Skip to main content

Notes on condensed mathematics

Kiran S. Kedlaya

Section 15 More on quasicoherent sheaves

In this section, we apply localization for quasicoherent sheaves for solid analytic rings to deduce some related results for Huber rings, including some whose statements make no reference to either solid rings or derived categories. We then set up a framework which will allow us to push beyond the solid case.

Reference.

This section is (rather loosely) based on [8], Lectures 9 and 10. See also [19], Lecture 1 (for the version of the story using Huber rings) and [1] (for the solid analogue); the latter draws some key ideas from [7].

Subsection 15.1 Subcategories of the derived \(\infty\)-category of quasicoherent sheaves

For \(A\) a sheafy solid analytic ring, we proved in Theorem 14.4.1 that the derived \(\infty\)-category \(\calD(\Mod_A)\) is a sheaf for the adic topology on \(\AnRing^{\op}_{\solid/A}\text{.}\) We can formally transfer this conclusion to several subcategories thereof.

Remark 15.1.1.

Theorem 14.4.1 is compatible with various categorical properties. For instance, the bounded-above and bounded-below derived categories are also subsheaves for the adic topology. However, we cannot argue that complexes in a fixed range \([a, \infty)\) or \((-\infty, b]\) also form a subsheaf for the adic topology: the argument in Remark 14.1.2 includes a step that disturbs the left end of the range (taking the derived tensor product with \(M\)) and a step that disturbs the right end of the range (taking the mapping cone).

Definition 15.1.2.

For \(\calA\) an abelian category, an object \(M\) in \(\calD(\calA)\) is compact if the functor \(R\iHom(M, \bullet)\) commutes with filtered colimits. The object \(M\) is pseudocoherent if for any \(a\text{,}\) the truncation to \([a, \infty)\) is compact. In the setting of Theorem 14.4.1, it is formal that both of these properties define subsheaves of \(\calD(\calA)\text{.}\)

Definition 15.1.3.

For \(A\) an analytic ring, for \(M \in \calD(\Mod_A)\text{,}\) write \(M^\vee\) for the dual object \(R\iHom_A(M, A)\text{.}\) We say \(M\) is reflexive if the canonical map \(M \to (M^\vee)^\vee\) is an isomorphism.
An object \(M\) of \(\calD(\Mod_A)\) is dualizable if there exist an object \(N\) of \(\calD(\Mod_A)\) and maps \(i\colon A \to N\otimes M\text{,}\) \(e\colon M \otimes N \to A\) for which the compositions
\begin{gather*} M \stackrel{1 \otimes i}{\to} M \otimes^L_A (N \otimes^L_A M) \cong (M \otimes^L_A N) \otimes^L_A M \stackrel{e \otimes 1}{\to} M\\ N \stackrel{i \otimes 1}{\to} (N \otimes^L_A M) \otimes^L_A N \cong N \otimes (M \otimes^L_A N) \stackrel{1 \otimes e}{\to} N \end{gather*}
are the identity maps. When this occurs, it follows as in [28], tag 0FFQ that for any \(P,Q \in \calD(\Mod_A)\text{,}\) there is a functorial isomorphism
\begin{equation} R\iHom_A(P, N \otimes^L_A Q) \cong R\iHom_A(M \otimes^L_A P, Q)\text{:}\tag{15.1} \end{equation}
the passage from left to right is given by tensoring with \(M\) and postcomposing with \(e\text{,}\) while the passage from right to left is tensoring with \(N\) and precomposing with \(i\text{.}\)
In particular, when \(M\) is dualizable, \(N\) is also dualizable and \(N \cong M^\vee\) and \(M \cong N^\vee\text{;.}\) Moreover, \(e\) is the evaluation map and \(i\) is the map corresponding via (15.1) to the map \(A \to \iHom_A(M, M)\) carrying \(1 \in A(*)\) to the identity map in \(\iHom_A(M,M)(*) = \Hom_A(M,M)\text{.}\) With this, in the setting of Theorem 14.4.1, it is formal that the property of being dualizable defines a subsheaf of \(\calD(\calA)\text{.}\)
Any dualizable object is reflexive, but not vice versa; see Remark 15.1.4 and Example 15.1.5.

Remark 15.1.4.

