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

This paper establishes that for a conilpotent coalgebra $C$, the inclusion of comodules into $C^*$-modules and the forgetful functor from contramodules to $C^*$-modules both induce fully faithful triangulated functors on appropriate bounded derived categories if and only if the Ext-groups $\operatorname{Ext}_C^n(k,k)$ are finite-dimensional for all $n \ge 0$, a condition the authors term "weakly finitely Koszul."

The Gist

Imagine you are an architect trying to understand the relationship between two different types of buildings: Comodules and Contramodules.

In the world of advanced mathematics (specifically algebra), these are abstract structures built on top of a "Coalgebra" (let's call it $C$). Think of $C$ as a unique blueprint or a set of rules for construction.

Now, there is a powerful, well-known tool called the Dual Algebra (let's call it $C^*$). This is like a giant, all-encompassing city where many different types of structures (called Modules) live.

The paper asks a very specific question: Can we perfectly translate the blueprints of our special buildings ($C$-structures) into the language of the giant city ($C^*$-structures) without losing any information?

There are two ways to do this translation:

1. The Inclusion (The "Comodule" path): We take a Comodule and say, "Look, this is also a Module in the big city."
2. The Forgetful (The "Contramodule" path): We take a Contramodule and say, "Forget the special rules for a second; just look at it as a Module in the big city."

The Core Problem: "Full-and-Faithfulness"

In math, a translation is fully faithful if it preserves everything.

• Faithful: It doesn't confuse two different buildings. (If Building A $\neq$ Building B, their translations are also different).
• Full: It doesn't miss any connections. (If two buildings in the big city are connected, that connection must have come from a connection between the original buildings).

The author, Leonid Positselski, wants to know: When does this translation work perfectly?

The Big Discovery: The "Weakly Finite Koszul" Rule

The paper proves a surprising symmetry. It turns out that the Comodule translation works perfectly if and only if the Contramodule translation works perfectly.

But what is the condition that makes this happen? It comes down to a specific measurement of the blueprint $C$.

Imagine the blueprint $C$ has a "complexity score" at every level of depth (Level 1, Level 2, Level 3, etc.).

• If the complexity at every level is finite (like a building with a finite number of rooms at every floor), then the translation is perfect.
• If the complexity at any level becomes infinite (like a floor with an infinite, uncountable number of rooms), the translation breaks.

The author calls coalgebras that pass this test "Weakly Finite Koszul."

The "Broken Translation" Metaphors

The paper explains what happens when the translation breaks, using some very clever analogies:

1. The Comodule Problem (The "Missing Link"):
If the complexity is infinite, the Comodule translation is like a translator who is too honest but incomplete.

• They can tell you that two buildings are different (they are "faithful").
• But they fail to see some connections that exist in the big city. They miss "extensions."
• Analogy: Imagine two simple houses in your village. In the big city, you can build a massive, complex skyscraper that connects them. The translator says, "I can't see that skyscraper; I only see the two houses." The connection exists in the big city, but the translator can't find the blueprint for it in the small village.

2. The Contramodule Problem (The "Fake Connection"):
If the complexity is infinite, the Contramodule translation is like a translator who is too eager but confused.

• They see connections that don't actually exist in the original village.
• Analogy: The translator looks at two houses in the big city and says, "Ah, these are connected!" But when you go back to the village, you realize those houses were never connected. The translator invented a bridge that doesn't exist.

The "Double Dual" Twist

The paper also reveals a weird mathematical quirk when things go wrong.

• When the Comodule translation fails, the "missing" information in the big city is so huge it's like the double mirror image of the original. It's not just a little bigger; it's exponentially larger (like taking a photo of a photo of a photo, and the resolution gets weirdly distorted).
• When the Contramodule translation fails, it's the opposite: the translator sees a "ghost" connection that is actually a reflection of a non-existent object.

Why Should You Care?

This might sound like pure abstract nonsense, but it's actually about consistency.

In mathematics, we often try to solve problems in a complicated world (Modules) by translating them into a simpler world (Comodules/Contramodules), solving them there, and translating back.

