Remark 20.1.1.
By convention, the symbol \(!\) in the following context is pronounced “shriek”.
Section 20 Six functors formalism for analytic rings
In this section, we introduce a Grothendieck topology on the opposite category of analytic rings which gives rise to a six functor formalism in the sense of Mann ([24], A.5).
Reference.
Subsection 20.1 Proper morphisms of analytic rings
We first define a key geometric property entering into the definition of a six functor formalism. Beware that the terminology is not quite aligned with the usage in algebraic geometry; see Remark 20.1.5.
Definition 20.1.2.
Let \(\calC, \calC'\) be symmetric monoidal categories. Let \(F \colon \calC \to \calC'\) be a functor. Let \(G \colon \calC' \to \calC\) be a functor which comes with a specified natural transformation (but not an isomorphism)
\begin{equation*}
G(M_1 \otimes_{\calC'} M_2) \cong G(M_1) \otimes_{\calC} G(M_2)
\qquad (M_1, M_2 \in \calC')\text{.}
\end{equation*}
Fix also a natural transformation \(G \circ F \to \id_{\calC}\text{,}\) e.g., by starting with an adjoint pair (in order).
In this setting, the projection formula holds if the composition of natural transformations
\begin{equation*}
G(F(M) \otimes_{\calC'} M') \to G(F(M)) \otimes_{\calC} G(M') \to
M \otimes_{\calC} G(M') \qquad (M \in \calC, M' \in \calC')
\end{equation*}
yields a natural isomorphism
\begin{equation}
G(F(M) \otimes_{\calC'} M') \cong M \otimes_{\calC} G(M') \qquad (M \in \calC, M' \in \calC')\text{.}\tag{20.1}
\end{equation}
For stable \(\infty\)-categories, we make the same definition except (as usual) the property of (20.1) being an isomorphism is replaced by the specification of an inverse (and suitable higher homotopies).
Example 20.1.3.
Let \(f\colon A \to B\) be a morphism in \(\Ring\text{.}\) Then the adjoint pair \(f^*\colon \Mod_A \to \Mod_B\) (extension of scalars) and \(f_*\colon \Mod_B \to \Mod_A\) (restriction of scalars) satisfies the projection formula: this expands to the statement that for \(M \in \Mod_A\text{,}\) \(N \in \Mod_B\text{,}\) we have a natural isomorphism
\begin{equation*}
(M \otimes_A B) \otimes_B N \cong M \otimes_A N
\end{equation*}
induced by the composition
\begin{align*}
(M \otimes_A B) \otimes_B N &\to (M \otimes_A B) \otimes_A N\\
&\to (M \otimes_A B) \otimes_A (B \otimes_B N)\\
&\to M \otimes_A (B \otimes_A B) \otimes_B N\\
&\to M \otimes_A B \otimes_B N\\
&\to M \otimes_A (B \otimes_B N) = M \otimes_A N \text{.}
\end{align*}
By the same token, the adjoint pair \(f^*\colon \calD(\Mod_A) \to \calD(\Mod_B)\) and \(f_*\colon \calD(\Mod_B) \to \calD(\Mod_A)\) also satisfies the projection formula,
We also have a miniature analogue of the proper base change property in etale cohomology To formulate this, let \(g \colon A \to A'\) be another morphism of rings; set \(B' := A' \otimes_A B\text{;}\) write \(g\) also for the resulting morphism \(B \to B'\text{;}\) and write \(f'\) for the resulting morphism \(A' \to B'\text{.}\) We can then interpret the comparison in \(\Mod_{A'}\)
\begin{equation*}
M \otimes_B B' \cong M \otimes_B (B \otimes_A A') \cong M \otimes_A A'
\end{equation*}
as giving rise to a natural isomorphism
\begin{equation*}
f'_* g* \cong g^* f_*\text{.}
\end{equation*}
This again holds at the derived level also.
Definition 20.1.4.