For \(\calA = \Mod_R\) with \(R\) a (discrete) commutative ring, an object of \(\calD(\Mod_R)\) is pseudocoherent if and only if it is represented by a bounded-above complex of finite projective modules (i.e., a pseudocoherent complex), while an object is compact if and only if it is represented by a bounded complex of finite projective modules (i.e., a perfect complex) if and only if it is dualizable.
In particular, an object of \(\Mod_R\) is dualizable if and only if it is finite projective. A simple example of a reflexive module which is not dualizable is the ideal \((x,y,z) \subset \QQ[x,y,z]\text{.}\)

Example 15.1.5.

Let \(A\) be the discrete solid commutative ring corresponding to \(\ZZ\text{.}\) For \(M := \bigoplus_\NN \underline{\ZZ}\) and \(N := \prod_\NN \underline{\ZZ}\) (placed in degree \(0\)), \(M\) and \(N\) are dual to each other and even satisfy (15.1), but neither is dualizable: we have
\begin{equation*} \Hom_{A}(A, M \otimes_{A} N) = \bigoplus_\NN \prod_\NN \ZZ \neq \prod_\NN \bigoplus_\NN \ZZ = \Hom_{A}(M, M) \end{equation*}
and crucially the identity matrix is missing from \(\Hom_{A}(A, M \otimes_A N)\) because we only see matrices with finitely many nonzero columns.

Subsection 15.2 Glueing for adic spaces

To help concretize the previous discussion, we deduce some consequences for glueing for solid analytic rings and sheafy Huber pairs. This depends on introducing one more categorical property of objects of \(\calD(\Mod_A)\text{,}\) this one serving as a mechanism to import a key idea from functional analysis (see Lemma 15.2.3).

Definition 15.2.1.

For \(A\) an analytic ring, an object \(M \in \calD(\Mod_A)\) is nuclear if for all \(S \in \Prof\text{,}\) the natural map
\begin{equation*} (A[\underline{S}]^\vee \otimes^L_A M)(*) \to \Hom_A(A[\underline{S}], M) = M(S) \end{equation*}
is an isomorphism in \(\calD(\Ab)\text{.}\) The class of nuclear objects is stable under arbitrary colimits. It is straightforward to check that every dualizable object is nuclear.
Although we do not use this directly, in the setting of Theorem 14.4.1, it is formal that the property of being nuclear defines a subsheaf of \(\calD(\Mod_A)\text{.}\)

Definition 15.2.2.

For \(A\) an analytic ring, a morphism \(f \colon P \to Q\) in \(\calD(\Mod_A)\) is trace class if there is some map \(g \colon A \to P^\vee \otimes^L_A Q\) such that the composition
\begin{equation*} P \stackrel{1 \otimes g}{\to} P \otimes^L_A P^\vee \otimes^L_A Q \stackrel{e \otimes 1}{\to} Q \end{equation*}
is equal to \(f\text{.}\)
The following argument is in some sense the point; it is a form of the Schwartz lemma ([21], Satz 1.2).

Proof.

Let us suppose first that \(A\) represents a Huber pair. By Lemma 7.3.7, we have \(A[\underline{S}] \cong \prod_\NN A\text{.}\) As in Example 15.1.5, we can represent the given element of \(\Hom_{A}(A, A[\underline{S}]^\vee \otimes^L_{A} A[\underline{S}])\) as an \(\NN \times \NN\) matrix over \(A^{\triangleright}(*)\) which is a convergent (for the operator topology) sum of matrices with finitely many nonzero columns. In particular, it can be written as a sum of two matrices \(N_0 + N_1\) where \(N_0\) has finitely many nonzero columns and \(N_1\) has entries in \(A^{\circ \circ}\text{.}\) In particular, \(1 - N_1\) is invertible and so the cone of \(f-1\) does not change if we replace \(N_0+N_1\) with
\begin{equation*} 1 - (1+N_1+N_1^2+\cdots)(1-N_0-N_1) = (1-N_1)^{-1}N_0\text{,} \end{equation*}
which is now itself a matrix with finitely many nonzero columns. We may thus replace the index set \(\NN\) with the subset corresponding to the nonzero columns without changing \(\cone(f-1)\text{.}\)
To handle the general case, note that the problem data can be described in terms of countably many elements of \(A^{\triangleright}(*)\text{;}\) consequently, any given problem instance can be obtained by base extension from a solid analytic ring \(A\) of the form
\begin{equation*} A^{\triangleright}\left( \prod_I \ZZ_\solid\right) = \prod_I R[T_j \colon j \in J] \end{equation*}
for \(R\) a discrete polynomial ring over \(\ZZ\) in countably many variables and \(J\) a countable index set. Using Example 9.2.11, we can interpret the given element of \(\Hom_{A}(A, A[\underline{S}]^\vee \otimes^L_{A} A[\underline{S}])\) as an \(\NN \times \NN\) matrix over \(R[T_j \colon j \in J]\) with the property that every monomial \(\prod_{j \in J} T_j^{e_j}\) (where \(e_j\) are nonnegative integers, all but finitely many equal to 0) appears in only finitely many columns. We now argue as before: separate the matrix as \(N_0 + N_1\) where \(N_0\) is again the matrix of constant terms (i.e., set each \(T_j\) to 0), and then \(1-N_1\) will be invertible. (Compare [1], Lemma 5.51.)