• If the translation is fully faithful, we can do this safely. We know the answer we get in the big city is exactly the same as the answer in the small village.
• If the translation is not fully faithful, we might get a "false positive" or a "false negative." We might think we solved a problem, but we actually solved a different one.

Summary in One Sentence

This paper proves that for a specific type of mathematical structure, the ability to perfectly translate "Comodules" into "Modules" is exactly the same as the ability to perfectly translate "Contramodules" into "Modules," and both depend entirely on whether the underlying blueprint has a finite complexity at every level. If the blueprint is too wild (infinite), the translation breaks, and we lose the ability to trust our mathematical maps.

Technical Summary

1. Problem Statement and Context

The paper investigates the relationship between the homological properties of a coalgebra $C$ over a field $k$ and the behavior of two specific functors connecting categories of modules and comodules/contramodules:

1. The Comodule Inclusion Functor ($\Upsilon$): $\Upsilon: C\text{--Comod} \to C^*\text{--Mod}$. This functor maps a $C$-comodule to a $C^*$-module (where $C^*$ is the dual algebra). It is known to be exact and fully faithful at the level of abelian categories. However, the question arises: does it induce a fully faithful functor on the derived categories?
2. The Contramodule Forgetful Functor ($\Theta$): $\Theta: C\text{--Contra} \to C^*\text{--Mod}$. This functor maps a $C$-contramodule to its underlying $C^*$-module. Unlike $\Upsilon$, $\Theta$ is generally not fully faithful.

The Core Question: Under what conditions on the coalgebra $C$ do these functors induce fully faithful triangulated functors on bounded (or half-bounded) derived categories? The paper focuses specifically on conilpotent coalgebras (also known as pointed irreducible coalgebras).

2. Methodology

The author employs a combination of homological algebra, coalgebra theory, and category theory. Key methodological components include:

• Minimal Resolutions: The construction of minimal injective coresolutions for comodules and minimal projective resolutions for contramodules over conilpotent coalgebras. These resolutions are characterized by the vanishing of differentials in the "socle" or "cosocle" (the $\gamma$-functor).
• Cobar Complexes: Utilizing the cobar complex to compute $\text{Ext}$ groups $\text{Ext}^*_C(k, k)$ and relating them to the structure of the dual algebra $C^*$.
• Duality and Double Duals: A central technique involves analyzing the relationship between $\text{Ext}$ spaces in the comodule category and the double duals of these spaces in the contramodule or module categories. The paper establishes that $\text{Ext}^n_{C^*}(k, k)$ is often isomorphic to the double dual $\text{Ext}^n_C(k, k)^{**}$.
• Filtration Arguments: Using the natural increasing filtration on comodules and decreasing filtration on contramodules (induced by the coaugmentation) to reduce complex Ext calculations to simpler cases involving the trivial module $k$.
• Derived Category Theory: Applying results regarding adjoint functors and their derived functors to link the full-and-faithfulness of the abelian functors to the vanishing of higher derived functors of the adjoints.

3. Key Definitions

• Weakly Finitely Koszul Coalgebra: A conilpotent coalgebra $C$ is defined as weakly finitely Koszul if the vector space $\text{Ext}^n_C(k, k)$ is finite-dimensional for all $n \geq 0$.
• Separated Contramodules: Contramodules $P$ where the natural map to the inverse limit of their filtration quotients is injective. The paper distinguishes between results holding for all contramodules versus those requiring separatedness.

4. Key Results and Theorems

The paper's main contribution is the equivalence of several seemingly distinct conditions, summarized in Theorem 1.1 and Theorem 1.2.

A. Equivalence of Conditions (Theorem 1.1)

For a conilpotent coalgebra $C$ and an integer $n \geq 1$, the following are equivalent:

1. The map $\text{Ext}^i_C(L, M) \to \text{Ext}^i_{C^*}(L, M)$ induced by $\Upsilon$ is an isomorphism for all comodules $L, M$ and $1 \leq i \leq n$.
2. The analogous map for right comodules is an isomorphism.
3. The map $\text{Ext}^i_{C}(P, Q) \to \text{Ext}^i_{C^*}(P, Q)$ induced by $\Theta$ is an isomorphism for all contramodules $P$, separated contramodules $Q$, and $0 \leq i \leq n-1$.
4. The vector space $\text{Ext}^i_C(k, k)$ is finite-dimensional for all $1 \leq i \leq n$.
5. The vector space $\text{Ext}^i_{C^*}(k, k)$ is finite-dimensional for all $1 \leq i \leq n$.

