Condensed Mathematics: Analytic Rings¶
Job: define analytic rings \((\mathcal{A},\mathcal{A}[-])\) in the sense of Clausen–Scholze, list the standard analytic rings (\(\overline{\mathbb{Z}}\), \(\mathbb{Z}[\tfrac{1}{p}]^{\triangleright}\), \(\mathbb{R}^{\triangleright}\), \(\mathbb{Q}_p^{\triangleright}\), \(A_{\inf}^{\blacksquare}\)), state the liquid tensor product theorem (the main theorem of Analytic), and reformulate the BMS \(A_{\inf}\) comparison in analytic-ring language.
Core¶
Analytic rings¶
An analytic ring (Clausen–Scholze) is a pair
where \(\mathcal{A}[-]\) is a functor satisfying the analytic-ring axioms: it sends a point to \(A\), sends finite disjoint unions to finite direct sums, and the induced "completion" localization \(\mathcal{A}[-]:D(\mathrm{Cond}(\mathrm{Ab}))\to D(\mathrm{Cond}(\mathrm{Ab}))\) is an idempotent, exact, symmetric-monoidal localization (a Bousfield localization in the stable \(\infty\)-categorical sense). A module over \(\mathcal{A}\) is a condensed \(A\)-module \(M\) with \(M\simeq M\otimes^{\mathbb{L}}_{A}\mathcal{A}[\ast]\), i.e. a local object of the localization; write \(D(\mathcal{A})\) for the localized category and \(\mathrm{Mod}_{\mathcal{A}}^{\heartsuit}\) for its heart.
The tensor product over \(\mathcal{A}\) is the monoidal structure on \(D(\mathcal{A})\):
the localized derived tensor. It is symmetric monoidal, closed, and the localization is exact — this is the content of "analytic ring" and is the theorem that repairs \(\mathrm{Cond}(\mathrm{Ab})\).
The standard analytic rings¶
| Analytic ring \(\mathcal{A}\) | Underlying \(A\) | Structure \(\mathcal{A}[-]\) | Coefficient category | Exactness | Side |
|---|---|---|---|---|---|
| \(\overline{\mathbb{Z}}\) | \(\mathbb{Z}\) | \(\mathbb{Z}[-]^{\blacksquare}\) (solid completion) | \(\mathrm{Solid}_{\mathbb{Z}}\) | AB5, stable, closed under \(\otimes_{\blacksquare}\) | non-archimedean, \(p\) not inverted |
| \(\mathbb{Z}[\tfrac{1}{p}]^{\triangleright}\) | \(\mathbb{Z}[\tfrac{1}{p}]\) | \(\mathbb{Z}[\tfrac{1}{p}][S]^{\triangleright}=\varprojlim_n\mathbb{Z}[\tfrac{1}{p}][S]_{(p)}\)-completion | liquid \(\mathbb{Z}[\tfrac{1}{p}]\)-modules | AB5 (liquid tensor product theorem) | \(p\) invertible, non-archimedean |
| \(\mathbb{R}^{\triangleright}\) | \(\mathbb{R}\) | \(\mathbb{R}[S]^{\triangleright}=\varprojlim\) of Banach \(\ell^1\)-type completions | liquid \(\mathbb{R}\)-modules (\(\mathrm{Liq}_{\mathbb{R}}\)) | AB5 — the hard theorem | archimedean |
| \(\mathbb{Q}_p^{\triangleright}\) | \(\mathbb{Q}_p\) | \(\mathbb{Q}_p[S]^{\triangleright}\) | liquid \(\mathbb{Q}_p\)-modules | AB5 | \(p\)-adic, \(p\) invertible |
| \(A_{\inf}^{\blacksquare}\) | \(A_{\inf}=W(R^\flat)\) | \(A_{\inf}[S]^{\blacksquare}\) (solid over \(A_{\inf}\)) | solid \(A_{\inf}\)-modules | AB5 (solid) | \(p\)-adic, links to prismatic |
The arrow \(\blacksquare\) denotes the solid (non-archimedean, \(p\)-adic) completion; \(\triangleright\) denotes the liquid completion (archimedean or \(p\)-invertible). The two completions are different analytic-ring structures on the same underlying animated ring when both are defined, and the choice of which to use is dictated by the problem: \(p\)-adic / non-archimedean \(\Rightarrow\) solid; archimedean or after inverting \(p\) \(\Rightarrow\) liquid.