A morphism \(f\colon \AnSpec B \to \AnSpec A\) in \(\AnRing^{\op}\) is proper if the analytic ring structure on \(B\) is the one induced from \(A\text{:}\)
\begin{equation*}
\Mod_B = \Mod_{B^{\triangleright}} \times_{\Mod_{A^{\triangleright}}} \Mod_A\text{.}
\end{equation*}
By Example 20.1.3, this is equivalent to requiring that the adjoint pair \(f^*\colon \calD(\Mod_A) \to \calD(\Mod_B)\) and \(f_*\colon \calD(\Mod_B) \to \calD(\Mod_A)\) satisfies the projection formula.
It is evident that the collection of proper morphisms is stable under composition and base extension. Moreover, if we set
\begin{equation*}
B' := (B^{\triangleright}, \Mod_{B^{\triangleright}} \times_{\Mod_{A^{\triangleright}}} \Mod_A)\text{,}
\end{equation*}
then we get the unique factorization \(\AnSpec B \to \AnSpec B' \to \AnSpec A\) such that \(\AnSpec B \to \AnSpec B'\) is an isomorphism of underlying condensed rings and \(\AnSpec B' \to \AnSpec A\) is proper; evidently, \(f\) is proper if and only if \(B' \cong B\text{.}\)
For \(f\) proper, we write \(f_!\) as a synonym for \(f_*\text{.}\) In this notation, we again have a form of proper base change: for \(g\colon \AnSpec A' \to \AnSpec A\) an arbitrary morphism in \(\AnRing^{\op}\text{,}\) if we set \(B' := A' \otimes_A B\text{,}\) write \(g\) also for the resulting morphism \(\AnSpec B' \to \AnSpec B\text{,}\) and write \(f'\) for the resulting morphism \(\AnSpec B' \to \AnSpec A'\text{,}\) we then have a natural isomorphism
\begin{equation*}
g^* f_! \cong f'_! g^*
\end{equation*}
of functors \(\calD(\Mod_B) \to \calD(\Mod_{A'})\text{.}\)
Remark 20.1.5.
Definition 20.1.4 agrees with Huber’s notion of properness when \(A\) and \(B\) are Tate, but it is of a somewhat different nature than the definition of properness in algebraic geometry. For example, every idempotent morphism is by definition proper; this includes loose rational localizations of solid analytic rings, loose dagger localizations of analytic rings, and dagger localizations of Tate analytic rings (by Remark 17.1.8).
Subsection 20.2 Open immersions
We first define a second key geometric property. Again, there is something of a disconnect with the usage in algebraic geometry; see Remark 20.2.2.
Definition 20.2.1.
A morphism \(f\colon \AnSpec B \to \AnSpec A\) in \(\AnRing^{\op}\) is an open immersion if the functor \(f^* \colon \calD(\Mod_A) \to \calD(\Mod_B)\) admits a fully faithful left adjoint \(f_!\) such that \((f_!,f^*)\) satisfies the projection formula. In particular, this implies that \(f^*\) commutes not only with colimits but also with limits.
Remark 20.2.2.
Beware that open immersion of schemes do not give rise to open immersions in the sense we have just defined, except in the special case of a closed-open immersion. In particular, if we take \(B = A[f^{-1}]\) for some \(f \in A^{\triangleright}\text{,}\) then the map \(\AnSpec B \to \AnSpec A\) is typically not an open immersion.
Example 20.2.3.
For any analytic ring \(A\) and any \(g \in A^{\triangleright}\text{,}\) we claim that the morphism
\begin{equation*}
j\colon (A^{\triangleright}, \Mod_A) \to (A^{\triangleright}_{\liquid g}, \Mod_{A \liquid g})\text{,}
\end{equation*}
is an open immersion. To check this, we must produce the left adjoint \(j_!\) for \(j^*\text{.}\) Before we do, recall that \(j_*\) is the inclusion of module categories whereas by Proposition 10.3.4, \(j^*\) has the form
\begin{equation*}
M \mapsto R\iHom_{A^{\triangleright}}(A^{\triangleright}[-1] \to \coker(\Delta_g)[0], M)\text{.}
\end{equation*}
Since we want
\begin{equation*}
\Hom_B(j_! M, N) \cong \Hom_C(M, j^* N)\text{,}
\end{equation*}
by tensor-Hom adjunction we satisfy the projection formula by taking
\begin{equation*}
j_! M = M \otimes_{A^{\triangleright}} (A^{\triangleright}[-1] \to \coker(\Delta_g)[0])\text{.}
\end{equation*}
Example 20.2.4.
