The homology of additive functors in prime characteristic

This paper computes specific Ext and Tor groups for general functors from a $\mathbb{Z}/p$-linear additive category to vector spaces by relating them to the corresponding groups for additive functors, thereby providing new methods for calculating the group homology of general linear groups.

The Gist

Imagine you are trying to understand the hidden structure of a massive, complex machine. In mathematics, this machine is often a "category" of objects (like shapes, numbers, or groups) and the rules that connect them.

This paper by Aurélien Djament and Antoine Touzé is essentially a master key for unlocking a specific type of mathematical puzzle involving "functors."

Here is the breakdown in simple, everyday language, using analogies to make it stick.

1. The Setup: Two Ways to Look at a Machine

Imagine you have a factory (let's call it Category A) that produces widgets. You have a team of inspectors (the Functors) who check these widgets.

The authors are interested in measuring the "distance" or "relationship" between two different inspectors, let's call them Inspector A and Inspector B. In math, this distance is measured by things called Ext and Tor groups. Think of these as "compatibility scores" or "tension levels" between the inspectors.

There are two ways to measure this tension:

1. The Strict Rulebook (Additive Functors): You only look at inspectors who follow a very rigid, linear rule: "If I see two widgets, I must check them exactly like I would check them separately and add the results." This is the Additive world. It's orderly, predictable, and easy to calculate.
2. The Wild World (All Functors): You look at any inspector, even the crazy ones who do weird things, like checking two widgets together and getting a totally different result than checking them apart. This is the General world. It's chaotic and much harder to calculate.

The Problem:
Usually, if you know the tension in the Strict Rulebook world, you can't easily guess the tension in the Wild World. They are different beasts. However, the authors discovered a special condition: If the factory operates in "Prime Characteristic" (a specific mathematical setting related to prime numbers, like 2, 3, 5...), there is a magical bridge between the two worlds.

2. The Big Discovery: The "Frobenius" Bridge

The paper proves that in this specific "Prime Characteristic" world, the Wild World isn't actually that wild. It turns out that the complex tension in the Wild World is just the simple tension from the Strict Rulebook plus a specific, predictable "noise" or "flavor."

The Analogy: The Perfect Smoothie
Imagine the "Strict Rulebook" tension is a plain, perfect glass of water.
The "Wild World" tension is a fancy smoothie.
The authors found that the smoothie is made by taking the plain water and blending it with a specific, pre-made "flavor pack" (which they call the algebra generated by classes $e_1, e_2, \dots$).

This "flavor pack" is completely independent of the inspectors you are comparing. It only depends on the rules of the factory itself.

• The Formula: Wild Tension = Strict Tension × Flavor Pack

This is huge because calculating the "Strict Tension" is easy (it's like standard algebra). Calculating the "Wild Tension" directly is a nightmare. Now, you just calculate the easy part and multiply by the known "flavor pack."

3. Why "Prime Characteristic" Matters

The authors specify that this only works when the math is done in "Prime Characteristic" (think of a clock that resets after 5 hours, or 7 hours, etc.).

• In "Zero Characteristic" (like normal clocks): The Wild World and the Strict World are actually the same. The smoothie is just water. (This was already known).
• In "Prime Characteristic": The Wild World is different, but the difference is perfectly structured. It's like the smoothie has a secret recipe that is always the same, no matter what ingredients (inspectors) you use.

4. The "Magic Ingredient": The ℵ-Additive Envelope

To prove this, the authors had to build a temporary "super-factory" (called the ℵ-additive envelope).

• The Metaphor: Imagine your factory is small. To prove a theorem about it, they built a giant, infinite warehouse next door where they could copy-paste widgets infinitely. This allowed them to use powerful mathematical tools (like "adjunctions") that don't work in the small factory. Once they proved the rule in the giant warehouse, they shrank it back down to the original factory, and the rule still held true.

5. Real-World Application: The General Linear Group

Why do we care about this? The paper mentions General Linear Groups.

• The Analogy: Think of a General Linear Group as a massive, shifting dance troupe. Mathematicians want to know the "homology" of this troupe—basically, how the dancers move together, where they get stuck, and how they form patterns.
• The "Inspectors" in the paper correspond to specific ways the dancers move.
• By using the authors' "Smoothie Formula," mathematicians can now calculate the complex movement patterns of these huge dance troupes by just doing simple math on the individual dancers and adding the "flavor pack."

