Spatial Kimberlites in p-adic Geometry

• Spatial kimberlites are structured v-sheaves with perfect scheme reduction and spatial diamond loci that bridge formal and analytic p-adic geometry.
• They employ a unique specialization map from generic to special fibers, ensuring robust reconstruction and cohomological consistency.
• Their applications include modeling local Shimura varieties and p-adic shtukas, influencing advances in the local Langlands program.

A spatial kimberlite is a valuative, formally separated v-sheaf with perfect scheme reduction and spatial diamond analytic locus, equipped with additional structural conditions that guarantee strong limits behavior and representability properties in the category of diamonds. Spatial kimberlites furnish the natural framework for the study of $p$-adic local models, the geometry of v-sheaves, and their interconnection with classical algebraic geometry over local fields. They are central objects in recent advances on the geometry and cohomology of local Shimura varieties, stacks of $p$-adic shtukas, and the interplay between schematic and diamond-theoretic structures in $p$-adic Hodge theory and the local Langlands program (Anschütz et al., 2022, Gleason, 2020, Gleason, 6 Jan 2026).

1. Definition and Structural Criteria

Let $O$ denote a complete discrete valuation ring with uniformizer $\pi$ and perfect residue field $k$ of characteristic $p>0$. Write $\mathrm{Spd}\,O$ for Scholze’s v-sheaf of untilts. A $p$-adic kimberlite is a small v-sheaf $X$ over $\mathrm{Spd}\,O$ that is:

• v-locally formal: there exists a cover by affinoid perfectoid charts $\mathrm{Spa}(R,R^+)\rightarrow X$ which factor through formal untilts $\mathrm{Spa}(R^+)\rightarrow X$;
• Formally adic: $X$ fits into a cartesian diagram with generic fiber $X_\eta$ and special fiber $X_s$;
• Generic fiber $$X_\eta:=X\times_{\mathrm{Spd}\,O}\mathrm{Spd}\,O[\pi^{-1}]$$ is a (locally) spatial diamond;
• Special fiber $X_s:=X\times_{\mathrm{Spd}\,O}\mathrm{Spd}\,k$ is representable by a perfect $k$-scheme of finite type;
• Flatness: every point in the special fiber lifts to a $\mathrm{Spa}(R, R^+)$-point for $R^+$ perfectoid.

A spatial kimberlite further requires that $X_\eta$ is a spatial diamond and, for the affine case, every geometric point $\mathrm{Spa}(C,C^+)\rightarrow X^{\mathrm{an}}$ admits a unique extension to a formal map $\mathrm{Spd}(C^+)\rightarrow X$, together with the existence of a single qcqs formally-adic v-cover by an affine formal v-sheaf $\mathrm{Spd}B\rightarrow X$ for some $I$-adic ring $B$. The thick-reduced decomposition expresses any spatial kimberlite as $X=X^{\mathrm{Red}}\cup X^{\mathrm{thi}}$, separating the reduced (formal) piece and the maximal thick (analytic) piece, with their intersection corresponding to the image of the specialization map on the analytic locus (Gleason, 6 Jan 2026, Anschütz et al., 2022, Gleason, 2020).

2. Specialization Principle and Full Faithfulness

A central result is the specialization faithfulness theorem: for a spatial $p$-adic kimberlite $X$ in the separated, quasi-compact, quasi-separated (qcqs) category over $\mathrm{Spd}\,O$, the data $(X_\eta, X_s, \mathrm{sp}: X_\eta\to X_s)$ determines $X$ fully faithfully. There exists a unique continuous, spectral specialization map $\mathrm{sp} : |X_\eta| \to |X_s|$ which extends the naive reduction of integral $\mathcal{O}_c$-points. Isomorphisms on fibers and compatibility of reduction maps imply isomorphism of the entire kimberlite, underscoring the sufficiency of the specialization triple for reconstruction. For small v-sheaves, v-closure coincides with weakly generalizing closure of topological spaces, ensuring the specialization principle is robust in this context (Anschütz et al., 2022, Gleason, 2020).