For any analytic ring \(A\) and any \(g \in A^{\triangleright}\text{,}\) we claim that the morphism \(j\colon A \to A[g^{-1}]_{\liquid g^{-1}}\) is an open immersion. In this case, \(j_* j^*\) has the form
\begin{align*}
M &\mapsto R\iHom_{A^{\triangleright}[g^{-1}]}(A^{\triangleright}[g^{-1}][-1] \to \coker(1-g^{-1}[1], P \otimes A^{\triangleright}[g^{-1}])[0], M)\\
&\cong R\iHom_{A^{\triangleright}}(A^{\triangleright}[-1] \to \coker(g-[1], P \otimes A^{\triangleright})[0], M)\text{.}
\end{align*}
Example 20.2.5.
Making Example 20.2.3 more explicit in the case \(A = (\underline{\ZZ}[T], \Mod_{\underline{\ZZ}[T]\solid})\text{,}\) \(g = T\text{,}\) we can write
\begin{align*}
\coker(\Delta_f)_\solid &\cong \coker(P \otimes \underline{\ZZ}[T] \stackrel{\Delta_T}{\to} P \otimes \underline{\ZZ}[T])_\solid\\
& \cong \coker(P_\solid \otimes_\solid \underline{\ZZ}[T] \stackrel{\Delta_T}{\to} P_\solid \otimes_\solid \underline{\ZZ}[T])\\
& \cong \coker(\underline{\ZZ} \llbracket U \rrbracket \otimes_\solid \underline{\ZZ}[T] \stackrel{1-TU}{\to}
\underline{\ZZ} \llbracket U \rrbracket \otimes_\solid \underline{\ZZ}[T])\\
& \cong \underline{\ZZ}((T^{-1}))\text{.}
\end{align*}
Consequently, we can write
\begin{equation*}
j_! M = M \otimes_{\underline{\ZZ[T]}} \underline{\ZZ}((T^{-1}))/\underline{\ZZ}[T]\text{.}
\end{equation*}
Similarly, in Example 20.2.4, if we take \(A = (\underline{\ZZ}[T], \Mod_{\underline{\ZZ}[T]\solid})\text{,}\) \(g = T\text{,}\) we get
\begin{equation*}
j_! M = M \otimes_{\underline{\ZZ[T]}} \underline{\ZZ}\llbracket T \rrbracket/\underline{\ZZ}[T]\text{.}
\end{equation*}
Remark 20.2.6.
By generalizing the proof of Example 20.2.4, one can show that for any analytic ring \(A\) and any \(f_1,\dots,f_n,g \in A^{\triangleright}(*)\) generating the unit ideal, the morphism \(A \to A[\tfrac{f_1}{g},\dots,\tfrac{f_n}{g}]_{\liquid f_1/g, \dots, f_n/g}\) is an open immersion. Note that this does not follow directly from our arguments so far because \(A \to A[\tfrac{1}{g}]\) is not an open immersion (Remark 20.2.2).
When \(A\) is solid and contains a topologically nilpotent unit, this morphism can be interpreted as a rational localization using Lemma 12.2.3. Even without this condition, we can still apply Remark 13.1.1 to infer that for a rational localization \(A \to B\) with associated loose rational localization \(A \to B'\text{,}\) the map \(\AnSpec B' \to \AnSpec B\) is an open immersion.
Example 20.2.7.
For any solid analytic ring \(A\) and any \(f \in A^{\triangleright}\text{,}\) the map from \(A\) to its derived \(f\)-completion is an open immersion. This follows by similar logic as in Example 20.2.3 once we recall from Proposition 9.2.9 that the right adjoint has the form
\begin{equation*}
M \mapsto R\iHom_{A^{\triangleright}}(A^{\triangleright} \to A^{\triangleright}_f, M)\text{.}
\end{equation*}
The following proposition explains the common shape of the previous examples.
Proposition 20.2.8.