The liquid tensor product theorem¶
The main theorem of Clausen–Scholze Analytic is:
Theorem (Liquid tensor product). For \(p\)-invertible \(\mathbb{R}\)-algebras \(A\) (in particular \(A=\mathbb{R}\), \(\mathbb{Z}[\tfrac{1}{p}]\), \(\mathbb{Q}_p\)), the liquid analytic ring \(A^{\triangleright}\) is well-defined: the functor \(A[-]^{\triangleright}\) satisfies the analytic-ring axioms, the localized category \(D(A^{\triangleright})\) is presentable stable symmetric-monoidal with exact localization, and the tensor product \(\otimes_{\triangleright}\) is the left derived monoidal product. In particular \(\mathrm{Mod}_{A^{\triangleright}}^{\heartsuit}\) (liquid \(A\)-modules) is an abelian AB5-exact category closed under all limits/colimits and \(\otimes_{\triangleright}\), and the natural functor \(\mathrm{Cond}(\mathrm{Ab})\to\mathrm{Mod}_{A^{\triangleright}}^{\heartsuit}\), \(M\mapsto M\otimes_{\triangleright}A^{\triangleright}\), is exact.
The proof is the deep part of Analytic: it goes via the archimedean liquid case \(A=\mathbb{R}\), where the structure \(A[S]^{\triangleright}\) is a completion modeled on a Banach-type norm (the "liquid norm", refining \(\ell^1\)), and the exactness of the localization is a non-trivial functional-analytic estimate (a controlled resolution by liquid objects, analogous to the way one proves exactness of a category of Banach spaces by an open-mapping / closed-graph argument, here reorganized as an exactness-of-localization statement in a stable \(\infty\)-category). The \(p\)-invertible cases (\(\mathbb{Z}[\tfrac{1}{p}]^{\triangleright}\), \(\mathbb{Q}_p^{\triangleright}\)) are deduced by base change from \(\mathbb{R}^{\triangleright}\) via the comparison \(\mathbb{Z}[\tfrac{1}{p}]\hookrightarrow\mathbb{R}\) and the archimedean-to-\(p\)-adic descent. The solid case \(\overline{\mathbb{Z}}\) is comparatively formal (a left completion), which is why the archimedean / liquid case is the theorem.
Reconnection to \(p\)-adic Hodge: \(A_{\inf}\) as an analytic ring¶
For a perfectoid field \(K\) with ring of integers \(\mathcal{O}_K\) and tilt \(K^\flat\), set
The pair \(A_{\inf}^{\blacksquare}=(A_{\inf},A_{\inf}[-]^{\blacksquare})\) is a solid analytic ring (solid over the \(p\)-adic ring \(A_{\inf}\)), and it is the bridge to prismatic cohomology:
- The BMS comparison (Bhatt–Morrow–Scholze, Integral \(p\)-adic Hodge Theory) — identifying \(p\)-complete \(K\)-theory with Nygaard-completed prismatic cohomology — is, in the condensed language, the statement that the relevant cohomology complexes are naturally modules over \(A_{\inf}^{\blacksquare}\) and that the étale / de Rham / crystalline comparison isomorphisms are base changes along the analytic-ring maps $$ A_{\inf}^{\blacksquare}\xrightarrow{\theta}\mathcal{O}K^{\blacksquare},\quad A. $$}^{\blacksquare}[\tfrac{1}{p}]\to B_{\mathrm{dR}}^{\triangleright},\quad A_{\inf}^{\blacksquare}\to\overline{\mathbb{Z}}\text{ (mod }\xi\text{)
- The prism \((A_{\inf},(\xi))\) of prismatic is the underlying ring-plus-ideal data of the analytic ring \(A_{\inf}^{\blacksquare}\); the Frobenius lift \(\varphi\) on \(A_{\inf}\) is an analytic-ring endomorphism, and the BMS Nygaard completion is a completion in \(D(A_{\inf}^{\blacksquare})\).
