Une conjecture $C_{\rm st}$ pour la cohomologie à support compact

This paper demonstrates that adjoining $p$-adic analogs of $\log p$ and $\log 2\pi i$ to the ring of analytic functions on the Fargues-Fontaine curve eliminates its Galois cohomology in degrees $\geq 1$, thereby enabling the formulation of $C_{\rm dR}$ and $C_{\rm st}$-type conjectures for the compact support cohomology of $p$-adic analytic varieties.

The Gist

Here is an explanation of the paper "A Cst Conjecture for Compact Support Cohomology" by Pierre Colmez, Sally Gilles, and Wiesława Nizioł, translated into everyday language with creative analogies.

The Big Picture: Cleaning Up a Messy Room

Imagine you are trying to study a complex, magical building (a p-adic analytic variety). You want to understand its shape and structure. Mathematicians have a powerful tool called Cohomology, which is like taking a high-resolution 3D scan of the building to see its holes, tunnels, and rooms.

There are two ways to scan this building:

1. Standard Scan: You look at the whole building, including the infinite space around it.
2. Compact Support Scan: You only look at the building itself, ignoring everything outside its walls. This is harder because the "edges" of the building behave strangely.

For a long time, mathematicians had a perfect recipe (a set of instructions) to translate the "Standard Scan" into a language we understand better (called de Rham and Hyodo-Kato cohomology). This recipe worked beautifully.

The Problem: When they tried to use the same recipe on the "Compact Support Scan" (the building with its edges), it failed. The translation came out garbled. Why? Because the "magic ink" (mathematical rings like $B_{dR}$ and $B_{st}$) used to do the translation had some ghosts in it. These ghosts were extra, unwanted noise appearing in the higher levels of the scan, making the final picture blurry and wrong.

The Solution: Adding a "Ghost-B-Gone" Spray

The authors of this paper discovered a clever fix. They realized that if they added a specific new ingredient to their magic ink, the ghosts would disappear.

The Ingredient: A "p-adic logarithm."
Think of this like adding a special detergent to your washing machine. The "dirty water" (the mathematical rings) had some stubborn stains (the unwanted cohomology in degrees $\ge 1$). By adding this specific detergent (which they call $\log t$ and $\log \tilde{p}$), the stains dissolve completely.

The Analogy of the "Logarithm":
Imagine you are trying to listen to a radio station, but there is a constant, annoying hum (the "ghosts") in the background.

• Old Method: You try to listen anyway, but the music is distorted.
• New Method: You invent a "noise-canceling headphone" (the new logarithm). When you put it on, the hum vanishes, and you hear the music perfectly clear.

In mathematical terms, they proved that by extending their rings with these logarithms, the "noise" (Galois cohomology in degrees 1 and higher) becomes zero. It vanishes.

Why This Matters: The New Recipe

Once the ghosts are gone, the authors could write a new recipe (a new conjecture) for the "Compact Support" scans.

• Before: The recipe was broken because the ingredients were contaminated.
• Now: With the "ghost-b-gone" logarithms added, the recipe works perfectly again.

They formulated a new rule (Conjecture 3.1) that says:

"If you take the compact support scan of a p-adic building, apply this new 'clean' magic ink, and do a specific type of math operation (called a derived Hom), you will get a perfect translation of the building's true shape."

The "Folklore" vs. The Surprise

The paper mentions two interesting side notes:

1. The Easy Part: For one type of magic ink ($B_{dR}$), mathematicians already knew this trick worked. It was "folklore" (common knowledge among experts).
2. The Surprise: For the other type of ink ($B_{st}$, related to the Fargues-Fontaine curve), they discovered that the trick works only if you use a specific "avatar" or version of the ink ($B_{FF}$). If you try to use the standard version, the ghosts don't go away. This was a surprising discovery that required a lot of heavy lifting to prove.

Summary in One Sentence

The authors found a way to "clean" the mathematical tools used to study p-adic shapes by adding a special logarithmic ingredient, which removes unwanted noise and allows them to finally write a perfect rule for understanding the shapes of these complex mathematical buildings when viewed from the "compact support" perspective.

Why Should You Care?