For a given analytic ring \(A\text{,}\) the open immersions \(j\colon \AnSpec B \to \AnSpec A\) correspond to idempotent commutative algebra objects \(C \in \calD(A)\) such that \(R\iHom_A(C, \bullet)[1]\) preserves colimits and preserves \(\calD_{\geq 0}(A)\text{.}\) The assignment is \(C := \cone(j_! B \to A)\text{.}\)
Corollary 20.2.9.
The property of a morphism \(j\) in \(\AnRing^{\op}\) being an open immersion is stable under base change. Moreover, \(j_!\) is compatible with base change in the same sense as for proper morphisms (Definition 20.1.4).
Example 20.2.10.
Define the analytic rings
\begin{align*}
A & := (\underline{\ZZ}, \Mod_{\underline{\ZZ} \liquid 1} = \Mod_{\underline{\ZZ} \solid})\\
B & := (\underline{\ZZ}[T], \Mod_{\underline{\ZZ}[T] \liquid 1})\\
C & := (\underline{\ZZ}[T], \Mod_{\underline{\ZZ}[T] \liquid \{1,T\}})\\
D & := (\underline{\ZZ}\langle T \rangle, \Mod_{\underline{\ZZ}\langle T \rangle \liquid \{1,T\}})
\end{align*}
so that we have a sequence of maps
\begin{equation*}
A \to B \to C \to D\text{.}
\end{equation*}
We analyze the various compositions as follows.
-
The map \(\AnSpec B \to \AnSpec A\) is proper (by inspection) but not an open immersion (pullback does not commute with infinite products).
-
The map \(\AnSpec C \to \AnSpec B\) is an open immersion (by Example 20.2.3) but not proper (by inspection).
-
The map \(\AnSpec D \to \AnSpec C\) is both proper and an open immersion (as a rational localization).
-
The map \(\AnSpec C \to \AnSpec A\) is neither an open immersion (pullback does not commute with infinite products) nor proper (because \(B \not\cong C\)).
-
The map \(\AnSpec D \to \AnSpec B\) is both proper and an open immersion (as a rational localization).
-
The map \(\AnSpec D \to \AnSpec A\) is proper (by factoring through \(\AnSpec B\)) but not an open immersion (pullback does not commute with products).
Subsection 20.3 \(!\)-able morphisms
We now put the two properties together to get the class of \(!\)-able morphisms.
Definition 20.3.1.
A morphism \(f\colon \AnSpec B \to \AnSpec A\) in \(\AnRing^{\op}\) is \(!\)-able (pronounced “shriekable”) if the factorization
\begin{equation*}
\AnSpec B \to \AnSpec B' \to \AnSpec A
\end{equation*}
of Definition 20.1.4 has the property that \(\AnSpec B \to \AnSpec B'\) is an open immersion (recall that \(\AnSpec B' \to \AnSpec A\) is automatically proper). Note the analogy with the definition of a quasiprojective morphism of schemes.
From this definition, it is not immediately obvious that a composition of \(!\)-able morphisms is again \(!\)-able; specifically, if \(f\colon \AnSpec B \to \AnSpec A\) is an open immersion and \(g\colon \AnSpec C \to \AnSpec B\) is proper, if we factor the composition \(f \circ g\colon \AnSpec C \to \AnSpec A\) through \(\AnSpec C'\) as above, we must check that the resulting map \(\AnSpec C \to \AnSpec C'\) is an open immersion. For this, note that on one hand \(\AnSpec B \otimes_{B'} C' \to \AnSpec C'\) is the base extension of an open immersion and hence is itself an open immersion; on the other hand, \(B \otimes_{B'} C' = C\) because \(g\) is proper.
Since we know how to define \(f_!\) when \(f\) is either proper or an open immersion, we can compose to define \(f_!\) for any \(!\)-able morphism. One must again do a bit of work to show that this is compatible with composition of arbitrary \(!\)-able morphisms.
Remark 20.3.2.
By Remark 20.2.6, a rational localization of a solid analytic ring is \(!\)-able: it factors as an open immersion followed by a proper map (the associated loose rational localization). Similarly, a dagger localization is \(!\)-able.
Remark 20.3.3.