- The Hodge–Tate specialization \(\overline{\triangle}_{X/A}\) is the base change to \(\mathcal{O}_K^{\blacksquare}=A_{\inf}^{\blacksquare}/\xi\); the étale comparison is the base change to the liquid \(B_{\mathrm{dR}}^{\triangleright}\)-category.
This is the precise sense in which condensed mathematics + analytic rings is the natural home of \(p\)-adic Hodge theory: the period rings \(A_{\inf}\), \(B_{\mathrm{dR}}\), \(B_{\mathrm{HT}}\) are analytic rings, and the comparison theorems are base-change statements in the analytic-ring category.
See it¶
The liquid tensor product computing \(\ell^2\). Take \(A=\mathbb{R}\) and the solid (underived) free module \(\mathbb{R}[S]\) for \(S=\{1,\ldots,n\}\) finite; the liquid completion \(\mathbb{R}[S]^{\triangleright}\) is the Banach \(\ell^1\)-space \(\ell^1(S,\mathbb{R})\cong\mathbb{R}^n\). Now consider the diagonal / Hilbert object: one wants to compute the liquid tensor product of two infinite-dimensional liquid \(\mathbb{R}\)-modules, say \(V=\ell^2(\mathbb{N})\) and \(V^\vee\), in the liquid category. The liquid tensor product theorem gives
and the natural map \(V\otimes_{\triangleright}V^\vee\to\mathbb{R}^{\triangleright}\) (the liquid evaluation) is well-defined because both sides are liquid and \(\otimes_{\triangleright}\) is exact — whereas in \(\mathrm{Cond}(\mathrm{Ab})\) the corresponding underived tensor is not defined (the AB5 failure). The liquid tensor product here recovers the projective Banach tensor of \(\ell^2\) with its dual, and the exactness of the liquid category is exactly what makes the Hilbert-space duality \(\ell^2\otimes\ell^2\to\mathbb{R}\) into an algebraic statement in \(\mathrm{Liq}_{\mathbb{R}}\). This is the prototypical "functional analysis becomes algebra" computation, and it is the reason the liquid tensor product theorem is the main theorem: without it, \(\ell^2\otimes\ell^2\) in \(\mathrm{Cond}(\mathrm{Ab})\) is unusable.
Check¶
Why is the liquid case (\(\mathbb{R}^{\triangleright}\)) the hard theorem, while the solid case (\(\overline{\mathbb{Z}}\)) is comparatively formal, and what exactly is the functional-analytic input that makes the liquid theorem true?
Solid modules over \(\overline{\mathbb{Z}}\) are a left completion: \(\mathbb{Z}[S]^{\blacksquare}=\varprojlim_n\mathbb{Z}[S_n]\), an inverse limit of free modules over finite sets, and the localization is the standard derived completion, exact by the general theory of derived completions (the kernel is \(L\)-complete acyclic). No estimate is needed. The liquid case \(\mathbb{R}^{\triangleright}\) requires an archimedean norm: \(\mathbb{R}[S]^{\triangleright}\) is the completion of \(\mathbb{R}[S]\) for a "liquid norm" that refines \(\ell^1\) and is non-discrete. Exactness of the localization \(D(\mathrm{Cond}(\mathrm{Ab}))\to D(\mathbb{R}^{\triangleright})\) is then a statement that the liquid norm resolves every condensed \(\mathbb{R}\)-module by liquid modules of bounded complexity — a non-trivial functional-analytic estimate (Clausen–Scholze Analytic), the analog of an open-mapping / closed-graph theorem phrased as exactness of a stable \(\infty\)-categorical localization. This is why \(\mathbb{R}^{\triangleright}\), not \(\overline{\mathbb{Z}}\), is where the theorem lives.
State the base-change diagram that recovers the BMS étale comparison as an analytic-ring base change, and identify the analytic ring at each node.
