Quasicoherent sheaves on adic spaces (or, How I learned to stop worrying and love the definition of a stable \infty-category)

Section 14 Quasicoherent sheaves on adic spaces (or, How I learned to stop worrying and love the definition of a stable \(\infty\)-category)

In this section, we establish localization for derived categories of quasicoherent sheaves for solid analytic rings.

Reference.

This section is loosely based on [8], Lectures 9 and 10.

Subsection 14.1 Glueing modules over analytic rings

Remark 14.1.1.

There are several different reasons why it is not reasonable to hope for a glueing statement about modules over analytic rings without any use of derived categories. We have seen two such reasons already: rational localizations are not flat Remark 13.1.7 and solid analytic rings are not necessarily sheafy (Example 13.1.6). To this we may add a third: the definition of a rational localization \(A \to B\) involves a change in the module category (because we insist on modules being solid with respect to the new elements \(f_i/g\)) and this involves another operation which is not exact.

We can fix these problems by considering derived categories of modules over solid analytic rings, except for the failure of sheafiness. For this, we need to allow ourselves to form the derived rational localization with respect to a sequence \(f_1,\dots,f_n,g\) of parameters; this amounts to taking the Koszul complex associated to the ring \(A^{\triangleright} \langle T_1,\dots,T_n \rangle\) and the elements \(f_1 - g T_1, \dots, f_n - g T_n\text{.}\) To handle this correctly requires deriving not just modules but rings, which we have already agreed not to do (Remark 10.1.8). We will thus resort to assuming that all of the solid analytic rings in the following discussion are sheafy, but keep in mind that this restriction is not really essential.

Remark 14.1.2.

Let \(A\) be a sheafy solid analytic ring and choose \(f,g \in A^{\triangleright}(*)\) generating the unit ideal. We give an argument that attempts to prove that the base extension functor

\begin{equation} D(\Mod_A) \to D(\Mod_{A \left\langle f/g \right\rangle}) \times_{D(\Mod_{A \left\langle f/g, g/f \right\rangle})} D(\Mod_{A \left\langle g/f \right\rangle})\tag{14.1} \end{equation}

is an equivalence of categories, but fails in a subtle way.

Each of the functors from \(D(\Mod_A)\) to one of the other categories has a right adjoint (restriction of scalars), so the same is true of the base extension. It thus remains to establish that the unit and the counit of this adjunction are equivalences. To simplify notation, let us rewrite (21.1) as

\begin{equation*} D(\Mod_A) \to D(\Mod_{A_1}) \times_{D(\Mod_{A_{12}})} D(\Mod_{A_{2}})\text{.} \end{equation*}

To check the unit, note that by Corollary 13.2.7, the sequence (13.2) is exact. For any \(M \in D(\Mod_A)\text{,}\) we may take the (derived) tensor product of this sequence with \(M\) to obtain a morphism from the mapping cone of

\begin{equation*} M \otimes_A A_1 \oplus M \otimes_A A_2 \to M \otimes_A A_{12} \end{equation*}

to \(M\) in \(D(\Mod_A)\) which inverts the adjunction.

For the counit, we first apply the universal property of localization to see that the multiplication maps

\begin{equation} A_{S_1} \otimes^L_A A_{S_2} \to A_{S_1 \cup S_2}\tag{14.2} \end{equation}

are isomorphisms for any subsets \(S_1, S_2\) of \(\{1,2\}\text{.}\) Using these isomorphisms, we see that if we start with \(M_i \in \Mod_{A_i}\) and isomorphisms \(M_i \otimes_{A_i} A_{12} \cong M_{12}\text{,}\) the counit is a map from \((M_1 \times_{M_{12}} M_2) \otimes^L_A A_i \in \Mod_{A_i}\) to \(M_i \otimes^L_A A_i\) and then to \(M_i\text{,}\) and using (14.2) we can produce inverses to both factors.

The subtle failure of this argument is the reference to “the” mapping cone. In fact any realization of a morphism in the derived category in terms of chain complexes gives rise to a mapping cone, and any two of the mapping cones constructed in this way are isomorphic in the derived category, but not canonically so (see [28], tag 014D); so we fail to produce a natural isomorphism between the unit and the identity functor.

