Pro-étale Site Overview

• Pro-étale site is a Grothendieck topology defined via weakly étale morphisms, generalizing the classical étale topology to include infinite Galois covers and constructible sheaves.
• It supports a robust derived category framework with a full six-functor formalism, enabling effective cohomological descent and motivic constructions.
• The pro-étale fundamental group bridges classical profinite and fpqc groups, offering a refined approach to both finite and infinite geometric coverings.

The pro-étale site is a Grothendieck topology on schemes, introduced by Bhatt and Scholze, designed to reconcile infinite and non-finite type phenomena in étale cohomology and the theory of local systems. It generalizes the classical étale topology and its associated Galois theory, providing a framework for infinite Galois covers, perfectoid structures, and a refined theory of constructible sheaves, fundamental groups, and motives. The underlying morphisms and covering families are defined using weakly étale morphisms, and the associated pro-étale fundamental group sits strictly between the usual profinite étale and fppf/fpqc groups, capturing a broader category of geometric coverings, including infinite ones (Lara, 2019).

1. Definition and Construction of the Pro-étale Site

Let $X$ be a scheme. The pro-étale site $X_{\mathrm{pro\text{-}\acute e t}}$ is defined as follows:

• Objects: All weakly étale $X$-schemes, i.e., morphisms $Y \to X$ such that $Y \to X$ is flat and the diagonal $\Delta: Y \to Y \times_X Y$ is flat (Jong et al., 2022). This is equivalent to being an ind-étale morphism—filtered colimit of étale morphisms (Bhatt et al., 2013, Ruimy et al., 12 Jan 2026).
• Coverings: A family $\{Y_i \to Y\}$ is a covering if it is an fpqc cover that can be refined by families either arising from étale surjective families or as cofiltered limits of finite étale surjections (pro-finite covers) (Jong et al., 2022).
• Morphisms: $X$-morphisms between weakly étale $X$-schemes.

The pro-étale topology is strictly finer than the classical étale topology and much coarser than the fpqc site—only families built from weakly étale maps are allowed (Lara, 2019).

The site admits a subcanonical topology and all cofiltered limits of representable objects (Achinger et al., 2021, Ruimy et al., 12 Jan 2026). There exists a basis of $w$-contractible affines, i.e., affines such that every weakly étale cover admits a section. This local contractibility property ensures the resulting topos is replete and supports well-behaved Postnikov towers and cohomological descent (Bhatt et al., 2013, Ruimy et al., 12 Jan 2026).

2. Weakly Étale Morphisms: Properties and Relevance

A morphism $X_{\mathrm{pro\text{-}\acute e t}}$0 is weakly étale if it satisfies any (and hence all) of:

• Flatness of $X_{\mathrm{pro\text{-}\acute e t}}$1 and of its diagonal $X_{\mathrm{pro\text{-}\acute e t}}$2;
• Flatness and formal unramifiedness;
• The Henselian lifting property: for any Henselian pair $X_{\mathrm{pro\text{-}\acute e t}}$3, every commutative square involving $X_{\mathrm{pro\text{-}\acute e t}}$4, $X_{\mathrm{pro\text{-}\acute e t}}$5 lifts uniquely to $X_{\mathrm{pro\text{-}\acute e t}}$6 (Jong et al., 2022).

The Henselian descent theorem states that any weakly étale, faithfully flat cover enables descent for arbitrary schemes, paralleling fpqc descent but utilizing Henselizations (Jong et al., 2022). Over excellent regular rings containing a field, all weakly étale algebras are ind-étale (Jong et al., 2022).

This lifting property is essential for ensuring the pro-étale site has enough points for cohomological effectiveness and for establishing comparison theorems with other sites (e.g., ind-étale and fpqc).

3. Locally Constant Sheaves and Geometric Coverings

On $X_{\mathrm{pro\text{-}\acute e t}}$7, a sheaf is locally constant if it becomes constant on some pro-étale cover. The category of locally constant sheaves coincides with the category of geometric coverings ($X_{\mathrm{pro\text{-}\acute e t}}$8), which comprise morphisms $X_{\mathrm{pro\text{-}\acute e t}}$9 that are étale (not necessarily of finite type) and satisfy the valuative criterion of properness (Lara, 2019, Achinger et al., 2021).

This generalizes the classical situation in which locally constant sheaves correspond to finite étale covers (profinite Galois theory). The pro-étale framework allows genuinely infinite étale covers (including covers of non-normal schemes and “universal covers”), vastly enlarging the class of geometric covers accessible to Galois-theoretic techniques (Bhatt et al., 2013, Lara, 2019).