Summary

The Paper in One Sentence:
The authors discovered that in a specific mathematical universe (prime numbers), the complex, chaotic relationships between mathematical functions can be perfectly predicted by taking the simple, orderly relationships and multiplying them by a universal, pre-calculated "flavor pack."

Why it's cool:
It turns a nightmare of complex calculation into a simple multiplication problem. It's like finding out that every time you try to bake a complicated cake, you don't need a new recipe; you just need to take your basic cake batter and add a specific, pre-mixed spice blend.

Technical Summary

Here is a detailed technical summary of the paper "The Homology of Additive Functors in Prime Characteristic" by Aurélien Djament and Antoine Touzé.

1. Problem Statement

The paper addresses a fundamental discrepancy in homological algebra regarding the computation of extension ($\text{Ext}$) and torsion ($\text{Tor}$) groups in the category of functors.

Let $A$ be an essentially small additive category and $k$ a field. There are two natural abelian categories of functors from $A$ to $k$-vector spaces ($V_k$):

1. $\text{Add}(A, k)$: The category of additive functors.
2. $\mathcal{F}(A, k)$: The category of all functors (not necessarily additive).

Since $\text{Add}(A, k)$ is a full abelian subcategory of $\mathcal{F}(A, k)$, there is a canonical map:
$$\Phi: \text{Ext}^*_{\text{Add}(A, k)}(\pi, \rho) \to \text{Ext}^*_{\mathcal{F}(A, k)}(\pi, \rho)$$
for additive functors $\pi, \rho$.

• In Characteristic 0: If $k$ has characteristic 0, $\Phi$ is an isomorphism (by [2]). The homological algebra of additive functors fully captures that of all functors.
• In Positive Characteristic: If $k$ has characteristic $p > 0$, $\Phi$ is generally not an isomorphism. The target $\text{Ext}^*_{\mathcal{F}(A, k)}$ can be non-zero in infinitely many degrees, whereas the source (often related to module categories) may vanish in high degrees.

The Core Question: How can one compute the "mysterious" $\text{Ext}$ and $\text{Tor}$ groups in the category of all functors $\mathcal{F}(A, k)$ using the simpler, better-understood groups in the category of additive functors $\text{Add}(A, k)$, specifically when $k$ is a field of positive characteristic?

2. Methodology

The authors employ a multi-layered approach combining category theory, homological algebra, and strict polynomial functors.

A. $\aleph$-Additive Envelopes

To handle representable functors that might take values in infinite-dimensional vector spaces, the authors introduce the $\aleph$-additive envelope $A^\aleph$ of the category $A$ (for a sufficiently large regular cardinal $\aleph$).

• This category formally adds $\aleph$-small coproducts to $A$.
• They utilize the fact that the left Kan extension along the inclusion $A \hookrightarrow A^\aleph$ is exact and induces isomorphisms on $\text{Ext}$ groups (Proposition 3.3).
• This allows them to reduce computations to cases where representable functors behave like free modules over a larger ring, facilitating adjunction arguments.

B. Strict Polynomial Functors and Frobenius Twists

The proof relies heavily on the theory of strict polynomial functors (introduced by Friedlander and Suslin, and developed by the authors in [8]).

• They use the category $\mathcal{P}_k$ of strict polynomial functors as an intermediate step.
• They analyze the self-extensions of the identity functor $I$ and its Frobenius twists $I^{(r)}$.
• A key algebraic structure identified is the graded algebra $E^*_\infty \cong k[e_1, e_2, \dots] / \langle e_i^p = 0 \rangle$, where $\deg(e_i) = 2p^i - 1$. This algebra arises from the self-extensions of the identity functor in the category of strict polynomial functors.

C. Reduction to the Base Case ($A = \mathbb{P}_{\mathbb{F}_p}$)

The proof strategy involves a series of reductions:

1. Base Case: Prove the result for $A = \mathbb{P}_{\mathbb{F}_p}$ (finite-dimensional vector spaces over $\mathbb{F}_p$). This relies on the known computation of $\text{Ext}^*_{\mathcal{F}(\mathbb{P}_{\mathbb{F}_p}, \mathbb{F}_p)}(I, I)$ from [7].
2. Extension of Scalars: Use the fact that $k$ is a perfect field to relate $\mathbb{F}_p$-linear structures to $k$-linear structures via the Hochschild-Kostant-Rosenberg (HKR) theorem. Specifically, they show that for perfect $k$, the Hochschild homology $HH_*(k)$ is trivial in positive degrees, which simplifies the comparison between additive and non-additive extensions.
3. $\delta$-functor Argument: They use a standard $\delta$-functor argument to extend the isomorphism from projective objects to all objects in the category.

