective modules

We construct examples of bounded below, noncontractible, acyclic complexes of finitely generated projective modules over some rings $S$, as well as bounded above, noncontractible, acyclic complexes of injective modules. The rings $S$ are certain rings of infinite matrices with entries in the rings of commutative polynomials or formal power series in infinitely many variables. In the world of comodules or contramodules over coalgebras over fields, similar examples exist over the cocommutative symmetric coalgebra of an infinite-dimensional vector space. A simpler, universal example of a bounded below, noncontractible, acyclic complex of free modules with one generator, communicated to the author by Canonaco, is included at the end of the paper.

math.RA↗

arXiv2024

Local, colocal, and antilocal properties of modules and complexes over commutative rings

This paper is a commutative algebra introduction to the homological theory of quasi-coherent sheaves and contraherent cosheaves over quasi-compact semi-separated schemes. Antilocality is an alternative way in which global properties are locally controlled in a finite affine open covering. For example, injectivity of modules over non-Noetherian commutative rings is not preserved by localizations, while homotopy injectivity of complexes of modules is not preserved by localizations even for Noetherian rings. The latter also applies to the contraadjustedness and cotorsion properties. All the mentioned properties of modules or complexes over commutative rings are actually antilocal. They are also colocal, if one presumes contraadjustedness. Generally, if the left class in a (hereditary complete) cotorsion theory for modules or complexes of modules over commutative rings is local and preserved by direct images with respect to affine open immersions, then the right class is antilocal. If the right class in a cotorsion theory for contraadjusted modules or complexes of contraadjusted modules is colocal and preserved by such direct images, then the left class is antilocal. As further examples, the class of flat contraadjusted modules is antilocal, and so are the classes of acyclic, Becker-coacyclic, or Becker-contraacyclic complexes of contraadjusted modules. The same applies to the classes of homotopy flat complexes of flat contraadjusted modules and acyclic complexes of flat contraadjusted modules with flat modules of cocycles.

math.AC↗

arXiv2023

Comodules and contramodules over coalgebras associated with locally finite categories

We explain how to attach a coalgebra $\mathcal C$ over a field $k$ to a small $k$-linear category $\mathsf E$ satisfying suitable finiteness conditions. In this context, we study full-and-faithfulness of the contramodule forgetful functor, and describe explicitly the categories of locally finite left $\mathcal C$-comodules and left $\mathcal C$-contramodules as certain full subcategories of the category of left $\mathsf E$-modules.

math.CT↗

arXiv2023

Semialgebras associated with nonunital algebras and $k$-linear subcategories

This paper is a sequel to arXiv:2307.13358 and arXiv:2308.16090. A construction associating a semialgebra with an algebra, subalgebra, and a coalgebra dual to the subalgebra played a central role in the author's book arXiv:0708.3398. In this paper, we extend this construction to certain nonunital algebras. The resulting semialgebra is still semiunital over a counital coalgebra. In particular, we associate semialgebras to locally finite subcategories in $k$-linear categories. Examples include the Temperley-Lieb and Brauer diagram categories and the Reedy category of simplices in a simplicial set.

math.CT↗

arXiv2023

Tensor-Hom formalism for modules over nonunital rings

We say that a ring $R$ is t-unital if the natural map $R\otimes_RR\rightarrow R$ is an isomorphism, and a left $R$-module $P$ is c-unital if the natural map $P\rightarrow\operatorname{Hom}_R(R,P)$ is an isomorphism. For a t-unital ring $R$, the category of t-unital left $R$-modules is a unital left module category over an associative, unital monoidal category of t-unital $R$-$R$-bimodules, while the category of c-unital left $R$-modules is opposite to a unital right module category over the same monoidal category. Any left s-unital ring $R$, as defined by Tominaga in 1975, is t-unital; and a left $R$-module is s-unital if and only if it is t-unital. For any (nonunital) ring $R$, the full subcategory of s-unital $R$-modules is a hereditary torsion class in the category of nonunital $R$-modules; and for rings $R$ arising from small preadditive categories, the full subcategory of c-unital $R$-modules is closed under kernels and cokernels. However, over a t-unital ring $R$, the full subcategory of t-unital modules need not be closed under kernels, and the full subcategory of c-unital modules need not be closed under cokernels in the category of nonunital modules. Nevertheless, the categories of t-unital and c-unital left $R$-modules are abelian and naturally equivalent to each other; they are also equivalent to the quotient category of the abelian category of nonunital modules by the Serre subcategory of modules with zero action of $R$. This is a particular case of the result from a 1996 manuscript of Quillen. We also discuss related flatness, projectivity, and injectivity properties; and study the behavior of t-unitality and c-unitality under the restriction of scalars for a homomorphism of nonunital rings. Our motivation comes from the theory of semialgebras over coalgebras over fields.

