[Paper Review] Drinfeld's Lemma for Perfectoid Spaces and Overconvergence of Multivariate $(\phi, \Gamma)$-Modules

Annie Carter, Kiran S. Kedlaya+1 more|arXiv (Cornell University)|Aug 12, 2018

Algebraic Geometry and Number Theory26 references7 citations

TL;DR

This paper establishes a multivariate generalization of Fontaine's (ϕ, Γ)-module theory for continuous p-adic Galois representations of products of absolute Galois groups of p-adic fields, using perfectoid spaces and a new form of Drinfeld's lemma. It constructs equivalences between such representations and projective étale (ϕ∆, Γ∆)-modules over multivariable power series rings, and proves that Galois cohomology is isomorphic to the cohomology of a multivariate Herr complex, extending classical results to higher-rank settings via Shapiro's lemma and base change over perfectoid fields.

ABSTRACT

Let $p$ be a prime, let $K$ be a finite extension of $\mathbb{Q}_p$, and let $n$ be a positive integer. We construct equivalences of categories between continuous $p$-adic representations of the $n$-fold product of the absolute Galois group $G_K$ and $(φ, Γ)$-modules over one of several rings of $n$-variable power series. The case $n=1$ recovers the original construction of Fontaine and the subsequent refinement by Cherbonnier--Colmez; for general $n$, the case $K = \mathbb{Q}_p$ had been previously treated by the third author. To handle general $K$ uniformly, we use a form of Drinfeld's lemma on profinite fundamental groups of products of spaces in characteristic $p$, but for perfectoid spaces instead of schemes. We also construct the multivariate analogue of the Herr complex to compute Galois cohomology; the case $K = \mathbb{Q}_p$ had been previously treated by Pal and the third author, and we reduce to this case using a form of Shapiro's lemma.

Motivation & Objective

• To develop a systematic theory of multivariate (ϕ, Γ)-modules for products of Galois groups of p-adic fields, generalizing Fontaine's classical theory.
• To establish an equivalence of categories between continuous p-adic representations of GK,∆ and projective étale (ϕ∆, Γ∆)-modules over multivariable rings such as OE∆(K), eOE∆(K), O†E∆(K), and eO†E∆(K).
• To construct a multivariate analogue of the Herr complex for computing Galois cohomology, and prove its isomorphism with Galois cohomology via base change and Shapiro's lemma.
• To extend the theory to products of Galois groups of distinct finite extensions of Qp, and to unify it with the case K=Qp via descent and ring-theoretic base change.

Proposed method

• Use a new form of Drinfeld's lemma for perfectoid spaces to relate the étale fundamental groups of products of perfectoid spaces in characteristic p to those in characteristic 0, generalizing the classical field of norms isomorphism.
• Construct multivariable rings of power series (e.g., OE∆(K)) with commuting actions of Frobenius ϕα and Galois groups ΓK,α for each factor in the product.
• Define projective étale (ϕ∆, Γ∆)-modules as finite free modules with commuting semilinear ϕα and ΓK,α actions, where ϕα∗ is an isomorphism.
• Introduce a multivariate Herr complex via the complex of continuous (ϕ∆, Γ∆)-cochains to compute Galois cohomology.
• Apply Shapiro's lemma for induced modules to reduce cohomological computations over general K to the case K=Qp, using faithful flatness of group ring extensions.
• Use base change along Zp[ΓK,∆] → Zp[ΓQp,∆] to descend isomorphisms of cohomology from the maximal cyclotomic extension to arbitrary K.

Experimental results

Research questions

• RQ1 Can the classical (ϕ, Γ)-module equivalence for one-dimensional Galois representations be generalized to products of Galois groups of p-adic fields?
• RQ2 How does Drinfeld's lemma, originally for schemes in positive characteristic, extend to the setting of perfectoid spaces?
• RQ3 Can the Herr complex for Galois cohomology be extended to multivariate (ϕ, Γ)-modules, and is it isomorphic to Galois cohomology?
• RQ4 How does the cohomology of (ϕ∆, Γ∆)-modules behave under base change and induction when the base field is not Qp?
• RQ5 Can the theory be extended to representations over rings like eOE∆(K) and eO†E∆(K), where ϕα is bijective?

Key findings

• The category of continuous representations of GK,∆ on finite free Zp-modules is canonically equivalent to the category of projective étale (ϕ∆, Γ∆)-modules over OE∆(K), eOE∆(K), O†E∆(K), and eO†E∆(K).
• For representations on finite-dimensional Qp-vector spaces, the equivalence holds over E∆(K), eE∆(K), E†∆(K), and eE†∆(K).
• Galois cohomology Hi(GK,∆, ·) is isomorphic to the cohomology hi−dΨ•(D(·)) of the multivariate Herr complex, with the shift by d = |∆|.
• The isomorphism between Galois cohomology and (ϕ∆, Γ∆)-cohomology descends from the case K=Qp via faithful flatness of the ring extension Zp[ΓK,∆] → Zp[ΓQp,∆].
• The cohomological isomorphism holds for both Zp- and Qp-representations, and extends to the overconvergent rings O†E∆(K) and E†∆(K) via base change.
• The results extend to products of Galois groups of distinct finite extensions of Qp, with analogous equivalences and cohomological isomorphisms established via Shapiro's lemma and induction.