Pro-étale Motives in Algebraic Geometry

• Pro-étale motives are a robust extension of étale motivic theories, merging condensed mathematics with a six-functor formalism.
• They incorporate arbitrary condensed ring spectra and enable well-behaved ℓ-adic and ℚℓ-adic realization functors inside presentable ∞-categories.
• Solidification within condensed categories achieves strict rigidity by embedding étale motives fully faithfully and bridging to solid sheaf contexts.

Pro-étale motives are a robust extension of étale motivic theories, formulated using pro-étale topologies, condensed mathematics, and stable $\infty$-categorical methods. This framework allows the incorporation of arbitrary condensed ring spectra as coefficients, encompasses the six operations of Grothendieck, achieves solid rigidity in the sense of Fargues–Scholze, and affords well-behaved realization functors, including $\ell$-adic and $\mathbb{Q}_\ell$-adic realizations, inside presentable categories. Over locally étale bounded bases, pro-étale motives strictly extend the theory of étale motives, embedding the latter fully faithfully and facilitating a seamless transfer to solid sheaf-theoretic contexts on schemes (Ruimy et al., 12 Jan 2026).

1. Definition and Structure of Pro-étale Motives

Consider a quasi-compact quasi-separated (qcqs) scheme $X$. Define $\mathrm{WSm}_X$ as the category of “weakly smooth” $X$-schemes, i.e., morphisms obtained by composing smooth and weakly étale maps. The pro-étale site is imposed on $\mathrm{WSm}_X$.

The $\infty$-category $\mathrm{Sh}_{\mathrm{pro}}(X)$ consists of hypercomplete anima-valued sheaves on the big pro-étale site of $X$, while $$\mathrm{Sh}_{\mathrm{pro}}(\mathrm{WSm}_X, \mathrm{Sp})$$ consists of hypercomplete spectral sheaves. Constructing the stable motivic $\infty$-category proceeds via:

• $$L_{\mathbb{A}^1}\mathrm{Sh}_{\mathrm{pro}}(\mathrm{WSm}_X, \mathrm{Sp})$$: the subcategory of $\mathbb{A}^1$-local objects.
• $$L_{\mathbb{A}^1}\mathrm{Sh}_{\mathrm{pro}}^{S^1}(X)$$: the stabilization under $S^1$-suspension.
• $\mathrm{SH}_{\mathrm{pro}}(X)$: the further stabilization inverting $S^{2,1} \simeq (\mathbb{P}^1,\infty)$.

A condensed ring spectrum $\Lambda$ is a commutative algebra object in Clausen–Scholze’s $\mathrm{Cond}(\mathrm{Sp})$. The $\Lambda$-linear pro-étale motivic coefficient system is provided by

$$\mathrm{SH}_{\mathrm{pro}}(-;\Lambda): \mathrm{Sch}^{op} \to \mathrm{CAlg}(\mathrm{Pr}),\quad X \mapsto \mathrm{Mod}_\Lambda(\mathrm{SH}_{\mathrm{pro}}(X)),$$

and by extension to

$$\mathrm{DM}_{\mathrm{pro}}(-;\Lambda) = \mathrm{SH}_{\mathrm{pro}}(-; H\Lambda),$$

where $H\Lambda$ is the Eilenberg–MacLane motivic spectrum.

The pro-étale realization functor

$$M_{\mathrm{proet}}(-;\Lambda) : (\mathrm{Sm}/-)_{\mathrm{pro}} \to \mathrm{SH}_{\mathrm{pro}}(-;\Lambda)$$

assigns to a weakly smooth $X$-scheme $Y$ the pro-étale representable sheaf, which is then $\mathbb{A}^1$-localized and $\mathbb{P}^1$-stabilized.

2. Six Functor Formalism for Pro-étale Motivic Spectra

For any morphism $f:Y\to X$ of qcqs schemes, there exists an adjoint pair

$$f^*:\mathrm{SH}_{\mathrm{pro}}(X)\rightarrow\mathrm{SH}_{\mathrm{pro}}(Y),\qquad f_*:\mathrm{SH}_{\mathrm{pro}}(Y)\rightarrow\mathrm{SH}_{\mathrm{pro}}(X),$$

with $f^*$ symmetric monoidal. If $f$ is finitely presented, adjoints

$$f_!:\mathrm{SH}_{\mathrm{pro}}(Y)\rightarrow\mathrm{SH}_{\mathrm{pro}}(X),\qquad f^!:\mathrm{SH}_{\mathrm{pro}}(X)\rightarrow\mathrm{SH}_{\mathrm{pro}}(Y)$$

exist, and there is a comparison isomorphism $f_! \cong f_*$ when $f$ is proper. Each $\mathrm{SH}_{\mathrm{pro}}(X)$ is stable, presentably symmetric monoidal with tensor $\otimes_X$ and internal Hom.