3. Key Contributions and Results

Theorem 1: The Main Isomorphism

Let $k$ be a perfect field of characteristic $p > 0$, and let $A$ be an essentially small $\mathbb{F}_p$-linear additive category. For any additive functors $\pi, \rho: A \to V_k$, there is a graded $k$-linear isomorphism:
$$\Psi: \text{Ext}^*_{\text{Add}(A, k)}(\pi, \rho) \otimes_k \text{Ext}^*_{\mathcal{F}(\mathbb{P}_k, k)}(I, I) \xrightarrow{\sim} \text{Ext}^*_{\mathcal{F}(A, k)}(\pi, \rho)$$
where the map is defined by Yoneda composition: $e \otimes e' \mapsto \Phi(e) \circ \pi^*(e')$.

• Structure of the "Correction" Factor: The term $\text{Ext}^*_{\mathcal{F}(\mathbb{P}_k, k)}(I, I)$ is isomorphic to the graded algebra:
  $$k[e_1, e_2, e_3, \dots] / \langle e_1^p, e_2^p, e_3^p, \dots \rangle$$
  where each generator $e_i$ has cohomological degree $2p^i - 1$.
• Significance: This theorem completely characterizes the "extra" homology appearing in the category of all functors. It is a tensor product of the additive homology with a specific "universal" algebra generated by Frobenius twists.

Corollary 2: The Tor Analogue

By dualizing Theorem 1, the authors obtain a similar result for Tor groups. Let $T_*$ be the graded vector space with dimension 1 in even degrees and 0 in odd degrees. For an $\mathbb{F}_p$-linear category $A$:
$$\text{Tor}^{\mathcal{F}(A, k)}_*(\pi, \rho) \cong \text{Tor}^{\text{Add}(A, k)}_*(\pi, \rho) \otimes_k T_*$$
This allows the computation of Tor in the category of all functors using the simpler additive Tor.

Computations for Non-Additive Functors (Section 8)

The paper extends these results to non-additive functors, specifically polynomial functors (in the sense of Eilenberg and Mac Lane).

• They provide a Künneth-type formula for the Tor of tensor products of additive functors.
• Example 3: They compute the Tor between functors involving symmetric powers ($S_d$) and general linear representations. If $d$ is invertible in $k$, the Tor between non-additive functors $\pi^* S_d$ and $\rho^* F_V$ is computed via the additive Tor of $\pi$ and $\rho$, twisted by the universal algebra $T_*$.

4. Significance and Applications

Application to General Linear Group Homology

The primary motivation for this work is the computation of the homology of general linear groups $GL_\infty(R)$.

• There is a known isomorphism relating the homology of $GL_\infty(R)$ with coefficients in polynomial functors to the Tor groups in the functor category:
  $$H_*(GL_\infty(R), F_\infty \otimes G_\infty) \cong H_*(GL_\infty(R), k) \otimes_k \text{Tor}^{\mathcal{F}(\mathbb{P}_R, k)}_*(F, G)$$
• Impact: By reducing the computation of $\text{Tor}^{\mathcal{F}}$ to $\text{Tor}^{\text{Add}}$ (which is computable via standard module theory) and the explicit algebra $E^*_\infty$, the authors provide a concrete, algorithmic method to compute the homology of $GL_\infty(R)$ for various coefficient systems in positive characteristic.

Theoretical Implications

• Perfectness Assumption: The results crucially depend on $k$ being a perfect field. Remark 6.2 explains that if $k$ is not perfect, the Hochschild homology $HH_1(k)$ is non-trivial, causing the isomorphism to fail. This highlights a deep connection between the perfection of the field and the structure of functor homology.
• Relation to Topological Hochschild Homology (THH): The paper notes that the target of the map $\Phi$ is related to THH. The results here provide a concrete algebraic description of this relationship for $\mathbb{F}_p$-linear categories, bridging the gap between classical homological algebra and topological methods.
• Generalization: The work generalizes previous results by Pirashvili and others, moving from specific cases (like $A = \mathbb{P}_{\mathbb{F}_p}$) to arbitrary $\mathbb{F}_p$-linear additive categories.

Summary

Djament and Touzé provide a complete structural description of the homological difference between additive and non-additive functors in prime characteristic. They prove that the "missing" homology is entirely captured by a universal graded algebra generated by Frobenius twists. This allows for the reduction of complex computations in the category of all functors to standard computations in additive categories, with direct applications to the stable homology of general linear groups.