Start with \(A_{\inf}^{\blacksquare}\) (solid over \(A_{\inf}=W(\mathcal{O}_{K^\flat})\)). The BMS prismatic complex \(\triangle_{X/A_{\inf}}\in D(A_{\inf}^{\blacksquare})\) carries a Frobenius-endomorphism (analytic-ring endomorphism \(\varphi\)). The comparison isomorphisms are: (i) Hodge–Tate \(=\,\) base change along \(A_{\inf}^{\blacksquare}\xrightarrow{\theta}\mathcal{O}_K^{\blacksquare}=A_{\inf}^{\blacksquare}/\xi\), giving \(\overline{\triangle}_{X/A_{\inf}}\in D(\mathcal{O}_K^{\blacksquare})\); (ii) étale \(=\,\) base change along \(A_{\inf}^{\blacksquare}[\tfrac{1}{p}]\to B_{\mathrm{dR}}^{\triangleright}\) to the liquid \(B_{\mathrm{dR}}\)-category, recovering \(R\Gamma_{\mathrm{\acute{e}t}}(X_{\overline{K}},\mathbb{Z}_p)\otimes B_{\mathrm{dR}}\); (iii) de Rham \(=\,\) base change along \(A_{\inf}^{\blacksquare}\to\mathcal{O}_K^{\blacksquare}\to K^{\triangleright}\) (the latter \(p\)-invertible liquid). Each node is an analytic ring, each arrow an analytic-ring map, and the comparison theorems are base-change compatibilities in \(D(\mathrm{Cond}(\mathrm{Ab}))\) — the condensed reformulation of the BMS dictionary (see prismatic p-adic Hodge).
Pitfalls¶
- Solid (\(\blacksquare\)) vs liquid (\(\triangleright\)): the arrow is not a notational preference — it specifies the analytic ring, hence the exactness theorem. Solid = \(p\) not inverted, non-archimedean, complete topological abelian groups (\(\mathbb{Z}_p\), profinite). Liquid = \(p\) invertible or archimedean, Banach/Hilbert (\(\ell^2\), \(\mathbb{R}\)-Banach spaces). Using \(\overline{\mathbb{Z}}\) for an archimedean problem gives \(\mathbb{R}^{\blacksquare}=0\) and a vacuous computation; the standard error.
- The liquid theorem is not formal: exactness of \(\mathrm{Liq}_{\mathbb{R}}\) is the main theorem of Analytic; do not assume it follows from the solid case by inverting \(p\) — the archimedean / \(\mathbb{R}\)-norm estimate is the genuine content, and the \(p\)-invertible cases are deduced from it (not the reverse).
- \(A_{\inf}^{\blacksquare}\neq A_{\inf}^{\triangleright}\): \(A_{\inf}\) is \(p\)-adically complete and \(p\) is not invertible in it; the relevant analytic ring is solid \(A_{\inf}^{\blacksquare}\), not liquid. Liquid \(A_{\inf}[\tfrac{1}{p}]^{\triangleright}\) appears only after inverting \(p\), where the étale comparison lives — base change to liquid is a post-hoc step, not the original structure.
- Analytic ring ≠ animated ring with topology: an analytic ring is the pair \((A,\mathcal{A}[-])\); forgetting the structure functor and treating \(A\) as a bare ring (or animated ring) loses the localization that makes the category exact. The "modules over \(A\)" in the condensed sense are not the same as condensed \(A\)-modules — they are the localized (solid/liquid) ones.
Refs¶
- Clausen–Scholze, Analytic (2019/2020), §I–IV (analytic rings; the liquid tensor product theorem)
- Clausen–Scholze, Condensed Mathematics (2019), §VII (the solid analytic ring \(\overline{\mathbb{Z}}\))
- Bhatt–Morrow–Scholze, Integral \(p\)-adic Hodge Theory (Publ. Math. IHÉS 2019), §9 (\(A_{\inf}\) comparison)
- Scholze, Lectures on Analytic Geometry (Tübingen 2019), §3–4 (the standard analytic rings)
Next¶
Pyknotic & Applications · Condensed index · Foundations · Solid Modules · Prismatic cohomology · p-adic Hodge · Functional analysis · Derived AG