Crucial Insight: If $\text{Ext}^n_C(k, k)$ is infinite-dimensional, the map $\Upsilon$ fails to be surjective on $\text{Ext}^n$, while the map $\Theta$ fails to be injective on $\text{Ext}^n$ (for specific objects). Specifically, $\text{Ext}^n_{C^*}(k, k)$ becomes isomorphic to the double dual $\text{Ext}^n_C(k, k)^{**}$, which is strictly larger if the original space is infinite-dimensional.

B. Derived Category Full-and-Faithfulness (Theorem 1.2)

The paper proves that the following eight conditions are equivalent for a conilpotent coalgebra $C$:

• The induced triangulated functors $\Upsilon^b: D^b(C\text{--Comod}) \to D^b(C^*\text{--Mod})$ and $\Upsilon^+: D^+(C\text{--Comod}) \to D^+(C^*\text{--Mod})$ are fully faithful.
• The induced triangulated functors $\Theta^b: D^b(C\text{--Contra}) \to D^b(C^*\text{--Mod})$ and $\Theta^-: D^-(C\text{--Contra}) \to D^-(C^*\text{--Mod})$ are fully faithful.
• The Condition: $\text{Ext}^n_C(k, k)$ is finite-dimensional for all $n \geq 0$ (i.e., $C$ is weakly finitely Koszul).

Note on Unbounded Categories: The paper explicitly states that it remains an open question whether these functors are fully faithful on the unbounded derived categories $D(C\text{--Comod})$ and $D(C\text{--Contra})$ in the general case, though it is known to be true for cocommutative coalgebras (via results on Noetherian local rings).

C. Specific Classes of Coalgebras

• Co-Noetherian and Cocoherent Coalgebras: Any left or right co-Noetherian (or finitely cogenerated cocoherent) conilpotent coalgebra is weakly finitely Koszul.
• Cocommutative Coalgebras: All finitely cogenerated cocommutative conilpotent coalgebras are weakly finitely Koszul. For these, the dual algebra $C^*$ is a complete Noetherian commutative local ring. In this specific case, the functors $\Upsilon$ and $\Theta$ induce equivalences between the derived categories of torsion/contramodule modules and the derived category of $C^*$-modules, ensuring full-and-faithfulness even for unbounded complexes.

5. Significance and Impact

1. Unification of Comodule and Contramodule Theory: The paper provides a symmetric framework linking the homological behavior of comodules and contramodules. It demonstrates that the "defect" in full-and-faithfulness for one functor (e.g., $\Upsilon$ failing surjectivity) is directly dual to the defect in the other (e.g., $\Theta$ failing injectivity), both governed by the finite-dimensionality of $\text{Ext}^*_C(k, k)$.
2. Clarification of "Koszulity": By introducing the term "weakly finitely Koszul," the author generalizes the classical notion of Koszul algebras to the coalgebra setting without requiring a grading. This captures a broader class of coalgebras where the derived category behaves "nicely."
3. Resolution of Open Questions: It settles the question of when the inclusion of rational modules into all modules induces a fully faithful derived functor, showing it depends entirely on the finite-dimensionality of the cohomology of the trivial module.
4. Counterexamples: The paper constructs explicit counterexamples (Example 6.4 and Example 8.2) showing that if the finite-dimensionality condition fails, the functors are not fully faithful, and the Ext spaces can become "too large" (double duals) or "too small" (non-injective).
5. Connection to Commutative Algebra: The results bridge coalgebra theory with the theory of complete Noetherian local rings, showing that the derived category of comodules over a cocommutative coalgebra is equivalent to the derived category of torsion modules over a local ring.

In summary, Positselski establishes that the homological "good behavior" of the derived categories of comodules and contramodules is strictly controlled by the finite-dimensionality of the cohomology of the trivial module, providing a precise criterion for when these categories embed fully faithfully into the module category of the dual algebra.