Proof.

We may start with a complex \(M\) of the form
\begin{equation*} \cdots \to A[\underline{S_1}] \to A[\underline{S_0}] \to 0 \end{equation*}
with all \(S_i \in \Prof\text{.}\) Since \(M\) is nuclear, the canonical map \(f'\colon A[\underline{S_0}] \to M\) can be factored as
\begin{equation*} A[\underline{S_0}] \stackrel{1 \otimes g}{\to} A[\underline{S_0}] \otimes^L_A A[\underline{S_0}]^\vee \otimes^L_{A} A[\underline{S_0}] \stackrel{e \otimes 1}{\to} A[\underline{S_0}] \end{equation*}
for some \(g'\colon A \to A[\underline{S_0}]^\vee \otimes^L_{A} M\text{.}\) Since \(A\) is projective in \(\Mod_{A}\text{,}\) we can further factor \(g'\) as
\begin{equation*} A \stackrel{g}{\to} A[\underline{S_0}]^\vee \otimes^L_{A} A[\underline{S_0}] \stackrel{1 \otimes f'}{\to} A[\underline{S_0}]^\vee \otimes^L_{A} M\text{;} \end{equation*}
that is, we have exhibited a trace class map \(f\colon A[\underline{S_0}] \to A[\underline{S_0}]\) such that \(f' \circ f = f'\text{,}\) and so \(f' \circ (1-f) = 0\text{.}\) We can then factor \(f'\) through a map \(\phi\colon \cone(1-f) \to M\text{.}\) By Lemma 15.2.3, \(\cone(1-f)\) can be represented by a perfect complex over \(A^{\triangleright}\text{;}\) we may then replace \(M\) with \(\cone(\phi)\) and induct to deduce the claim. (Compare [1], Theorem 5.50.)

Proof.

We may reduce to checking glueing for a standard binary covering. In light of Theorem 14.4.1, we may further assume that we start with a complex in \(\calD(\Mod_{A})\) whose base change to each \(\calD(\Mod_{A_i})\) is represented by a discrete pseudocoherent complex of \(A^{\triangleright}\)-modules. In particular, the object in \(\calD(\Mod_{A_i})\) is dualizable (by Remark 15.1.4) and pseudocoherent. The original complex in \(\calD(\Mod_{A})\) is then dualizable (hence nuclear) and pseudocoherent, hence discrete by Theorem 15.2.4; that is, we have succeeded in glueing together a discrete pseudocoherent complex of \(A^{\triangleright}\)-modules.
So far we have not yet said anything about amplitudes; we address that next.

Proof.

Since \(\coker(h)\) is finitely generated, \(\coker(h) \otimes_A A \left\langle \frac{f}{g} \right\rangle \) automatically belongs to \(\Mod_{A \langle f/g \rangle}\text{;}\) that is, the result is already in \(\Mod_{A^{\triangleright}[f/g]_{\liquid f/g}}\) without any analytic completion required. Consequently, we may directly apply Lemma 21.4.4.

Proof.

Proof.

This follows by applying Theorem 15.2.7 twice, once to a suitable shift of the original complex (to force the amplitude into \((-\infty, b])\)) and once to a suitable shift of the dual (to force the amplitude into \([a, +\infty)\)).

Proof.

Remark 15.2.10.

Corollary 15.2.9 was known previously when \(A\) is a locally Tate Huber ring; see [19], Theorem 1.4.2. It implies in particular that the category of vector bundles on \(\Spa A\) does not depend on \(A^+\text{,}\) whereas the full category of quasicoherent sheaves definitely does (Proposition 11.3.13). In particular, the fact in that quasicoherent sheaves on affine schemes are colimits of vector bundles has no analogue in the analytic setting.