Base change and projection formulas hold:

• Proper base change: $f^* p_* \simeq p'_* (f')^*$ for Cartesian squares.
• Projection formulas for proper/smooth maps: $(p_*M)\otimes N\simeq p_*(M\otimes p^*N)$.

Localization for a closed immersion $i:Z \to X$ and open complement $j:U \to X$ yields cofiber sequences

$$j_! j^* \to \mathrm{Id} \to i_* i^*,\qquad i_! i^! \to \mathrm{Id} \to j_* j^*.$$

Purity and ambidexterity are available:

• Purity: For regular closed immersion $s:Z\to X$ of codimension $d$, $s^! \simeq \Sigma^{N_s} s^*$, with $N_s$ the normal bundle.
• Ambidexterity: For $f:X\to S$ finitely presented, smooth, and proper, with virtual tangent bundle $T_f$, $f_*\simeq f_! \Sigma^{-T_f}$ and $f^! \simeq \Sigma^{T_f} f^*$. These constitute a full six-functor formalism [(Ruimy et al., 12 Jan 2026), Thm. 3.18].

3. Embedding Étale Motives into Pro-étale Motives

Let $\mathrm{DM}_{\mathrm{et}}(-;\Lambda)$ denote Voevodsky’s étale motivic spectra. For schemes $X$ that are locally étale bounded (finite Krull dimension and bounded Galois cohomological dimension at every residue field), and any ring spectrum $\Lambda$ in which the relevant residue characteristics are invertible, the functor

$$\nu^*: \mathrm{DM}_{\mathrm{et}}(X;\Lambda) \rightarrow \mathrm{DM}_{\mathrm{pro}}(X;\Lambda)$$

is fully faithful.

This fully faithfulness holds at the level of unstable $\mathbb{A}^1$-local objects by reduction to sheaves of sets and cohomological dimension arguments. It persists through $\mathbb{A}^1$-invariant and $\mathbb{P}^1$-stable objects due to the commutation of the appropriate Hom and stabilization functors, thereby embedding $\mathrm{DM}_{\mathrm{et}}$ fully into $\mathrm{DM}_{\mathrm{pro}}$ in this context [(Ruimy et al., 12 Jan 2026), Thm. 2.25]. A plausible implication is the extension of motivic phenomena previously restricted to étale settings into the strictly larger pro-étale context.

4. Condensed Categories and Solidification

A condensed $\infty$-category $C$ is a sheaf of $\infty$-categories on the site of profinite sets ($\mathrm{ProFin}$), $C: \mathrm{ProFin}^{op} \to \mathrm{Cat}$, satisfying descent. If every $C(S)$ is presentable and the base-change $s^*$ for $s:S'\to S$ possesses a left adjoint $s_!$ compatible with further base-change, then $C$ is presentable. The tensor product $-\otimes^{\mathrm{cond}}-$ endows the category of presentable condensed categories with a symmetric monoidal structure.

Notable examples include:

• $\underline{\mathrm{Sh}}(X_{\mathrm{pro}})$: $S\mapsto \mathrm{Sh}((X\times S)_{\mathrm{pro}})$,
• $\underline{D}(X_{\mathrm{pro}}, \Lambda)$,
• $\underline{\mathrm{DM}}_{\mathrm{pro}}(X, \Lambda)$,
• $$\underline{\mathrm{Solid}}_\Lambda: S \mapsto D(S, \Lambda)^\sharp$$,
• $\underline{\mathrm{Mod}}_\Lambda$.

Given a $\underline{\mathrm{Mod}}_\Lambda$-linear presentable condensed category $C$, its solidification is

$$C^\sharp := C\otimes^{\mathrm{cond}}_{\underline{\mathrm{Mod}}_\Lambda} \underline{\mathrm{Solid}}_\Lambda.$$

For the derived category, $$D(X, \Lambda)^\sharp \simeq \underline{D}(X_{\mathrm{pro}}, \Lambda)^\sharp(*)$$, recapturing the abelian/derived solid sheaves, with the embedding

$$\rho^\sharp: D(X, \Lambda)^\sharp \rightarrow D(\mathrm{WSm}_X, \Lambda)^\sharp$$

fully faithful, exact, and preserving all colimits and f_! for $f$ weakly étale [(Ruimy et al., 12 Jan 2026), Prop. 4.28, Thm. 5.4].