math.RA↗

arXiv2023

Quasi-coherent torsion sheaves, the semiderived category, and the semitensor product: Semi-infinite algebraic geometry of quasi-coherent sheaves on ind-schemes

We construct the semi-infinite tensor structure on the semiderived category of quasi-coherent torsion sheaves on an ind-scheme endowed with a flat affine morphism into an ind-Noetherian ind-scheme with a dualizing complex. The semitensor product is "a mixture of" the cotensor product along the base and the derived tensor product along the fibers. The inverse image of the dualizing complex is the unit object. This construction is a partial realization of the Semi-infinite Algebraic Geometry program, as outlined in the introduction to arXiv:1504.00700.

math.AG↗

arXiv2023

Fp-projective periodicity

The phenomenon of periodicity, discovered by Benson and Goodearl, is linked to the behavior of the objects of cocycles in acyclic complexes. It is known that any flat $\mathsf{Proj}$-periodic module is projective, any fp-injective $\mathsf{Inj}$-periodic module is injective, and any $\mathsf{Cot}$-periodic module is cotorsion. It is also known that any pure $\mathsf{PProj}$-periodic module is pure-projective and any pure $\mathsf{PInj}$-periodic module is pure-injective. Generalizing a result of Saroch and Stovicek, we show that every $\mathsf{FpProj}$-periodic module is weakly fp-projective. The proof is quite elementary, using only a strong form of the pure-projective periodicity and the Hill lemma. More generally, we prove that, in a locally finitely presentable Grothendieck category, every $\mathsf{FpProj}$-periodic object is weakly fp-projective. In a locally coherent category, all weakly fp-projective objects are fp-projective. We also present counterexamples showing that a non-pure $\mathsf{PProj}$-periodic module over a regular finitely generated commutative algebra (or a hereditary finite-dimensional associative algebra) over a field need not be pure-projective.

math.CT↗

arXiv2023

Topologically semiperfect topological rings

We define topologically semiperfect (complete, separated, right linear) topological rings and characterize them by equivalent conditions. We show that the endomorphism ring of a module, endowed with the finite topology, is topologically semiperfect if and only if the module is decomposable as an (infinite) direct sum of modules with local endomorphism rings. Then we study structural properties of topologically semiperfect topological rings and prove that their topological Jacobson radicals are strongly closed and the related topological quotient rings are topologically semisimple. For the endomorphism ring of a direct sum of modules with local endomorphism rings, the topological Jacobson radical is described explicitly as the set of all matrices of nonisomorphisms. Furthermore, we prove that, over a topologically semiperfect topological ring, all finitely generated discrete modules have projective covers in the category of modules, while all lattice-finite contramodules have projective covers in both the categories of modules and contramodules. We also show that the topological Jacobson radical of a topologically semiperfect topological ring is equal to the closure of the abstract Jacobson radical, and present a counterexample demonstrating that the topological Jacobson radical can be strictly larger than the abstract one. Finally, we discuss the problem of lifting idempotents modulo the topological Jacobson radical and the structure of projective contramodules for topologically semiperfect topological rings.

math.RA↗

arXiv2023

An explicit self-dual construction of complete cotorsion pairs in the relative context

Let $R\to A$ be a homomorphism of associative rings, and let $(\mathcal F,\mathcal C)$ be a hereditary complete cotorsion pair in $R\mathsf{-Mod}$. Let $(\mathcal F_A,\mathcal C_A)$ be the cotorsion pair in $A\mathsf{-Mod}$ in which $\mathcal F_A$ is the class of all left $A$-modules whose underlying $R$-modules belong to $\mathcal F$. Assuming that the $\mathcal F$-resolution dimension of every left $R$-module is finite and the class $\mathcal F$ is preserved by the coinduction functor $\operatorname{Hom}_R(A,-)$, we show that $\mathcal C_A$ is the class of all direct summands of left $A$-modules finitely (co)filtered by $A$-modules coinduced from $R$-modules from $\mathcal C$. Assuming that the class $\mathcal F$ is closed under countable products and preserved by the functor $\operatorname{Hom}_R(A,-)$, we prove that $\mathcal C_A$ is the class of all direct summands of left $A$-modules cofiltered by $A$-modules coinduced from $R$-modules from $\mathcal C$, with the decreasing filtration indexed by the natural numbers. A combined result, based on the assumption that countable products of modules from $\mathcal F$ have finite $\mathcal F$-resolution dimension bounded by $k$, involves cofiltrations indexed by the ordinal $ω+k$. The dual results also hold, provable by the same technique going back to the author's monograph on semi-infinite homological algebra arXiv:0708.3398. In addition, we discuss the $n$-cotilting and $n$-tilting cotorsion pairs, for which we obtain better results using a suitable version of a classical Bongartz-Ringel lemma. As an illustration of the main results of the paper, we consider certain cotorsion pairs related to the contraderived and coderived categories of curved DG-modules.

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