nLab solid module

Definition

Definition

(Mann (2022), Definition 1.6.3)
Let $A$ be a ring and let $\mathcal{D}(A)$ be the (“derived”) $\infty$-category of chain complexes of condensed $A$-modules?. For any profinite set $S=\underset{\underset{i}{\leftarrow}}\lim S_{i}$ consider the $\infty$-(co)limit

$$A_{\square}[S] \;\coloneqq\; \underset {\underset{ A'\subseteq A }{\rightarrow}} {\lim} \; \underset {\underset{i}{\leftarrow}} {\lim} \; A'\big[S_{i}\big] \,,$$

where $A'$ ranges over all subrings of $A$ which are of finite type over $\mathbb{Z}$.

An object $M \in \mathcal{D}(A)$ is called solid if for any profinite set $S$ the canonical map of mapping objects

$$\underline{Hom} \big( A_{\square}[S] ,\, M \big) \longrightarrow \underline{Hom} \big( A[S] ,\, M \big)$$

is an equivalence in $\mathcal{D}(A)$.

The full subcategory of solid modules in $\mathcal{D}(A)$ is denoted $\mathcal{D}_{\square}(A)$.

Globalization

Using the notion of adic spaces, we can glue together solid modules and consider the derived category $\mathcal{D}((\mathcal{O}_{X},\mathcal{O}_{X}^{+})_{\square})$, for $X$ an adic space. By passing to the homotopy category we get the triangulated category $D((\mathcal{O}_{X},\mathcal{O}_{X}^{+})_{\square})$ (ScholzeLCM, Lectures IX and X).

Coherent duality

There exists a notion of coherent duality (analogous to Grothendieck duality) for solid modules (ScholzeLCM, Theorem 11.1).

For brevity, given a scheme $X$ with associated adic space $X^{\ad}$, let us define $D(\mathcal{O}_{X,\square}):=D((\mathcal{O}_{X^{\ad}},\mathcal{O}_{X^{\ad}}^{+})_{\square})$.

Theorem

Let $F:X\to \mathrm{Spec}(R)$ be a separated and smooth map of finite type, of dimension $d$. Let $\omega_{X/R}=\bigwedge^{d}\Omega_{X/R}^{1}$. There is a canonical functor

$$f_{!}:D(\mathcal{O}_{X,\square})\to D(R_{\square})$$

that agrees with $R\Gamma(X,-)$ in the case that $f$ is proper. It preserves compact objects. There is a natural trace map

$$f_{!}\omega_{X/R}[d]\to R$$

such that for all $C\in D(\mathcal{O}_{X,\square})$, the natural map

$$R\Hom_{\mathcal{O}_{X}}(C,\omega_{X/R})[d]\to R\Hom_{R}(f_{!}C,R)$$

is an isomorphism.

Six-functor formalism

The category $D(\mathcal{O}_{X,\square})$ admits the six operations (ScholzeLCM, Lecture XI). The first four functors $-\otimes -$, $\Hom(-,-)$, $f_{*}$, $f^{*}$ are classical and do not require condensed mathematics. The functor $f_{!}$ has to be constructed, and $f^{!}$ will be defined to be its right adjoint.

Let $f:X\to Y$ be a separated map of finite type. Then the corresponding map of adic spaces $f_{\ad}:X^{\ad}\to Y^{\ad}$ factors as $f_{\ad}:X^{\ad}\xrightarrow{j} X^{\ad/ Y} \xrightarrow{f^{\ad /Y}} Y^{\ad}$, where the map $j$ is an open immersion and an isomorphism if $f$ is proper.

Proposition

(ScholzeLCM Proposition 11.2)

The functor

$$j^{*}:D(\mathcal{O}_{X^{\ad /Y}},\mathcal{O}_{X^{\ad /Y}}^{+})\to D(\mathcal{O}_{X^{\ad}},\mathcal{O}_{X^{\ad}}^{+})$$

admits a left adjoint

$$j_{!}:D((\mathcal{O}_{X^{\ad}},\mathcal{O}_{X^{\ad }}^{+})_{\square})\to D(\mathcal{O}_{X^{\ad} /Y},\mathcal{O}_{X^{\ad /Y}}^{+})_{\square}).$$

Definition

(ScholzeLCM Definition 11.3) The functor $f_{!}:D(\mathcal{O}_{X},\square)\to D(\mathcal{O}_{Y,\square})$ is defined to be the composition

$$f_{!}=f_{*}^{\ad /Y}\circ j_{!}.$$

It turns out that the functor $f_{!}$ commutes with all direct sums and therefore admits a right adjoint, which will be the sixth functor we call $f^{!}$.

Solid $\mathcal{O}_{X}^{+}/\pi$ almost-modules

In Mann22, Mann combines the theory of solid modules with the theory of almost modules to construct the derived $\infty$-category (in the sense of 1.3.2 of LurieHA) $\mathcal{D}_{\square}^{\a}(\mathcal{O}_{X}^{+}/ \pi)$ of solid $\mathcal{O}_{X}^{+}/ \pi$ almost modules and with it the six operations on rigid analytic spaces, in order to prove the following “mod p” version of Poincare duality:

Proposition

(Mann22, Theorem 1.1.1)

Let $K$ be an algebraically closed field of characteristic $0$ whose residue field is of characteristic $p$. Let $X$ be a proper smooth rigid-analytic variety of pure dimension $d$ over $K$. Then for all $i\in\mathbb{Z}$ there is a natural perfect pairing

$$H_{et}^{i}(X,\mathbb{F}_{p})\otimes_{\mathbb{F}}H_{et}^{2d-i}(X,\mathbb{F}_{p})\to\mathbb{F}_{p}(-d).$$

“mod p” Poincare duality for rigid analytic spaces had also previously been proven by Zavyalov in Zavyalov21, using different methods.

The theory of almost modules is necessary in order to make the structure sheaf $\mathcal{O}_{X}^{+}/\pi$ acyclic? on affinoid perfectoid spaces (compare the analogous classical situation for abelian sheaf cohomology or Cech cohomology for schemes).

In the course of proving Poincare duality for rigid analytic spaces, Mann also proves a version of a p-torsion Riemann-Hilbert correspondence for small v-stacks (Mann22, Theorem 3.9.23).

Related concepts

• condensed mathematics

• analytic ring

• six operations

References

• Lucas Mann, A p-Adic 6-Functor Formalism in Rigid-Analytic Geometry [arXiv:2206.02022]

• Peter Scholze, Lectures on condensed mathematics, pdf

Zavyalov’s proof of Poincare duality for rigid analytic spaces can be found in

• Bogdan Zavyalov, Mod-p Poincaré Duality in p-adic Analytic Geometry, arXiv:2111.01830

• Jacob Lurie, Higher Algebra, (pdf)