Subsection 15.3 Idempotent morphisms

Let us isolate the key point of the previous construction. This will give us a notion of an “open immersion” which applies to analytic rings which are not necessarily solid. (We persist in retaining sheafiness conditions because we do not expand upon Remark 10.1.8.)

Definition 15.3.1.

A morphism \(A \to B\) of analytic rings is idempotent if the map \(B^{\triangleright} \otimes^L_A B^{\triangleright} \to B^{\triangleright}\) is an isomorphism. An equivalent condition is that the composition
\begin{equation} B^{\triangleright} \otimes^L_A B^{\triangleright} \to B^{\triangleright} \to B^{\triangleright} \otimes^L_A B^{\triangleright}\text{,}\tag{15.2} \end{equation}
where the second map is \(b \mapsto b \otimes 1\text{,}\) is an isomorphism.

Example 15.3.2.

In the category of rings, any morphism \(A \to B\) which is a localization at an arbitrary multiplicative subset is idempotent. Consequently, such a morphism corresponds to an idempotent morphism of discrete analytic rings.

Remark 15.3.3.

If \(A \to B \to C\) are two idempotent morphisms, then so is \(A \to C\) because
\begin{equation*} C^{\triangleright} \otimes^L_A C^{\triangleright} = C^{\triangleright} \otimes^L_{B} (B^{\triangleright} \otimes^L_A B^{\triangleright}) \otimes^L_B C^{\triangleright} \end{equation*}
using both \(B^{\triangleright}\)-module structures on \(B^{\triangleright} \otimes^L_A B^{\triangleright}\text{.}\) Conversely, if \(A \to B\) and \(A \to B \to C\) are both idempotent, then so is \(B \to C\) because
\begin{equation*} C^{\triangleright} \otimes^L_B C^{\triangleright} = (C^{\triangleright} \otimes^L_A C^{\triangleright}) \otimes_{B^{\triangleright} \otimes^L_A B^{\triangleright}} B^{\triangleright}\text{.} \end{equation*}

Remark 15.3.4.

If \(A \to B\) is idempotent, then for any morphism \(A \to C\text{,}\) \(C \to B \otimes_A C\) is idempotent provided that \(B \otimes_A C = B \otimes^L_A \otimes C\text{.}\)

Example 15.3.5.

By base extension from the case \(\ZZ[f] \to \ZZ[f^{\pm}]\) of Example 15.3.2, for analytic ring \(A\) and any \(f \in A^{\triangleright}(*)\text{,}\) the morphism \(A \to A_f := A[T]/(1-fT)\) is idempotent.

Example 15.3.6.

Let \(A\) be a sheafy solid analytic ring. Let \(A \to B\) be a rational localization defined by some parameters \(f_1,\dots,f_n,g\) that generate the unit ideal. Then \(A \to B\) is idempotent.
Consequently, any composition of rational localizations is idempotent, whether or not it is again a rational localization (Remark 12.2.8). Similarly, any tensor product of rational localizations is idempotent (provided that it is underived).

Remark 15.3.7.

In Example 15.3.6, if we drop the condition that \(A\) is sheafy and put ourselves in a context where we can consider a derived rational localization \(A \to B\) (Remark 13.1.7), then \(A \to B\) is still idempotent; that is, (15.2) is still an isomorphism.
Let us work this out in the special case where \(B = A \langle f \rangle\) for some \(f \in A^{\triangleright}\text{,}\) the general case being similar. In this case \(B^{\triangleright}\) is represented by the Koszul complex associated to the element \(T-f\) of \(A^{\triangleright} \langle T \rangle\text{,}\) and so \(B^{\triangleright} \otimes^L_A B^{\triangleright}\) is represented by the Koszul complex associated to the elements \(T-f,U-f\) of \(A^{\triangleright} \langle T,U \rangle\text{.}\) The Koszul complex does not change if we replace the sequence with \(T-U,T-f\text{;}\) it thus suffices to observe that the sequence
\begin{equation*} 0 \to A^{\triangleright}\langle T,U \rangle \stackrel{\times T-U} A^{\triangleright}\langle T,U \rangle \to A^{\triangleright}\langle T \rangle \end{equation*}
is split exact.

Example 15.3.8.

