arXiv subjects

Leonid Positselski

Publications and source records attributed to Leonid Positselski.

At least 19 recordsLinked to original sources

arXiv2026

The contraherent version of the theorem of Slavik and Stovicek

This is a paper about contraherent cosheaves on non-semi-separated schemes. We prove two theorems, a negative one and a positive one. On the negative side, let $X$ be a quasi-compact, quasi-separated scheme that is not semi-separated. We present a locally cotorsion contraherent cosheaf on $X$ that does not have an admissible monomorphism into any locally injective locally contraherent cosheaf. The construction and proof follow the arguments of Slavik and Stovicek in arXiv:1902.05740. On the positive side, let $X$ be a Noetherian scheme of finite Krull dimension. We prove that every $\mathbf W$-locally contraherent cosheaf on $X$ has an admissible monomorphism into a locally cotorsion $\mathbf W$-locally contraherent cosheaf. Moreover, the cokernel is a flat contraherent cosheaf. The proof is based on the theorem of Raynaud-Gruson about the projective dimensions of flat modules and Enochs' classification of flat cotorsion modules.

math.AG↗

arXiv2026

Coderived and contraderived categories for a cotorsion pair, flat-type cotorsion pairs, and relative periodicity

Given a hereditary complete cotorsion pair $(\mathsf A,\mathsf B)$ generated by a set of objects in a Grothendieck category $\mathsf K$, we construct a natural equivalence between the Becker coderived category of the left-hand class $\mathsf A$ and the Becker contraderived category of the right-hand class $\mathsf B$. We show that a nested pair of cotorsion pairs $(\mathsf A_1,\mathsf B_1)\le(\mathsf A_2,\mathsf B_2)$ provides an adjunction between the related co/contraderived categories, which is induced by a Quillen adjunction between abelian model structures. Then we specialize to the cotorsion pairs $(\mathsf F,\mathsf C)$ sandwiched between the projective and the flat cotorsion pairs in a module category, and prove that the related co/contraderived categories for $(\mathsf F,\mathsf C)$ are the same as for the projective and flat cotorsion pairs if and only if two periodicity properties hold for $\mathsf F$ and $\mathsf C$. The same applies to the cotorsion pairs sandwiched between the very flat and the flat cotorsion pairs in the category of quasi-coherent sheaves over a quasi-compact semi-separated scheme. More generally, we define and discuss cotorsion pairs of the very flat type and of the flat type in Grothendieck categories (as well as exact categories of the flat type), and work with a cotorsion pair sandwiched between one of the very flat type and one of the flat type. The motivating examples of the classes of flaprojective modules and relatively cotorsion modules for a ring homomorphism are discussed, and periodicity conjectures formulated for them.

math.CT↗

arXiv2026

Contraherent cosheaves of contramodules on Noetherian formal schemes

We define the exact category of contraherent cosheaves of contramodules on a locally Noetherian formal scheme, as well as the exact categories of locally contraherent cosheaves of contramodules (with respect to a given open covering). We also construct the direct image and inverse image functors of locally contraherent cosheaves of contramodules under morphisms of locally Noetherian formal schemes, and discuss the functors of contraherent $\mathfrak{Hom}$ and contratensor product of quasi-coherent torsion sheaves and contraherent cosheaves of contramodules. At the end, we have a discussion of projective, antilocally flat, coflasque, and flat contraherent cosheaves of contramodules. The exposition in the section of preliminaries in adic commutative algebra is worked out in the greater generality of arbitrary commutative rings with adic topologies (of finitely generated ideals).

math.AG↗

arXiv2026

Torsion modules and differential operators in infinitely many variables

This paper grew out of the author's work on arXiv:2504.18460. Differential operators in the sense of Grothendieck acting between modules over a commutative ring can be interpreted as torsion elements in the bimodule of all operators with respect to the diagonal ideal in the tensor square of the ring. Various notions of torsion modules for an infinitely generated ideal in a commutative ring lead to various notions of differential operators. We discuss differential operators of transfinite orders and differential operators having no global order at all, but only local orders with respect to specific elements of the ring. Many examples are presented. In particular, we prove that every ordinal can be realized as the order of a differential operator acting on the algebra of polynomials in infinitely many variables over a field. We also discuss extension of differential operators to localizations of rings and modules, and to colocalizations of modules.

