arxiv: 2603.27732 · v3 · submitted 2026-03-29 · 🧮 math.AG · math.AC· math.CT

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Contraherent cosheaves of contramodules on Noetherian formal schemes

Leonid Positselski

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Pith reviewed 2026-05-14 21:51 UTC · model grok-4.3

classification 🧮 math.AG math.ACmath.CT

keywords contraherent cosheavescontramodulesNoetherian formal schemeslocally Noetherian formal schemesadic topologiesdirect image functorsinverse image functorscontratensor product

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The pith

The paper defines an exact category of contraherent cosheaves of contramodules on locally Noetherian formal schemes and constructs direct and inverse image functors for them.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper sets out to define contraherent cosheaves of contramodules as an exact category on locally Noetherian formal schemes. It also introduces locally contraherent versions relative to open coverings and builds functors for direct and inverse images under morphisms between such schemes. Additionally, it develops contraherent Hom and contratensor product operations between quasi-coherent torsion sheaves and these cosheaves. A reader would care because these constructions provide a framework for working with contramodules in the context of formal schemes, extending classical sheaf theory. The preliminaries cover adic topologies on general commutative rings to support the definitions.

Core claim

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 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 Hom and contratensor product of quasi-coherent torsion sheaves and 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拓

What carries the argument

The exact category of contraherent cosheaves of contramodules, which serves as the setting in which direct and inverse image functors, contraherent Hom, and contratensor products are defined and shown to preserve the relevant structures.

If this is right

Direct and inverse image functors map (locally) contraherent cosheaves to (locally) contraherent cosheaves.
The contraherent Hom and contratensor product supply internal operations between quasi-coherent torsion sheaves and contraherent cosheaves.
The same definitions extend verbatim to the locally contraherent setting once an open covering is fixed.
The underlying adic commutative algebra holds for arbitrary commutative rings equipped with adic topologies of finitely generated ideals.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The framework supplies a dual language to ordinary quasi-coherent sheaves that may be useful for computing cohomology on formal schemes.
One could test the constructions explicitly on the formal spectrum of a complete local ring or on the formal completion of a scheme along a closed subscheme.
The generality of the adic preliminaries suggests the definitions may extend to certain non-Noetherian formal schemes when the ideal topologies remain well-behaved.

Load-bearing premise

The locally contraherent condition with respect to a given open covering forms an exact category and the adic topologies on arbitrary commutative rings behave sufficiently well for the image functors to be well-defined.

What would settle it

A concrete morphism of locally Noetherian formal schemes together with a contraherent cosheaf whose direct image fails to remain contraherent, or an explicit computation where the contratensor product fails to be associative or to preserve exactness.

read the original abstract

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. 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).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines exact categories of contraherent cosheaves of contramodules on locally Noetherian formal schemes plus direct/inverse image functors and related operations, with a useful generalization of adic algebra.

read the letter

The paper defines the exact category of contraherent cosheaves of contramodules on a locally Noetherian formal scheme, along with locally contraherent versions tied to open coverings. It constructs direct image and inverse image functors for these under morphisms, and covers the contraherent Hom and contratensor product with quasi-coherent torsion sheaves. The preliminaries extend adic algebra to arbitrary commutative rings with finitely generated ideal topologies. This setup is new in its specific combination for contramodules on formal schemes. The work does well by providing explicit functor constructions that should support homological algebra in this area. It builds directly on earlier contramodule and cosheaf results, which fits since those form the foundation. The main soft spot is that the abstract and description focus on definitions without showing the full verification of exactness or functor properties. The assumption that the topologies work for arbitrary rings and that coverings yield exact categories needs solid proof in the text. If those hold, the central claims are fine; otherwise there could be gaps in the technical details. This paper is for specialists in homological algebra and algebraic geometry who deal with formal schemes and contramodules. A reader already comfortable with Positselski's prior papers on these topics will get the most value, as it extends the framework in a targeted way. It deserves serious peer review because the definitions organize the material at a useful level and the constructions appear consistent with the stated hypotheses. I recommend sending it to referees rather than a desk reject.

Referee Report

2 major / 2 minor

Summary. The paper defines 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. It constructs the direct image and inverse image functors of locally contraherent cosheaves of contramodules under morphisms of locally Noetherian formal schemes, and discusses the functors of contraherent Hom and contratensor product of quasi-coherent torsion sheaves and contraherent cosheaves of contramodules. The preliminaries generalize adic commutative algebra to arbitrary commutative rings with adic topologies of finitely generated ideals.