Let \(A\) be a solid analytic ring, choose \(f \in A(*)\text{,}\) and let \(B\) denote the derived \(f\)-completion of \(A\text{;}\) that is, for
\begin{equation*} A_f := \coker(1-fT, A^{\triangleright}[T])\text{,} \end{equation*}
we have \(B^{\triangleright} = \iExt^1_A(A_f/A, B)\) and \(\Mod_B := \Mod_{B^{\triangleright}} \times_{\Mod_{A^{\triangleright}}} \Mod_A\text{.}\) If \(B^{\triangleright}\) is also the derived derived \(f\)-completion of \(A^{\triangleright}\) (that is, \(\iHom_A(A_f/A, B) = 0\)), then \(A \to B\) is idempotent: we may deduce this from the universal case \(A = \underline{\ZZ[T]}\text{,}\) \(f = T\) which follows from Lemma 9.3.1.
Note however that the objects of \(B\) are not themselves derived \(f\)-complete; that is, so far we did not restrict \(\Mod_B\) to consist only of those objects in \(\Mod_{B^{\triangleright}} \times_{\Mod_{A^{\triangleright}}} \Mod_A\) such that \(R\iHom_{B^{\triangleright}}(A_f, M) = 0\text{.}\) If we were to impose that additional restriction, we would obtain a distinct morphism of analytic rings, which would again be idempotent.

Example 15.3.9.

For \(P = \ZZ[\underline{\NN_\infty}]/\ZZ[\underline{\infty}]\) the sequence space, equip \(\ZZ[q]\) with the discrete topology and form the morphism \(\underline{\ZZ[q]} \to P\) of condensed rings taking \(q\) to \([1]\text{.}\) Then the corresponding morphism of discrete analytic rings is idempotent. (Remember, taking \(P\) to be a “discrete” analytic ring makes sense even though \(P\) is not a discrete ring; it just means take the module category to be all of \(\Mod_P\text{.}\))
To see that \(P \otimes_{\underline{\ZZ[q]}}^L P \cong P\text{,}\) first apply (5.1) and Remark 5.2.4 to write
\begin{equation*} \ZZ[\underline{\NN_\infty}] \otimes^L_{\underline{\ZZ}} \ZZ[\underline{\NN_\infty}] \cong \ZZ[\underline{\NN_\infty \times \NN_\infty}]\text{.} \end{equation*}
Then set \(Z := (\NN_\infty \times \NN_\infty) \setminus (\NN \times \NN)\text{;}\) using Proposition 5.4.2, we obtain
\begin{equation*} (\ZZ[\underline{\NN_\infty}] \otimes^L_{\underline{\ZZ}} \ZZ[\underline{\NN_\infty}])/\ZZ[\underline{Z}] \cong \ZZ[\underline{(\NN \times \NN)_\infty}]/\ZZ[\underline{\infty}] \end{equation*}
where \((\NN \times \NN)_\infty\) is the one-point compactification of \(\NN \times \NN\text{.}\) Finally, note that
\begin{equation*} (\ZZ[\underline{(\NN \times \NN)_\infty}]/\ZZ[\underline{\infty}])/(q \otimes 1 - 1 \otimes q) \cong P\text{.} \end{equation*}

Definition 15.3.10.

We define the idempotent topology as the Grothendieck topology on \(\AnRing^{\op}\) generated by finite families of idempotent morphisms \(\{\AnSpec A_i \to \AnSpec A\}_{i \in I}\) where the complex
\begin{equation} 0 \to A^{\triangleright} \to \bigoplus_i A_i^{\triangleright} \to \bigoplus_{i \lt j} A_i^{\triangleright} \otimes^L_A A_j^{\triangleright} \to \cdots\tag{15.3} \end{equation}
is acyclic in \(\calD(\Mod_A)\text{.}\)

Remark 15.3.11.

We have deliberately been a bit careless in stating Definition 15.3.10: the tensor products \(A_i^{\triangleright} \otimes^L_A A_j^{\triangleright}\) and so on need not be underived. A correct formulation would require disambiguating what object of \(\calD(\Mod_A)\) is meant by (15.3).

Proof.

Example 15.3.13.

For \(A\) a solid analytic ring, the family
\begin{equation*} \AnSpec A \llbracket T \rrbracket \to \AnSpec A[T], \qquad \AnSpec A [T^{\pm}] \to \AnSpec A[T] \end{equation*}
is a covering for the idempotent topology. In this case, Theorem 15.3.12 can be used to recover the Beauville–Laszlo theorem for glueing vector bundles, as well as analogous statements for perfect and pseudocoherent complexes.
PrevTopNext