Even if you aren't a mathematician, this is a story about problem-solving. Sometimes, a system fails not because the main idea is wrong, but because there is hidden "noise" or "contamination" in the tools you are using. By identifying the source of the noise and inventing a simple "detergent" to remove it, you can unlock a whole new level of understanding. This paper is a masterclass in finding that detergent.

Technical Summary

Here is a detailed technical summary of the paper "A Cst Conjecture for Compact Support Cohomology" by Pierre Colmez, Sally Gilles, and Wiesława Nizioł.

1. Problem Statement

The paper addresses a fundamental gap in $p$-adic Hodge theory regarding the formulation of comparison conjectures for compact support cohomology of $p$-adic analytic varieties.

• Context: The classical $C_{dR}$ and $C_{st}$ conjectures (Colmez–Nizioł) provide isomorphisms between the pro-étale cohomology of a proper (or partially proper) variety $Y$ and its de Rham or Hyodo-Kato cohomology. These are formulated using continuous Galois-equivariant homomorphisms ($\text{Hom}_{G_K}$) into period rings like $B_{dR}$ and $B_{st}$.
• The Obstacle: These standard recipes fail for compact support cohomology ($H^*_{c}$).
  1. Derived Functors: For non-proper varieties, one must use the derived Hom functor ($R\text{Hom}$) rather than the standard Hom.
  2. Parasitic Terms: The Galois cohomology of the period rings $B_{dR}$ and $B_{st}$ is non-trivial in degree 1 (specifically, $H^1(G_K, B_{dR}) \cong K \cdot \log \chi_{cycl}$). When computing $R\text{Hom}$, these non-vanishing $H^1$ terms create "parasitic" contributions that prevent the recovery of the correct de Rham or Hyodo-Kato cohomology groups.
• Goal: The authors aim to construct modified period rings where the Galois cohomology vanishes in degrees $\ge 1$, thereby allowing a clean formulation of $C_{dR}$ and $C_{st}$ conjectures for compact support cohomology.

2. Methodology

The core strategy involves killing the Galois cohomology of the relevant period rings by adjoining specific transcendental elements (logarithms).

A. The Case of $B_{dR}$ (Folklore)

• Mechanism: The authors recall that $H^1(G_K, B_{dR})$ is generated by the class of the cyclotomic character's logarithm ($\log \chi_{cycl}$).
• Solution: By adjoining a formal variable $\log t$ (where $t$ is Fontaine's $p$-adic $2\pi i$) to$B_{dR}$, they define$B_{pdR} = B_{dR}[\log t]$.
• Action: The Galois group $G_K$ acts on $\log t$ via $\sigma(\log t) = \log t + \log \chi_{cycl}(\sigma)$.
• Result: This extension trivializes the $H^1$ class. The paper proves (Theorem 1.4) that for $\Lambda = B_{pdR}$, $H^i(G_K, \Lambda) = 0$ for all $i \ge 1$.

B. The Case of the Fargues-Fontaine Curve (Main Novelty)

• Context: The ring $B$ is defined as the ring of analytic functions on the Fargues-Fontaine curve $Y_{FF}$ (specifically $B = \mathcal{O}(Y_{FF})$). This ring plays a role analogous to $B_{st}$ in the context of the curve.
• Challenge: Unlike $B_{dR}$, the structure of $B$ is more complex. The authors must handle the cohomology of $B$ and its extensions $B[1/t]$.
• Construction:
  1. They introduce a transcendental element $\log \tilde{p}$ satisfying specific Frobenius ($\phi$) and Galois ($\sigma$) relations:
     • $\phi(\log \tilde{p}) = p \log \tilde{p}$
     • $\sigma(\log \tilde{p}) = \log \tilde{p} + \text{Kum}(\sigma) \cdot t$ (where $\text{Kum}$ is the Kummer cocycle).
  2. They define the extended rings:
     • $B_{FF}^+ = B[\log \tilde{p}]$
     • $B_{FF} = B_{FF}^+[1/t]$
     • $B_{pFF}^+ = B_{FF}^+[\log t]$
     • $B_{pFF} = B_{pFF}^+[1/t]$
• Key Technical Step: The proof relies on analyzing the quotient $B/tB$, which decomposes into a product of copies of $\mathbb{C}$ (Proposition 2.2, Lemma 2.1). By using induction on the degree of polynomials in $\log \tilde{p}$ and $\log t$, and leveraging the vanishing of cohomology for the base ring $B$ in high degrees, they show that the cohomology of the extended rings vanishes in degrees $\ge 1$.
• Result: Theorem 2.4 establishes that for $\Lambda = B_{pFF}^+$ or $B_{pFF}$, $H^i(G_K, \Lambda) = 0$ for all $i \ge 1$.