math.AC↗

arXiv2026

Homomorphisms of topological rings and change-of-scalar functors

We consider homomorphisms of complete, separated right or two-sided linear topological rings with countable bases of neighborhoods of zero $\mathfrak f\colon\mathfrak R\to\mathfrak S$. Taut maps of right linear topological rings, strongly right taut maps of two-sided linear topological rings, left proflat continuous ring maps, and topological ring epimorphisms are discussed. For a left proflat topological ring epimorphism $\mathfrak f$, we show that the functor of restriction of scalars on the categories of left contramodules $\mathfrak f_\sharp\colon\mathfrak S{-}\mathsf{Contra}\longrightarrow\mathfrak R{-}\mathsf{Contra}$ is fully faithful. Assuming that the contramodule-to-module forgetful functor $\mathfrak R{-}\mathsf{Contra}\longrightarrow\mathfrak R{-}\mathsf{Mod}$ is fully faithful and the topological ring map $\mathfrak f$ is left proflat, we prove that the commutative square of forgetful functors between the left contramodule and module categories over $\mathfrak S$ and $\mathfrak R$ is a pseudopullback diagram. This provides a description of the essential image of $\mathfrak f_\sharp$ under the conjunction of the respective assumptions. The left adjoint functor to $\mathfrak f_\sharp$ always exists, but is not exact even when $\mathfrak f$ is (pro)flat. A right adjont functor to $\mathfrak f_\sharp$ does not always exist, but for a left proflat map $\mathfrak f$ we construct it explicitly and show that it has good exactness properties. This work is motivated by the theory of contraherent cosheaves of contramodules on formal schemes.

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arXiv2026

A relative version of Bass' theorem about finite-dimensional algebras

As a special case of Bass' theory of perfect rings, one obtains the assertion that, over a finite-dimensional associative algebra over a field, all flat modules are projective. In this paper we prove the following relative version of this result. Let $R\rightarrow A$ be a homomorphism of associative rings such that $A$ is a finitely generated projective right $R$-module. Then every flat left $A$-module is a direct summand of an $A$-module filtered by $A$-modules $A\otimes_RF$ induced from flat left $R$-modules $F$. In other words, a left $A$-module is cotorsion if and only if its underlying left $R$-module is cotorsion. The proof is based on the cotorsion periodicity theorem.

math.RA↗

arXiv2026

Roos axiom holds for quasi-coherent sheaves

Let $X$ be either a quasi-compact semi-separated scheme, or a Noetherian scheme of finite Krull dimension. We show that the Grothendieck abelian category $X{-}\mathsf{Qcoh}$ of quasi-coherent sheaves on $X$ satisfies the Roos axiom $\mathrm{AB}4^*$-$n$: the derived functors of infinite direct product have finite homological dimension in $X{-}\mathsf{Qcoh}$. In each of the two settings, two proofs of the main result are given: a more elementary one, based on the Cech coresolution, and a more conceptual one, demonstrating existence of a generator of finite projective dimension in $X{-}\mathsf{Qcoh}$ in the semi-separated case and using the co-contra correspondence (with contraherent cosheaves) in the Noetherian case. The hereditary complete cotorsion pair (very flat quasi-coherent sheaves, contraadjusted quasi-coherent sheaves) in the abelian category $X{-}\mathsf{Qcoh}$ for a quasi-compact semi-separated scheme $X$ is discussed.