Significance. If the exactness properties and functoriality hold as claimed, the work supplies a coherent framework for contramodule-based homological algebra on formal schemes, extending prior constructions of contraherent cosheaves to the formal setting and broadening the scope of adic algebra. The parameter-free nature of the core definitions and the explicit treatment of direct/inverse images constitute a clear technical advance for derived-category applications involving torsion.

major comments (2)

[Preliminaries] Preliminaries section: the generalization of adic topologies to arbitrary commutative rings is asserted to support the contramodule structures, but the verification that the inverse-image functor preserves the contraherent condition (under the locally Noetherian hypothesis) requires an explicit check that the finitely generated ideal topology remains compatible with the completion and exactness axioms used later for the cosheaf category.
[Locally contraherent cosheaves] Section on locally contraherent cosheaves: the claim that the category with respect to a given open covering is exact rests on the assumption that restriction to the covering elements preserves short exact sequences of contramodules; a concrete diagram chase or reference to the relevant exactness lemma in the adic algebra preliminaries is needed to confirm this is load-bearing for the global exact category.

minor comments (2)

[Functor constructions] Notation for the contratensor product is introduced without an explicit comparison to the classical tensor product of quasi-coherent sheaves; a short remark clarifying the distinction would improve readability.
The abstract mentions 'Noetherian formal schemes' while the body uses 'locally Noetherian'; a uniform choice of terminology throughout would eliminate potential confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the specific suggestions for improving the clarity of the preliminaries and the exactness arguments. We address each major comment below and will revise the manuscript accordingly to incorporate the requested verifications and references.

read point-by-point responses

Referee: Preliminaries section: the generalization of adic topologies to arbitrary commutative rings is asserted to support the contramodule structures, but the verification that the inverse-image functor preserves the contraherent condition (under the locally Noetherian hypothesis) requires an explicit check that the finitely generated ideal topology remains compatible with the completion and exactness axioms used later for the cosheaf category.

Authors: We agree that an explicit verification would improve the exposition. In the revised manuscript we will insert a short paragraph in the preliminaries section that directly checks preservation of the contraherent condition under inverse image, confirming compatibility of the finitely generated ideal topology with the relevant completion and exactness axioms when the base is locally Noetherian. revision: yes
Referee: Section on locally contraherent cosheaves: the claim that the category with respect to a given open covering is exact rests on the assumption that restriction to the covering elements preserves short exact sequences of contramodules; a concrete diagram chase or reference to the relevant exactness lemma in the adic algebra preliminaries is needed to confirm this is load-bearing for the global exact category.

Authors: We will add an explicit reference to the exactness lemma for restrictions of contramodules (already proved in the adic-algebra preliminaries for finitely generated ideals) together with a brief diagram chase showing that the restriction functors preserve short exact sequences. This will make the exactness of the locally contraherent category with respect to a covering fully self-contained. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines new exact categories of contraherent cosheaves of contramodules on locally Noetherian formal schemes and constructs direct/inverse image functors plus Hom and contratensor operations. These rest on preliminaries in adic commutative algebra for arbitrary rings with finitely generated ideal topologies, stated explicitly with respect to a given open covering. No load-bearing step reduces a claimed prediction or uniqueness result to a fitted parameter, self-citation chain, or definitional loop; the central content consists of independent constructions and functoriality statements under the listed hypotheses, making the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper rests on standard background from category theory, adic commutative algebra, and prior work on contramodules; no free parameters are introduced, and the new entities are the defined categories themselves.

axioms (1)

standard math Standard axioms of exact categories and commutative rings with adic topologies
Invoked in the preliminaries section for arbitrary commutative rings with adic topologies of finitely generated ideals.

invented entities (1)

contraherent cosheaves of contramodules no independent evidence
purpose: To form an exact category on locally Noetherian formal schemes
Newly defined structure whose exactness and functoriality are the main claims.

pith-pipeline@v0.9.0 · 5425 in / 1284 out tokens · 36198 ms · 2026-05-14T21:51:58.610827+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Homomorphisms of topological rings and change-of-scalar functors
math.RA 2026-03 unverdicted novelty 6.0

For left proflat topological ring epimorphisms, restriction of scalars on contramodules is fully faithful and the forgetful square is a pseudopullback.

Reference graph

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