4. Pro-étale Fundamental Group and Infinite Galois Categories

The automorphism group of the fiber functor from the infinite Galois category $X$0 is the pro-étale fundamental group $X$1, which is a Noohi group—a Hausdorff topological group with a basis of open subgroups and Raïkov-complete topology (Lara, 2019, Achinger et al., 2021, Bhatt et al., 2013).

This group satisfies key comparison properties:

• Its profinite completion recovers the classical étale fundamental group $X$2 of SGA1.
• Its pro-discrete completion recovers the SGA3 fppf/fpqc fundamental group.
• In particular, $X$3 canonically interpolates between existing fundamental groups and goes strictly beyond them, controlling all locally constant sheaves and infinite local systems (e.g., ℓ-adic sheaves on non-normal schemes) (Lara, 2019, Bhatt et al., 2013).

The theory includes an explicit homotopy exact sequence for geometrically connected schemes $X$4 of finite type over a field $X$5, generalizing the classical Galois sequence: $X$6 with the left map a topological embedding and the right map a surjective open quotient, connecting the geometric and arithmetic fundamental groups (Lara, 2019).

The van Kampen theorem and Künneth formula generalize to the pro-étale context, with the van Kampen theorem expressing $X$7 as a Noohi free product of the fundamental groups of covering pieces, modulo relations encoding compatibility and cocycle conditions (Lara, 2019).

5. Derived Categories, Constructible Sheaves, and Functorial Tools

The pro-étale site supports a robust theory of derived categories, particularly for constructible and ℓ-adic sheaves. On $X$8, one can define sheaves with coefficients in $X$9, $Y \to X$0, or adèles directly, with locally constant and adically complete objects forming the constructible derived category. There is a full six-functor formalism (pullback, pushforward, proper and exceptional functors, internal Hom, tensor product, dualizing complexes), and unbounded cohomological descent always converges (Bhatt et al., 2013, Ruimy et al., 12 Jan 2026).

For motivic categories, the pro-étale site allows the definition of pro-étale motives and pro-étale motivic spectra, with coefficients in any condensed ring spectrum (Ruimy et al., 12 Jan 2026). The pro-étale motivic stable homotopy category $Y \to X$1 enhances classical motivic homotopy theory, embedding étale motivic spectra fully faithfully over locally étale bounded schemes.

Solidification processes connect the pro-étale motives to the abelian category of solid sheaves (in the sense of Fargues–Scholze), yielding rigidity theorems and a “solid realization” functor that recovers the classical ℓ-adic realization while remaining within presentable categories (Ruimy et al., 12 Jan 2026).

6. Comparison with Other Fundamental Groups and Specialization

The pro-étale fundamental group functorially fits into specialization sequences when relating generic and special fibers of formal schemes—crucially, the construction of the specialization map

$Y \to X$2

relates de Jong’s rigid fundamental group to the Bhatt-Scholze pro-étale group on the special fiber, extending the classical profinite specialization (Achinger et al., 2021). The specialization map is constructed using admissible blowups, normalizations, and Berthelot tubes; under normality assumptions, the image is dense.

In tame situations (e.g., over discretely valued fields with residue characteristic zero), every finite étale cover in the rigid setting extends to a de Jong covering space, leading to exact compatibility with pro-étale descent (Achinger et al., 2021).

7. Applications and Examples

The pro-étale site underlies contemporary developments in:

• Cohomology Theory: Recovery of Galois cohomology, p-adic Hodge theory, perfectoid cohomology, and a natural setting for constructing ℓ-adic and adèle-valued sheaves (Jong et al., 2022, Bhatt et al., 2013).
• Motivic Theory: Construction of pro-étale motives, solid realization functors, full six-functor formalisms on solid sheaves, and comparison with classical motivic categories (Ruimy et al., 12 Jan 2026).
• Infinite Coverings: Pro-étale covers naturally include infinite Galois towers and perfectoid towers, which are not accessible via finite étale or even fpqc topologies.
• Homotopy Theory and Fundamental Groups: Expression of profinite and prodiscrete homotopy types, explicit computation for singular, non-normal, or nodal curves—yielding more refined local systems than are visible from the profinite viewpoint (Bhatt et al., 2013, Lara, 2019).

In summary, the pro-étale site provides a comprehensive unifying framework, subsuming existing Galois-theoretic, cohomological, and motivic tools, and proving suitable for both finite and infinite constructions encountered in modern arithmetic geometry and homotopy theory (Lara, 2019, Achinger et al., 2021, Jong et al., 2022, Bhatt et al., 2013, Ruimy et al., 12 Jan 2026).