math.AG↗

arXiv2026

Resolutions as directed colimits

A general principle suggests that "anything flat is a directed colimit of countably presentable flats". In this paper, we consider resolutions and coresolutions of modules over a countably coherent ring $R$ (e.g., any coherent ring or any countably Noetherian ring). We show that any $R$-module of flat dimension $n$ is a directed colimit of countably presentable $R$-modules of flat dimension at most $n$, and any flatly coresolved $R$-module is a directed colimit of countably presentable flatly coresolved $R$-modules. If $R$ is a countably coherent ring with a dualizing complex, then any F-totally acyclic complex of flat $R$-modules is a directed colimit of F-totally acyclic complexes of countably presentable flat $R$-modules. The proofs are applications of an even more general category-theoretic principle going back to an unpublished 1977 preprint of Ulmer. Our proof of the assertion that every Gorenstein-flat module over a countably coherent ring is a directed colimit of countably presentable Gorenstein-flat modules uses a different technique, based on results of Saroch and Stovicek. We also discuss totally acyclic complexes of injectives and Gorenstein-injective modules, obtaining various cardinality estimates for the accessibility rank under various assumptions.

math.AC↗

arXiv2026

On pure monomorphisms and pure epimorphisms in accessible categories

In all $κ$-accessible additive categories, $κ$-pure monomorphisms and $κ$-pure epimorphisms are well-behaved, as shown in our previous paper arXiv:2311.02418. This is known to be not always true in $κ$-accessible nonadditive categories. Nevertheless, mild assumptions on a $κ$-accessible category are sufficient to prove good properties of $κ$-pure monomorphisms and $κ$-pure epimorphisms. In particular, in a $κ$-accessible category with finite products, all $κ$-pure monomorphisms are $κ$-directed colimits of split monomorphisms, while in a $κ$-accessible category with finite coproducts, all $κ$-pure epimorphisms are $κ$-directed colimits of split epimorphisms. We also discuss what we call Quillen exact classes of monomorphisms and epimorphisms, generalizing the additive concept of one-sided exact category.

math.CT↗

arXiv2025

Pseudo-dualizing complexes of torsion modules and semi-infinite MGM duality

This paper is an MGM version of arXiv.org:1703.04266 and arXiv:1907.03364, and a follow-up to Section 5 of arXiv:1503.05523. In the setting of a commutative ring $S$ with a weakly proregular finitely generated ideal $J\subset S$, we consider the maximal, abstract, and minimal corresponding classes of $J$-torsion $S$-modules and $J$-contramodule $S$-modules with respect to a given pseudo-dualizing complex of $J$-torsion $S$-modules $L^\bullet$, and construct the related triangulated equivalences. As a special case, we obtain an equivalence of the semiderived categories for an $I$-adically coherent commutative ring $R$ with a weakly proregular ideal $I\subset R$, a dualizing complex of $I$-torsion $R$-modules $D^\bullet$, and a ring homomorphism $f\colon R\rightarrow S$ such that $f(I)\subset J$ and $S$ is a flat $R$-module. (If the ring $S$ is not Noetherian, then a certain further assumption, which we call quotflatness of the morphism of pairs $f\colon (R,I)\rightarrow(S,J)$, needs to be imposed.) In that case, the pseudo-dualizing complex $L^\bullet$ is constructed as a complex of $J$-torsion $S$-modules quasi-isomorphic to the tensor product of $D^\bullet$ with the infinite dual Koszul complex for some set of generators of the ideal $J\subset S$.

math.AC↗

arXiv2025

Contraherent cosheaves on schemes

Contraherent cosheaves are globalizations of contraadjusted or cotorsion modules over commutative rings obtained by gluing together over a scheme, with the colocalization functors $\operatorname{Hom}_R(S,{-})$ used for the gluing (where $S$ is the ring of functions on an affine open subscheme in $\operatorname{Spec}R$). The category of contraherent cosheaves over a scheme is a Quillen exact category with exact functors of infinite product. Over a quasi-compact semi-separated scheme or a Noetherian scheme of finite Krull dimension (in a different version - over any locally Noetherian scheme), it also has enough projectives. We construct the derived co-contra correspondence over a scheme in two forms. The "naive" one is an equivalence of the conventional derived categories of quasi-coherent sheaves and contraherent cosheaves, valid over any quasi-compact semi-separated scheme. The more sophisticated version is an equivalence between the coderived category of quasi-coherent sheaves and the contraderived category of contraherent cosheaves over a Noetherian scheme with a dualizing complex. The former point of view allows us to obtain an explicit construction of the Lipman-Neeman extraordinary inverse image functor $f^!$ for a morphism of quasi-compact semi-separated schemes $f\colon Y\to X$. The latter approach provides an expanded version of the covariant Serre-Grothendieck duality theory and leads to the Hartshorne-Deligne extraordinary inverse image functor $f^!$ (which we denote by $f^+$) for a morphism of finite type $f$ between Noetherian schemes. We also construct a derived semico-semicontra correspondence, mounting the "naive" version along the fibers on top of the one depending on a dualizing complex on the base of a flat fibration. Noncommutative analogues of Noetherian stacks, affine Noetherian formal schemes, and ind-affine ind-schemes are briefly discussed in the appendices.

