{'fraction_non_alphanumeric': 0.07421253579382756, 'fraction_numerical': 0.016942411708558702, 'mean_word_length': 3.6241724290127997, 'pattern_counts': {'":': 0, '<': 5, '<?xml version=': 0, '>': 20, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 17, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Faithfully flat descent of almost perfect complexes in rigid geometry

We prove a version of faithfully flat descent in rigid analytic geometry, for almost perfect complexes and without finiteness assumptions on the rings involved. This extends results of Drinfeld for vector bundles.


Introduction
We begin with the statement of faithfully flat descent in rigid analytic geometry. Let K be a complete nonarchimedean field, and let A → A ′ be a faithfully flat map of K-affinoid algebras. In ordinary algebra, faithfully flat descent [2,Exp. VIII] states that the category of A-modules can be described as the category of A ′ -modules with descent data, which take place over the tensor products In rigid geometry, one obtains a similar conclusion, but with the tensor products replaced with completed tensor products and only for finitely generated modules.
Given a K-affinoid algebra B, we recall that B is noetherian, and we let Coh(B) denote the category of finitely generated B-modules. One has the following result, due to Bosch-Görtz [13]; the discretely valued case was previously known to Gabber (cf. [34,Theorem 1.9]). See also [15] and [3,Sec. 5.11] for accounts. Theorem 1.1 (Bosch-Görtz-Gabber). Let A → A ′ be a faithfully flat map of K-affinoid algebras. We have an equivalence of categories . . . In other words, to specify a finitely generated A-module is equivalent to specifying a finitely generated A ′ -module with descent data over the completed tensor products There are at least two ways in which one could hope to generalize Theorem 1.1. The first is to work with the derived ∞-category; in the algebraic setting, one can generalize faithfully flat descent to derived ∞-categories [31,Sec. D.6.3]. In the rigid analytic case, one imposes the following finiteness condition, introduced in [1, Exp. I].
Definition 1.2 (Almost perfect complexes). For a ring R, we consider the derived ∞-category D(R). An object M ∈ D(R) is called almost perfect (or pseudocoherent ) if it can be represented (up to quasi-isomorphism) by a chain complex M • such that M i = 0 for i ≪ 0 and each M i is finitely generated projective. We let APerf(R) ⊂ D(R) be the full subcategory spanned by almost perfect objects. Example 1.3. Suppose R is noetherian. Then M ∈ D(R) is almost perfect if and only if H i (M ) is finitely generated for each i and vanishes for i ≪ 0. In particular, APerf(R) is the (homologically) bounded-below derived ∞-category of the abelian category of finitely generated R-modules.
From Theorem 1.1 and in view of Example 1.3, one may deduce the following extension of faithfully flat descent to almost perfect complexes; compare Hennion-Porta-Vezzosi [25,Sec. 3] for closely related results.
Theorem 1.4. The construction A → APerf(A) satisfies flat hyperdescent on K-affinoid algebras. 1 In particular, if A → A ′ is a faithfully flat map of K-affinoid algebras, then → . . . . The second way one may attempt to generalize Theorem 1.1 is to allow more general rings than K-affinoid algebras. The context of rigid geometry imposes strong finiteness assumptions: in particular, the rings A, A ′ are noetherian. A result of Drinfeld shows that descent for vector bundles (rather than finitely generated modules) holds very generally. We next formulate a version of Drinfeld's theorem.
Definition 1.5 (The site Alg ♭ OK ). Let O K ⊂ K be the ring of integers, and let π ∈ O K denote a nonzero nonunit. Let Alg ♭ OK denote the category of O K -algebras R which are π-torsion-free and π-adically complete. We say that a map R → R ′ in Alg ♭ OK is π-completely faithfully flat if R/π → R ′ /π is faithfully flat; this defines the π-completely flat topology on (Alg ♭ OK ) op . For any ring A, we let Vect(A) denote the category of finitely generated projective A-modules. Then one has the following result. Compare [16, Th. 3.11] and [17,Prop. 3.5.4].