Remark 14.1.3.

We can reinterpret the failure of Remark 14.1.2 in terms of the fact that the ordinary derived category of an abelian category does not have limits: one has to first lift a given diagram up to the level of chain complexes, and it is not always possible to make consistent choices to do that. In Remark 14.1.2, we are taking the limit of a contractible diagram and so we don’t see this issue until we remember about the requirement of naturality. However, already for a three-term covering the diagram is far from contractible:

and one cannot lift such a diagram in general from the derived category to the category of chain complexes. So in this case one cannot hope to lift even an individual object of \(D(\Mod_{A_1}) \times_{D(\Mod_{A_{12}})} D(\Mod_{A_{2}})\) without keeping track of some higher compatibilities among chain homotopies.

Subsection 14.2 Enter \(\infty\)-categories

With Remark 14.1.3 compelling us to go forward, we dip our toes into the vast sea that is [23], Chapter 1. But first we need some background from the formalism of homotopy theory.

Remark 14.2.1.

By way of motivation, we first recall the classical hierarchy of categories. A category in the usual sense can also be called a \(1\)-category: it has components at level \(0\) (the objects) and level \(1\) (the morphisms).

If one seeks to put categories themselves into a category, the result is a \(2\)-category: one has components at level \(0\) (the objects), level \(1\) (the functors), and level \(2\) (the natural transformations). In particular, an equality between two given morphisms of objects is no longer a property but a structure: one must specify the choice of a particular isomorphism between them, and this choice gets carried along when composing.

Another way to specify a \(2\)-category is as a category enriched in \(1\)-categories: the morphisms between objects are not a set but rather a \(1\)-category. We may similarly define a n-category for any \(n\) by saying that a category enriched in \(n\)-categories is an \((n+1)\)-category.

The notion of an \(\infty\)-category is meant to take a “limit over \(n\)” in this process, but one can in fact jump directly to the limit.

Definition 14.2.2.

Let \(\Delta\) be the category of finite ordered sets. That is, \(\Delta\) contains the objects \([0], [1], [2],\dots\) where \([n] = \{0,1,2,\dots,n\}\text{,}\) and a morphism \([n] \to [m]\) is a nondecreasing map of sets.

For any category \(\calC\text{,}\) a simplicial object in \(\calC\) is a contravariant functor \(\Delta \to \calC\) (i.e., a “presheaf” on \(\Delta\) valued in \(\calC\)). These form a category with morphisms being natural transformations of functors. For example, a simplicial object in sets will be called a simplicial set.

Example 14.2.3.

The nerve of a small category \(\calC\) is the simplicial set defined as follows. The image of \([n]\) is the set of all chains \(x_0 \stackrel{f_0}{\to} x_1\stackrel{f_1}{\to} \cdots \stackrel{f_{n-1}}{\to} x_n\) of composable morphisms in \(\calC\text{;}\) in particular, \([0]\) maps to the set of objects and \([1]\) to the set of morphisms. The map \([n] \to [m]\) corresponds to “reparametrizing” the chain by inserting identity maps and/or composing.

This example has the inner horn filling property, which can be described as follows. For \(n \geq 0\text{,}\) let \(\Delta^n\) be the simplicial set represented by \([n] \in \Delta\) (i.e., \([k] \mapsto \Hom_\Delta([k],[n])\) (the standard \(n\)-simplex). For \(0 \leq i \leq n\text{,}\) let \(\Lambda_i^n\) be the simplicial subset obtained from \(\Delta^n\) by omitting all morphisms with image containing \(\{0,\dots,n\} \setminus \{i\}\) (the \(i\)-th horn of \(\Delta^n\)). Then for any \(i\) with \(0 \lt i \lt n\text{,}\) any morphism of simplicial sets from \(\Lambda_i^n\) into the nerve extends uniquely to \(\Delta^n\text{.}\) For example, for \(n=2, i=1\) this is saying that any pair of composable arrows admits a unique composition. By contrast, filling the horn with \(i=0\) would say that any two morphisms with the same source have the property that each one factors through the other, which is almost never true.