math.CT↗

arXiv2025

A contramodule generalization of Neeman's flat and projective module theorem

This paper builds on top of arXiv:2306.02734. We consider a complete, separated topological ring $\mathfrak R$ with a countable base of neighborhoods of zero consisting of open two-sided ideals. The main result is that the homotopy category of projective left $\mathfrak R$-contramodules is equivalent to the derived category of the exact category of flat left $\mathfrak R$-contramodules, and also to the homotopy category of flat cotorsion left $\mathfrak R$-contramodules. In other words, a complex of flat $\mathfrak R$-contramodules is contraacyclic (in the sense of Becker) if and only if it is an acyclic complex with flat $\mathfrak R$-contramodules of cocycles, and if and only if it is coacyclic as a complex in the exact category of flat $\mathfrak R$-contramodules. These are contramodule generalizations of theorems of Neeman and of Bazzoni, Cortes-Izurdiaga, and Estrada.

math.RA↗

arXiv2025

Pseudo-dualizing complexes and pseudo-derived categories

The definition of a pseudo-dualizing complex is obtained from that of a dualizing complex by dropping the injective dimension condition, while retaining the finite generatedness and homothety isomorphism conditions. In the specific setting of a pair of associative rings, we show that the datum of a pseudo-dualizing complex induces a triangulated equivalence between a pseudo-coderived category and a pseudo-contraderived category. The latter terms mean triangulated categories standing "in between" the conventional derived category and the coderived or the contraderived category. The constructions of these triangulated categories use appropriate versions of the Auslander and Bass classes of modules. The constructions of derived functors providing the triangulated equivalence are based on a generalization of a technique developed in our previous paper arXiv:1503.05523.

math.CT↗

arXiv2025

Exact DG-categories and fully faithful triangulated inclusion functors

We construct an "almost involution" assigning a new DG-category to a given one, and use this construction to recover, say, the abelian category of graded modules over the graded ring $R^*$ from the DG-category of DG-modules over a DG-ring $(R^*,d)$. This provides an appropriate technical background for the definition and discussion of abelian and exact DG-categories. In the setting of exact DG-categories, derived categories of the second kind are defined in the maximal natural generality. We develop the related abstract category-theoretic language and use it to formulate and prove several full-and-faithfulness theorems for triangulated functors induced by the inclusions of fully exact DG-subcategories. Such functors are fully faithful for derived categories of the second kind more often than for the conventional derived categories. Examples and applications range from the categories of complexes in abelian/exact categories to matrix factorization categories, and from curved DG-modules over curved DG-rings to quasi-coherent CDG-modules over quasi-coherent CDG-quasi-algebras over schemes.

math.CT↗

arXiv2025

Contraderived categories of CDG-modules

For any CDG-ring $B^\bullet=(B^*,d,h)$, we show that the homotopy category of graded-projective (left) CDG-modules over $B^\bullet$ is equivalent to the quotient category of the homotopy category of graded-flat CDG-modules by its full triangulated subcategory of flat CDG-modules. The contraderived category (in the sense of Becker) $\mathsf D^{\mathsf{bctr}}(B^\bullet{-}\mathbf{Mod})$ is the common name for these two triangulated categories. We also prove that the classes of cotorsion and graded-cotorsion CDG-modules coincide, and the contraderived category of CDG-modules is equivalent to the homotopy category of graded-flat graded-cotorsion CDG-modules. Assuming the graded ring $B^*$ to be graded right coherent, we show that the contraderived category $\mathsf D^{\mathsf{bctr}}(B^\bullet{-}\mathbf{Mod})$ is compactly generated and its full subcategory of compact objects is anti-equivalent to the full subcategory of compact objects in the coderived category of right CDG-modules $\mathsf D^{\mathsf{bco}}(\mathbf{Mod}{-}B^\bullet)$. Specifically, the latter triangulated category is the idempotent completion of the absolute derived category of finitely presented right CDG-modules $\mathsf D^{\mathsf{abs}}(\mathbf{mod}{-}B^\bullet)$.