The definition of a \(!\)-able morphism corresponds roughly to Huber’s notion of a “\(+\)-weakly finite type morphism”.
Remark 20.3.4.
For \(i=1,\dots,n\text{,}\) let \(f_i \colon \AnSpec B_i \to \AnSpec A\) be a morphism of analytic rings. Let \(f \colon \AnSpec \bigoplus_i B_i \to \AnSpec A\) be the product. Then \(f\) is \(!\)-able if and only if \(f_1,\dots,f_n\) are all \(!\)-able.
Subsection 20.4 A six functors formalisms
The “six functors formalism” for analytic rings will be expressed in terms of the following six functors. For any object \(A \in \AnRing^{\op}\text{,}\) we have the two functors
\begin{align*}
R\Hom_A(\bullet, \bullet) &\colon \calD(\Mod_A)^{\op} \times \calD(\Mod_A) \to \calD(\Mod_A)\\
\bullet \otimes_A^L \bullet &\colon \calD(\Mod_A) \times \calD(\Mod_A) \to \calD(\Mod_A)
\end{align*}
which form an adjoint pair in that order. Moreover, \(\otimes_A^L\) defines a symmetric monoidal structure on \(\calD(\Mod_A)\text{.}\)
For a morphism \(f\colon \AnSpec B \to \AnSpec A\) in \(\AnRing^{\op}\text{,}\) we have two more functors:
\begin{gather*}
f^* \colon \calD(\Mod_A) \to \calD(\Mod_B)\\
f_* \colon \calD(\Mod_B) \to \calD(\Mod_A)
\end{gather*}
which form an adjoint pair in that order, and whose formation is compatible with composition. Moreover, \(f^*\) is symmetric monoidal with respect to tensor product.
Finally, for a \(!\)-able morphism \(f\colon \AnSpec B \to \AnSpec A\) in \(\AnRing^{\op}\text{,}\) we have two more functors:
\begin{gather*}
f_! \colon \calD(\Mod_B) \to \calD(\Mod_A)\\
f^! \colon \calD(\Mod_A) \to \calD(\Mod_B)
\end{gather*}
which form an adjoint pair in that order, and whose formation is compatible with composition. Moreover, \(f_!\) satisfies base change and the projection formula. (Recall that \(f_! = f_*\) when \(f\) is proper and \(f^! = f^*\) when \(f\) is an open immersion.)
Subsection 20.5 \(!\)-descent
Definition 20.5.1.
Let \(f\colon Y = \AnSpec B \to X = \AnSpec A\) be a morphism in \(\AnRing^{\op}\text{.}\) We say that \(f\) satisfies \(*\)-descent if \(f^*\) defines an isomorphism of \(\calD(\Mod_A)\) with the limit of
\begin{equation*}
\calD(\Mod_B) \rightrightarrows \calD(\Mod_{B \otimes_A B}) \cdots
\end{equation*}
where the maps are all of the form \(\bullet^*\text{.}\)
Suppose now that \(f\) is \(!\)-able. We similarly say that \(f\) satisfies \(!\)-descent if \(f^!\) defines an isomorphism of \(\calD(\Mod_A)\) with the limit of
\begin{equation*}
\calD(\Mod_B) \rightrightarrows \calD(\Mod_{B \otimes_A B}) \cdots
\end{equation*}
where the maps are all of the form \(\bullet^!\text{.}\)
We say that \(f\) satisfies universal \(*\)-descent (resp. universal \(!\)-descent) if any base change of \(f\) satisfies \(*\)-descent (resp. \(!\)-descent). It can be shown that any \(!\)-able morphism that satisfies \(!\)-descent also satisfies universal \(!\)-descent and (universal) \(*\)-descent.
We will equip the category \(\AnRing^{\op}\) with the Grothendieck topology defined by finite disjoint unions and \(!\)-able maps that satisfy \(!\)-descent. For example, a dagger localization of a Tate analytic ring \(f\) is both proper (Remark 20.1.5) and an open immersion (Remark 20.2.6), so \(f^! = f^*\) and we can deduce \(!\)-descent directly from \(*\)-descent (Theorem 21.4.5).