3. Key Contributions

1. Vanishing Theorems: The paper provides rigorous proofs that adjoining $\log t$ (and $\log \tilde{p}$ for the Fargues-Fontaine context) kills the Galois cohomology of the period rings $B_{dR}$ and $B$ in degrees $\ge 1$.
   • Specifically, $H^1(G_K, B_{pFF}) = 0$.
2. Distinction from $B_{st}$: The authors highlight a crucial difference: while $B_{st}$ can be modified to $B_{st}[\log t]$, the cohomology of this specific ring does not vanish in degree 1. Therefore, the standard $B_{st}$ cannot be used for compact support conjectures. Instead, the "avatar" $B_{pFF}$ (derived from the Fargues-Fontaine curve) is the correct object to use.
3. Formulation of New Conjectures: The vanishing results allow the authors to formulate precise $C_{dR}$ and $C_{st}$ conjectures for compact support cohomology.

4. Main Results (The Conjectures)

The paper proposes the following conjectures for a partially proper analytic variety $Y$ over a finite extension $K$ of $\mathbb{Q}_p$:

• Compact Support $C_{dR}$ Conjecture:
  $$R\Gamma_{dR}(Y) \simeq R\text{Hom}_{G_K}(R\Gamma_{\text{pro\'et}, c}(Y_{\mathbb{C}_p}, \mathbb{Q}_p(d))[2d], B_{pdR})$$
  Here, $B_{pdR} = B_{dR}[\log t]$. The shift $[2d]$ and twist $(d)$ are standard Poincaré duality adjustments.

• Compact Support $C_{st}$ Conjecture:
  $$R\Gamma_{HK}(Y) \simeq \varinjlim_{[L:K]<\infty} R\text{Hom}_{GL}(R\Gamma_{\text{pro\'et}, c}(Y_{\mathbb{C}_p}, \mathbb{Q}_p(d))[2d], B_{pFF})$$
  Here, $B_{pFF}$ is the ring constructed from the Fargues-Fontaine curve with $\log \tilde{p}$ and $\log t$ adjoined.

Why these work:
Because $H^i(G_K, B_{pdR}) = 0$ and $H^i(G_K, B_{pFF}) = 0$ for $i \ge 1$, the derived Hom complex $R\text{Hom}$ reduces effectively to a single term (or a complex with no higher extension obstructions), ensuring the isomorphism recovers exactly the de Rham or Hyodo-Kato cohomology without "parasitic" $H^1$ terms.

5. Significance

• Unification of Proper and Non-Proper Cases: This work extends the scope of $p$-adic Hodge theory comparison isomorphisms to non-proper (compact support) settings, which are essential for arithmetic applications involving open varieties.
• Resolution of Technical Obstructions: It solves the specific problem of "parasitic terms" arising from the non-vanishing $H^1$ of period rings, a problem that had previously prevented a clean conjectural framework for compact support.
• New Period Rings: It identifies $B_{pFF}$ as the correct period ring for $C_{st}$ in the compact support setting, distinguishing it from the classical $B_{st}$. This deepens the understanding of the relationship between the Fargues-Fontaine curve and classical $p$-adic period rings.
• Foundation for Future Work: These conjectures provide the necessary theoretical framework for proving comparison theorems for specific classes of non-proper $p$-adic varieties, potentially impacting the study of $L$-functions and special values in the non-proper context.

In summary, Colmez, Gilles, and Nizioł demonstrate that by carefully extending period rings with logarithmic elements, one can "kill" the Galois cohomology obstructions, thereby enabling a robust formulation of $p$-adic comparison conjectures for compact support cohomology.