math.RA↗

arXiv2025

$\mathcal D$-$Ω$ duality on the contra side

Given a smooth morphism of schemes $X\rightarrow T$, denote by $\mathcal D_{X/T}^{\mathsf{cr}}$ the sheaf of rings of fiberwise crystalline differential operators on $X$ relative to $T$ and by $Ω^\bullet_{X/T}$ the de Rham sheaf of DG-algebras of relative differential forms on $X$ over $T$. Assume that the scheme $X$ is quasi-compact and semi-separated. We construct a commutative square diagram of triangulated equivalences between four triangulated categories: the derived category of quasi-coherent sheaves of $\mathcal D_{X/T}^{\mathsf{cr}}$-modules, the reduced coderived category of quasi-coherent DG-modules over $Ω_{X/T}^\bullet$, the derived category of contraherent cosheaves of $\mathcal D_{X/T}^{\mathsf{cr}}$-modules, and the reduced contraderived category of contraherent DG-modules over $Ω_{X/T}^\bullet$. The equivalence involving the contraderived category was previously known for affine varieties only; we use contraherent cosheaves in order to obtain a nonaffine generalization of the "contra side" of the story. The exposition is written in the generality of finite locally free twisted Lie algebroids $(\mathfrak g,\widetilde{\mathfrak g})$ over quasi-compact semi-separated schemes $X$, the quasi-coherent twisted universal enveloping quasi-algebras of $(\mathfrak g,\widetilde{\mathfrak g})$, and the Chevalley-Eilenberg quasi-coherent CDG-quasi-algebras of $(\mathfrak g,\widetilde{\mathfrak g})$. The equivalence between the derived categories of quasi-coherent and contraherent $\mathcal A$-modules, called the "naive co-contra correspondence", is proved quite generally for any quasi-coherent quasi-algebra $\mathcal A$ over $X$.

math.AG↗

arXiv2025

The categories of corings and coalgebras over a ring are locally countably presentable

For any commutative ring $R$, we show that the categories of $R$-coalgebras and cocommutative $R$-coalgebras are locally $\aleph_1$-presentable, while the categories of $R$-flat $R$-coalgebras are $\aleph_1$-accessible. Similarly, for any associative ring $R$, the category of $R$-corings is locally $\aleph_1$-presentable, while the category of $R$-$R$-bimodule flat $R$-corings is $\aleph_1$-accessible. The cardinality of the ring $R$ can be arbitrarily large. We also discuss $R$-corings with surjective counit and flat kernel. The proofs are straightforward applications of an abstract category-theoretic principle going back to Ulmer. For right or two-sided $R$-module flat $R$-corings, our cardinality estimate for the accessibility rank is not as good. A generalization to comonoid objects in accessible monoidal categories is also considered.

math.RA↗

arXiv2025

Homological full-and-faithfulness of comodule inclusion and contramodule forgetful functors

In this paper we consider a conilpotent coalgebra $C$ over a field $k$. Let $Υ\colon C\textsf{-Comod}\longrightarrow C^*\textsf{-Mod}$ be the natural functor of inclusion of the category of $C$-comodules into the category of $C^*$-modules, and let $Θ\colon C\textsf{-Contra}\longrightarrow C^*\textsf{-Mod}$ be the natural forgetful functor. We prove that the functor $Υ$ induces a fully faithful triangulated functor on bounded (below) derived categories if and only if the functor $Θ$ induces a fully faithful triangulated functor on bounded (above) derived categories, and if and only if the $k$-vector space $\operatorname{Ext}_C^n(k,k)$ is finite-dimensional for all $n\ge0$. We call such coalgebras "weakly finitely Koszul".

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