Definition 14.2.4.

A small \(\infty\)-category is a simplicial set \(\calC\) with the inner horn filling property: for \(0 \lt i \lt n\text{,}\) every map of simplicial sets from the horn \(\Lambda_i^n\) into \(\calC\) can be extended, not necessarily uniquely, across the simplex \(\Delta^n\text{.}\) Using the nerve construction, we may view every small category as a small \(\infty\)-category. We use that example to set language; we refer to elements of \([0]\) as objects in the \(\infty\)-category, elements of \([1]\) as morphisms or \(1\)-morphisms, and so on.

An \(\infty\)-category is the same thing but without the set-theoretic restrictions. This requires some care with foundations which I will not exercise here.

Definition 14.2.5.

A zero object in an \(\infty\)-category \(\calC\) is an object which is both initial and final. We say \(\calC\) is pointed if it admits a zero object, which we then conventionally call \(0\text{.}\)

A triangle in a pointed \(\infty\)-category \(\calC\) is a diagram \(\Delta^1 \times \Delta^1 \to \calC\) of the form

where \(0 \in \calC\) is a zero object. We say that a triangle is a fiber sequence if the diagram is a pullback square; in this case we call the triangle a fiber of \(Y \to Z\text{.}\) Dually, we say that a triangle is a cofiber sequence if the diagram is a pushout square; in this case we call the triangle a cofiber of \(X \to Y\text{.}\)

Definition 14.2.7.

An \(\infty\)-category \(\calC\) is stable if the following conditions hold.

\(\calC\) is pointed.
Every morphism in \(\calC\) admits a fiber and a cofiber.
A triangle in \(\calC\) is a fiber sequence if and only if it is a cofiber sequence.

Remark 14.2.8.

Note that in Definition 14.2.7, the last condition is desirable for derived categories because we want the concept of a triangle to be rotationally symmetric; in other words, we want the mapping cone of a morphism to satisfy a universal property on both sides.

Subsection 14.3 Derived \(\infty\)-categories

Remark 14.3.1.

Let \(\calA\) be an abelian category. The traditional approach to constructing the derived category of \(\calA\) is to start with chain complexes valued in \(\calA\text{,}\) pass to the homotopy category (i.e., quotienting by any map homotopic to zero), then invert quasi-isomorphisms (maps that induces isomorphisms on cohomology groups). The trouble is that some operations we would want to perform on complexes are sensitive to how the “forgotten” homotopies fit together.

A better approach would be to view the chain complexes as objects in an \(\infty\)-category, where \(1\)-morphisms are morphisms of chain complexes, \(2\)-morphisms are chain homotopies between morphisms, and higher morphisms record higher compatibilies among chain homotopies. Taking the homotopy category of the result should then have the effect of inverting quasi-isomorphisms to get us back to the familiar setting.

Definition 14.3.2.

A differential graded category is a category enriched over the category of chain complexes in \(\Ab\text{.}\) For \(\calC\) a differential graded category, the differential graded nerve \(N_{\dg}(\calC)\) is an \(\infty\)-category defined as in [23], Construction 1.3.1.6 and verified to be an \(\infty\)-category in Proposition 1.3.1.10. While the general description is a bit dense, we can expand the low-order terms as in [23], Example 1.3.1.8:

An object of \(N_{\dg}(\calC)\) is an object of \(\calC\text{.}\)
A \(1\)-morphism of \(N_{\dg}(\calC)\) is a pair of objects \(X,Y \in \calC\) together with a \(0\)-cocycle in \(\Hom_{\calC}(X,Y)\text{.}\)
A \(2\)-morphism of \(N_{\dg}(\calC)\) is a triple of objects \(X,Y,Z \in \calC\text{,}\) a triple of \(1\)-morphisms \(f\colon X \to Y\text{,}\) \(g\colon Y \to Z\text{,}\) \(h \colon X \to Z\text{,}\) and a \(1\)-cochain in \(\Hom_{\calC}(X,Z)\) with differential \((g\circ f) - h\text{.}\)

When \(\calC\) is the category of chain complexes of an additive category, \(N_{\dg}(\calC)\) is stable ([23], Proposition 1.3.2.10).

