nLab solid module

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Definition

Definition

(Mann (2022), Definition 1.6.3)
Let AA be a ring and let π’Ÿ(A)\mathcal{D}(A) be the (β€œderived”) ∞ \infty -category of chain complexes of condensed A A -modules?. For any profinite set S=lim←iS iS=\underset{\underset{i}{\leftarrow}}\lim S_{i} consider the ∞ \infty -(co)limit

A β–‘[S]≔limβ†’Aβ€²βŠ†Alim←iAβ€²[S i], A_{\square}[S] \;\coloneqq\; \underset {\underset{ A'\subseteq A }{\rightarrow}} {\lim} \; \underset {\underset{i}{\leftarrow}} {\lim} \; A'\big[S_{i}\big] \,,

where Aβ€²A' ranges over all subrings of AA which are of finite type over β„€ \mathbb{Z} .

An object Mβˆˆπ’Ÿ(A)M \in \mathcal{D}(A) is called solid if for any profinite set SS the canonical map of mapping objects

HomΜ²(A β–‘[S],M)⟢HomΜ²(A[S],M) \underline{Hom} \big( A_{\square}[S] ,\, M \big) \longrightarrow \underline{Hom} \big( A[S] ,\, M \big)

is an equivalence in π’Ÿ(A)\mathcal{D}(A).

The full subcategory of solid modules in π’Ÿ(A)\mathcal{D}(A) is denoted π’Ÿ β–‘(A)\mathcal{D}_{\square}(A).

Globalization

Using the notion of adic spaces, we can glue together solid modules and consider the derived category π’Ÿ((π’ͺ X,π’ͺ X +) β–‘)\mathcal{D}((\mathcal{O}_{X},\mathcal{O}_{X}^{+})_{\square}), for XX an adic space. By passing to the homotopy category we get the triangulated category D((π’ͺ X,π’ͺ X +) β–‘)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 XX with associated adic space X adX^{\ad}, let us define D(π’ͺ X,β–‘):=D((π’ͺ X ad,π’ͺ X ad +) β–‘)D(\mathcal{O}_{X,\square}):=D((\mathcal{O}_{X^{\ad}},\mathcal{O}_{X^{\ad}}^{+})_{\square}).

Theorem

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

f !:D(π’ͺ X,β–‘)β†’D(R β–‘)f_{!}:D(\mathcal{O}_{X,\square})\to D(R_{\square})

that agrees with RΞ“(X,βˆ’)R\Gamma(X,-) in the case that ff is proper. It preserves compact objects. There is a natural trace map

f !Ο‰ X/R[d]β†’Rf_{!}\omega_{X/R}[d]\to R

such that for all C∈D(π’ͺ X,β–‘)C\in D(\mathcal{O}_{X,\square}), the natural map

RHom π’ͺ X(C,Ο‰ X/R)[d]β†’RHom R(f !C,R)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(π’ͺ X,β–‘)D(\mathcal{O}_{X,\square}) admits the six operations (ScholzeLCM, Lecture XI). The first four functors βˆ’βŠ—βˆ’-\otimes -, Hom(βˆ’,βˆ’)\Hom(-,-), f *f_{*}, f *f^{*} are classical and do not require condensed mathematics. The functor f !f_{!} has to be constructed, and f !f^{!} will be defined to be its right adjoint.

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

Proposition

(ScholzeLCM Proposition 11.2)

The functor

j *:D(π’ͺ X ad/Y,π’ͺ X ad/Y +)β†’D(π’ͺ X ad,π’ͺ X ad +)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((π’ͺ X ad,π’ͺ X ad +) β–‘)β†’D(π’ͺ X ad/Y,π’ͺ X ad/Y +) β–‘).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(π’ͺ X,β–‘)β†’D(π’ͺ Y,β–‘)f_{!}:D(\mathcal{O}_{X},\square)\to D(\mathcal{O}_{Y,\square}) is defined to be the composition

f !=f * ad/Y∘j !.f_{!}=f_{*}^{\ad /Y}\circ j_{!}.

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

Solid π’ͺ X +/Ο€\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) π’Ÿ β–‘ a(π’ͺ X +/Ο€)\mathcal{D}_{\square}^{\a}(\mathcal{O}_{X}^{+}/ \pi) of solid π’ͺ X +/Ο€\mathcal{O}_{X}^{+}/ \pi almost modules and with it the six operations on rigid analytic spaces, in order to prove the following β€œmod p” version of Poincare duality:

Proposition

(Mann22, Theorem 1.1.1)

Let KK be an algebraically closed field of characteristic 00 whose residue field is of characteristic pp. Let XX be a proper smooth rigid-analytic variety of pure dimension dd over KK. Then for all iβˆˆβ„€i\in\mathbb{Z} there is a natural perfect pairing

H et i(X,𝔽 p)βŠ— 𝔽H et 2dβˆ’i(X,𝔽 p)→𝔽 p(βˆ’d).H_{et}^{i}(X,\mathbb{F}_{p})\otimes_{\mathbb{F}}H_{et}^{2d-i}(X,\mathbb{F}_{p})\to\mathbb{F}_{p}(-d).

β€œmod 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 π’ͺ X +/Ο€\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).

References

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

Last revised on November 25, 2022 at 20:46:43. See the history of this page for a list of all contributions to it.