3. Henselianity and Cohomological Properties

Spatial kimberlites, particularly thick and proper ones, admit a henselianity property analogous to that of formal schemes. If $X$ is a thick, proper spatial kimberlite with perfectly proper reduced fiber and finite-dimensional analytic fiber, then for any $\ell\neq p$-torsion coefficient $\Lambda$ and $A\in D_\et(X^\an,\Lambda)$, both $\pi_{X,!}(j_*A)=0$ and $\pi_{X,*}(j_!A)=0$, where $j:X^\an\hookrightarrow X$ is the inclusion of the analytic locus. This vanishing of extension-by-zero and nearby cycles demonstrates a precise formal/analytic dichotomy and ensures well-behaved cohomological functors, crucial for the analysis of $p$-adic shtukas and related stacks (Gleason, 6 Jan 2026).

4. Comparison with Prekimberlite and Classical Kimberlite Theory

Spatial kimberlites strengthen the original kimberlite setup by imposing global v-formalizing (unique geometric point formalizations) and single qcqs formally-adic covers. This enhancement guarantees closure under finite limits, stability under fiber products and formal completions, and automatic representability of all maps in locally spatial diamonds. In contrast, classical kimberlites lack such robustness, with limitations on limits, representability, and formal completion stability. Spatial kimberlites thus form a rigid, robust subcategory of v-sheaves, aligning formal and analytic $p$-adic geometry in a way compatible with diamonds and perfectoid spaces (Gleason, 6 Jan 2026, Gleason, 2020).

5. Examples and Applications

Prominent examples include:

• Banach–Colmez v-sheaves: For an isocrystal $$D=\oplus_{\lambda\geq0}\mathcal{O}(\lambda)^{m_\lambda}$$, the Banach–Colmez v-sheaf $\mathrm{BC}(D)$ is (locally) a spatial kimberlite, with the locus for zero slopes forming a pointed spatial kimberlite.
• Automorphism group sheaves of isocrystals: The group-sheaf $\widetilde{G}_b$ for $b\in B(G)$, including its pro-unipotent part, is locally spatial or pointed spatial by analysis of its structure and embedding into endomorphism Banach–Colmez spaces.
• Beilinson–Drinfeld Grassmannians and local models: The local model $\mathcal{M}_{G,\mu}$ defined as a v-closure of the Schubert diamond in the affine Grassmannian context, forms a thick, proper, formally-adic spatial kimberlite, with its special fiber identified as the $p$-admissible locus in the corresponding Witt vector partial affine flag variety (Anschütz et al., 2022).
• Shtuka fibers: For the Newton polygon map $$\sigma : \mathrm{Sht}_{G,\mu}\rightarrow \mathrm{Bun}_G$$, fixed-$b$ fibers are locally spatial kimberlites, providing an interface between integral local Shimura varieties and perfectoid generic fibers (Gleason, 6 Jan 2026).

Spatial kimberlites provide the natural context for the formulation and proof of fundamental conjectures in $p$-adic geometry, such as the Scholze–Weinstein conjecture for local Shimura varieties and the Haines–Kottwitz test function conjecture. Their properties underpin the cohomological and categorical correspondences central to the local Langlands program.

6. Site Structures and Topology

Spatial kimberlites admit intrinsic Zariski and étale sites. The Zariski site consists of open immersions, compatible with the classical Zariski site of the reduced special fiber, while the étale site is formed from formally adic, qcqs, separated v-sheaves étale over the analytic locus, reducing étale over $F_s$. Both sites are invariant under formal or perfect thickenings, ensuring compatibility with classical algebraic geometry when the kimberlite arises from a formal scheme (Gleason, 2020).

The topological space underlying a spatial kimberlite is locally spectral, reflecting Hochster’s theory, and is usually noetherian in applications, such as the integral Beilinson–Drinfeld Grassmannians where rank-one points are dense, and the specialization map is spectral, surjective, and specializing.

7. Connections to Diamonds and Local Models

Every spatial kimberlite is representable in (locally) spatial diamonds (Theorem 3.32 in (Gleason, 6 Jan 2026)), and any map between spatial kimberlites is representable in this sense. This diamondal representability, together with the thick–reduced decomposition and the specialization principle, enables a precise understanding of the relationship between rigid/diamond geometry and perfect scheme geometry in the $p$-adic regime. The spatial kimberlite framework is thus essential for comparison theorems and equivalences between different categories of moduli spaces in arithmetic geometry, crucial for advances in $p$-adic cohomology and the theory of shtukas (Gleason, 6 Jan 2026, Anschütz et al., 2022, Gleason, 2020).