Definition 14.3.3.

An abelian category \(\calA\) is Grothendieck if it is presentable and the collection of monomorphisms is closed under small filtered colimits. This condition guarantees that \(\calA\) has enough injectives.

The category of condensed abelian groups is Grothendieck; more generally, for any analytic ring \(A\text{,}\) \(\Mod_A\) is Grothendieck (using the analytifications of \(A^{\triangleright}[\underline{S}]\) for \(S \in \Prof\) as generators).

Definition 14.3.4.

Let \(\calA\) be a Grothendieck abelian category. Let \(\Ch(\calA)\) denote the category of chain complexes valued in \(\calA\text{.}\) Let \(\Ch(\calA)^\circ\) be the full subcategory of fibrant objects in \(\calA\) for the model structure described in [23], Proposition 1.3.5.3; per [23], Proposition 1.3.5.6, any fibrant object is a complex of injective objects, and any bounded above complex of injective objects is fibrant. Let \(\calD(\calA) := N_{\dg}(\Ch(\calA)^{\circ})\text{;}\) this \(\infty\)-category is stable ([23], Proposition 1.3.5.9). It can also be viewed as the underlying \(\infty\)-category of the model category \(\Ch(\calA)\) ([23], Proposition 1.3.5.15). We refer to \(\calD(\calA)\) as the derived \(\infty\)-category associated to \(\calA\text{.}\)

Subsection 14.4 Glueing for quasicoherent sheaves

We now state the descent theorem for the derived \(\infty\)-category of quasicoherent sheaves.

Theorem 14.4.1.

For any sheafy solid analytic ring \(A\text{,}\) the assignment \(A \mapsto \calD(\QCoh(A))\) is a sheaf on \(\AnRing^{\op}_{\solid/A}\) for the adic topology.

Proof.

By Proposition 12.3.8 it suffices to check glueing for a two-term covering, for which the attempted proof in Remark 14.1.2 becomes an actual proof. However, to demonstrate this more forcefully, we opt to directly check descent for a covering as in Lemma 12.3.7.

Given \(f_1,\dots,f_n \in A^{\triangleright}\) generating the unit ideal, set \(A_i := A \left\langle \tfrac{f_1}{f_i},\dots,\tfrac{f_n}{f_i} \right\rangle\) and for any subset \(S\) of \(\{1,\dots,n\}\text{,}\) let \(A_S\) be the tensor product of \(A_i\) over \(A\) for \(i\) ranging over \(S\text{.}\) By Theorem 13.2.6, we have an exact sequence

\begin{equation} 0 \to A \to \bigoplus_i A_i \to \bigoplus_{i \lt j} A_{ij} \to \cdots\text{.}\tag{14.3} \end{equation}

What we need to check is that \(\calD(\QCoh_A)\) is the limit of the diagram formed by the categories \(\calD(\QCoh_{A_S})\) for \(S\) running over all nonempty subsets of \(\{1,\dots,n\}\text{,}\) with an arrow \(\calD(\QCoh_{A_S}) \to \calD(\QCoh_{A_T})\) whenever \(S\) is a subset of \(T\text{.}\) The point is that since we are now working with derived \(\infty\)-categories, the functor from \(\calD(\QCoh_A)\) to the diagram again has a right adjoint given by taking a (finite) limit over the diagram, and we can argue as in Remark 14.1.2. Namely, we see that the unit is an isomorphism by tensoring with (14.3), and we see that the counit is an isomorphism using the equality (14.2).

Remark 14.4.2.

Since we are now working in terms of stable \(\infty\)-categories, it is straightforward (and ultimately desirable) to relax the hypothesis in Theorem 14.4.1 that \(A\) must be sheafy, by viewing the derived localizations of \(A\) as ring objects in a suitable sense (as per Remark 10.1.8). However, we will not pursue this point further here.

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