Homomorphisms of topological rings and change-of-scalar functors

1. Problem Statement and Motivation

The paper addresses a fundamental gap in the theory of contraherent cosheaves on formal schemes. While the theory of quasi-coherent sheaves on schemes is well-established (glued via localization/extension of scalars), the dual theory of contraherent cosheaves on formal schemes requires gluing via colocalization (coextension of scalars).

• The Core Issue: Given a continuous homomorphism of topological rings $f: R \to S$, one wishes to define a functor that "glues" contramodules. In the module case, this is the coextension of scalars functor $\text{Hom}_R(S, -)$. However, for topological rings, the category of contramodules ($R\text{-Contra}$) behaves differently than the category of modules.
  • The forgetful functor $f^\sharp: S\text{-Contra} \to R\text{-Contra}$ (restriction of scalars) always has a left adjoint (contraextension of scalars), but this functor is generally not exact, even for "flat" morphisms.
  • Crucially, a right adjoint to $f^\sharp$ (which would correspond to $\text{Hom}_R(S, -)$) does not always exist in the category of contramodules because the forgetful functor does not preserve infinite direct sums.
• The Goal: To construct an explicit, well-behaved right adjoint to the restriction of scalars functor $f^\sharp$ and to characterize the conditions under which the restriction functor is fully faithful. This is essential for defining the category of contraherent cosheaves of contramodules on non-affine formal schemes.

2. Methodology and Framework

The author works within the framework of complete, separated right or two-sided linear topological rings with countable bases of neighborhoods of zero.

• Contramodules: The paper utilizes the definition of contramodules as modules over a monad $R[[ - ]]$ (infinite summation operations).
• Key Classes of Morphisms: The paper introduces and analyzes specific classes of continuous ring maps:
  • Taut Maps: Maps where the closure of the image of an open ideal is open.
  • Strongly Right Taut Maps: A stricter condition for two-sided linear rings, ensuring the induced maps on discrete quotients behave well.
  • Left Proflat Maps: Maps $f: R \to S$ that can be represented as a projective limit of flat morphisms of discrete rings $f_n: R_n \to S_n$, satisfying specific bimodule isomorphism conditions ($R_n \otimes_{R_{n+1}} S_{n+1} \cong S_n$).
  • Topological Ring Proepimorphisms: Maps where the induced maps on discrete quotients $R/I \to S/J$ are ring epimorphisms.
• Cotorsion Pairs: The methodology relies heavily on the theory of cotorsion pairs in abelian categories. Specifically, the pair (flat contramodules, cotorsion contramodules) is used to resolve arbitrary contramodules into short exact sequences, allowing the extension of functors from flat/separated objects to general objects.

3. Key Contributions and Results

A. Full-and-Faithfulness of Restriction of Scalars

The paper establishes precise conditions under which the restriction of scalars functor $f^\sharp: S\text{-Contra} \to R\text{-Contra}$ is fully faithful.

• Theorem: For a left proflat map $f$, the functor $f^\sharp$ is fully faithful if and only if $f$ is a topological ring proepimorphism.
• This result generalizes the classical algebraic fact that a ring homomorphism is an epimorphism if and only if the restriction of scalars on modules is fully faithful.
• The paper proves that for such maps, the essential image of $f^\sharp$ consists of those $R$-contramodules whose underlying $R$-modules arise from $S$-modules.

B. The Pseudopullback Diagram

A major structural result characterizes the relationship between contramodules and modules.

• Theorem: Assuming the forgetful functor $R\text{-Contra} \to R\text{-Mod}$ is fully faithful and $f$ is left proflat, the commutative square of forgetful functors:
  $$\begin{array}{ccc} S\text{-Contra} & \xrightarrow{f^\sharp} & R\text{-Contra} \\ \downarrow & & \downarrow \\ S\text{-Mod} & \xrightarrow{f^*} & R\text{-Mod} \end{array}$$
  is a pseudopullback diagram (up to equivalence).
• Significance: This implies that an $R$-contramodule comes from an $S$-contramodule if and only if its underlying $R$-module comes from an $S$-module. This provides a concrete description of the essential image of the restriction functor.

C. Construction of the Right Adjoint (Coextension of Scalars)

The most significant technical achievement is the explicit construction of the right adjoint to $f^\sharp$, denoted $f^\flat: R\text{-Contra} \to S\text{-Contra}$, defined by $P \mapsto \text{Hom}_R(S, P)$.

• Existence: While this functor does not exist in general, the author proves it exists for left proflat maps.
• Construction Strategy:
  1. For separated contramodules, the structure is constructed via projective limits of discrete modules.
  2. For general (non-separated) contramodules, the author uses a "flat special precover" resolution (a short exact sequence $0 \to C \to F \to P \to 0$ where $F$ is flat and $C$ is cotorsion).
  3. Since $S$ is a flat left $R$-contramodule (under the proflat assumption), the functor $\text{Hom}_R(S, -)$ preserves the exactness of this sequence, allowing the $S$-contramodule structure on $\text{Hom}_R(S, P)$ to be defined as the cokernel of the map between the structures on $\text{Hom}_R(S, F)$ and $\text{Hom}_R(S, C)$.
• Exactness Properties: The constructed functor $f^\flat$ is exact on the category of cotorsion contramodules. This is crucial because the category of contraherent cosheaves is often defined on the exact category of "contraadjusted" or "cotorsion" objects.

D. Counterexamples and Limitations

The paper provides counterexamples to show that:

• The right adjoint to restriction of scalars does not exist for general continuous maps (e.g., for the inclusion $k[x] \to k[[x]]$).
• The left adjoint (contraextension of scalars) is not exact even for proflat maps, refuting naive expectations that flatness implies exactness for all contramodules.

4. Significance

1. Foundations of Contraherent Cosheaves: This work provides the necessary algebraic machinery to define contraherent cosheaves of contramodules on formal schemes. Without the existence and exactness properties of the coextension functor $f^\flat$, the gluing of these objects over non-affine formal schemes would be ill-defined.
2. Duality in Algebraic Geometry: It establishes a rigorous duality between the theory of quasi-coherent sheaves (glued via localization) and contraherent cosheaves (glued via colocalization), extending the scope of homological algebra to formal geometry.
3. Generalization of Ring Epimorphisms: The paper successfully generalizes the theory of ring epimorphisms to the topological setting, introducing "topological ring proepimorphisms" and linking them to the full-and-faithfulness of contramodule restriction functors.
4. Technical Innovation: The use of transfinite induction and flat/cotorsion resolutions to construct the $S$-contramodule structure on $\text{Hom}_R(S, P)$ for non-separated objects is a novel and powerful technique in the theory of topological rings.

In summary, Positselski resolves the obstruction to defining coextension of scalars for contramodules over topological rings, proving that under the "left proflat" condition, this functor exists, is right adjoint to restriction, and possesses the necessary exactness properties to support the theory of contraherent cosheaves.