5. Solid Rigidity and Identification with Solid Sheaves

Solid rigidity establishes equivalences between categories of solidified motives and solid sheaves:

• Effective solidity and torsion rigidity: After inverting $Z/nZ(1)\to Z/nZ[\mu_n]$ (for $n$ coprime to the residue characteristics), $$D(X,\Lambda)^\sharp \simeq D^{\mathbb{A}^1}(\mathrm{WSm}_X, \Lambda)^\sharp$$, identifying $D(X,n)^\sharp$ with effective solid pro-étale motives.
• Full rigidity: Further inverting the Tate twist $M(1)\to M\otimes \mu_{\infty, \mathcal{P}}$ leads to

$$D^{\mathbb{A}^1}(\mathrm{WSm}_X, \Lambda)^\sharp \simeq \mathrm{DM}^{\mathrm{eff}}(X, \Lambda)^\sharp$$

and

$$D(X, \Lambda)^\sharp \simeq \mathrm{DM}_{\mathrm{pro}}(X, \Lambda)^\sharp \simeq \mathrm{DM}(X, \Lambda)^\sharp,$$

i.e., solidified pro-étale motives align with the modified Fargues–Scholze solid sheaf categories for schemes [(Ruimy et al., 12 Jan 2026), Thm. 5.11, Prop. 5.12].

The following table summarizes the main identifications:

Category	After Inverting…	Identified With
$D(X, \Lambda)^\sharp$	$Z/nZ(1)\to Z/nZ[\mu_n]$	Effective solid pro-étale motives
$D^{\mathbb{A}^1}(\mathrm{WSm}_X, \Lambda)^\sharp$	Tate twist $M(1)\to M\otimes \mu_{\infty, \mathcal{P}}$	$\mathrm{DM}^{\mathrm{eff}}(X, \Lambda)^\sharp$
$D(X, \Lambda)^\sharp$	Stabilization	$\mathrm{DM}_{\mathrm{pro}}(X, \Lambda)^\sharp$

6. Solid Realization Functors and $\ell$-adic Comparison

There is a symmetric monoidal functor of six-functor formalisms

$$\rho_\sharp: \mathrm{DM}_{\mathrm{et}}(-; \mathbb{Z}) \rightarrow D(-, \mathbb{Z}_\ell)^\sharp,$$

that also extends to $\mathbb{Q}_\ell$ and more general coefficients via change of scalars.

A commutative square of adjointable symmetric monoidal functors relates motives and solid sheaves:

$$\begin{array}{ccc} \mathrm{DM}(X;\mathbb{Z}) & \xrightarrow{\rho_\ell} & D(X;\mathbb{Z}_\ell)^\sharp \ \downarrow \otimes \mathbb{Q} & & \downarrow \otimes \mathbb{Q} \ \mathrm{DM}(X; \mathbb{Q}) & \xrightarrow{\rho_{\mathbb{Q}_\ell}} & D(X; \mathbb{Q}_\ell)^\sharp \end{array}$$

On compact (geometric) objects, this construction recovers the established $\ell$-adic and $\mathbb{Q}_\ell$-adic realization functors of Cisinski–Déglise, Bachmann–Cisinski, and Huber–Kebekus–Olsson, but generalized to the context of solid sheaves and presentable $\infty$-categories [(Ruimy et al., 12 Jan 2026), Cor. 5.17, Cor. 5.18]. This framework is notably compatible with coefficient changes.

7. Summary and Significance

Pro-étale motives provide a comprehensive enhancement of étale motivic homotopy theory, operating over locally étale bounded schemes and supporting a full six-functor formalism. Solidification within the condensed category paradigm produces a rigid identification with solid sheaf categories, effectively transferring the motivic formalism into the solid context. This enables robust functorial realization theories—including $\ell$-adic and $\mathbb{Q}_\ell$-adic realizations—within presentable $\infty$-categories, with full compatibility for coefficient extensions and comparison theorems (Ruimy et al., 12 Jan 2026).