Theorem 1.6 (Drinfeld). The construction R → Vect(R[1/π]) is a sheaf of categories on Alg ♭ OK . That is, given R → R ′ in Alg ♭ OK which is π-completely faithfully flat, the natural functor is an equivalence of categories.
In this note, we will prove various common generalizations of Theorem 1.6 and Theorem 1.4, using some simplifications that occur when one works in the derived context. An instance of the result is the following.
Theorem 1.7. The construction R → APerf(R[1/π]) is a hypercomplete sheaf of ∞-categories on Alg ♭ OK . In particular, given R → R ′ in Alg ♭ OK which is π-completely faithfully flat, the natural functor induces an equivalence . This remains true if we replace APerf with APerf ≥0 ⊂ APerf (the subcategory of connective objects) or Perf ⊂ APerf (the subcategory of perfect complexes), or the subcategories Perf The passage from Theorem 1.1 to Theorem 1.4 is facilitated by the t-structure on APerf in the Kaffinoid case. By contrast, in the setting of Theorem 1.7, there is no t-structure on APerf(R[1/π]). As a consequence, we do not know how to deduce Theorem 1.7 from Theorem 1.6, and we will give a different argument: we will construct a stable ∞-category M(R) into which APerf(R[1/π]) embeds, and such that M(R) admits a t-structure that enables one to prove descent results. Our 1 One also has descent for hypercovers instead only forČech covers; this is additional information when one works with ∞-categories. main ingredients are the monadicity theorem of Barr-Beck-Lurie and a tool that exists only in the derived sense: the equivalence between torsion and complete modules [19,Th. 2.1]. Although it will not strictly be necessary to the proof, our argument is inspired by a result of Bhatt [7] that in this case states that any π-complete O K -module M such that M [1/π] = 0 is in fact annihilated by a fixed power of π.
In fact, we will prove two versions of our descent result (Theorem 7.8 and Theorem 5.8 below). The first is a generalization of Theorem 1.7 to the case of a finitely generated ideal (rather than simply inverting an element). The second is one where π-complete faithful flatness is replaced by universal descent in the sense of [32,Sec. 3], which for finitely presented maps is equivalent to being a v-cover [37].
Part of the difficulty in proving descent results such as the above is the lack of a well-behaved theory of quasicoherent (rather than coherent) sheaves in analytic (e.g., rigid) geometry. Recently Clausen and Scholze have, using condensed mathematics, defined a category of quasicoherent sheaves on an extremely large class of analytic spaces [38]. We expect that our results can be obtained using their framework. Compare also recent work of Andreychev [4], which uses condensed mathematics to prove some related descent statements (for the analytic rather than flat topology).
Conventions. We will generally use the notations and conventions of [30,31], and formulate our results for connective E ∞ -rings rather than for ordinary commutative rings. In many cases this is essential even for questions that begin with discrete rings, because the derived completion process may introduce higher homotopy groups. In particular, all tensor products will be derived tensor products.
Given a connective E ∞ -ring R, we write Mod(R) for the ∞-category of R-module spectra. When R is an ordinary commutative ring, this recovers the unbounded derived ∞-category D(R). The stable ∞-category Mod(R) is equipped with a t-structure, and we let Mod(R) ≥0 , Mod(R) ≤0 ⊂ Mod(R) denote the subcategories of connective and coconnective objects. We let Mod(R) ♥ denote the (usual) abelian category of discrete π 0 (R)-modules, which is the heart of this t-structure. We will use homological indexing conventions.
We will freely use the theory of t-structures on stable ∞-categories as in [30, Sec. 1.2.1] (after [6] for triangulated categories). Given a stable ∞-category C equipped with a t-structure, we say that C is right-bounded if C = n C ≥n and left-complete if C ≃ lim ← −n C ≤n (where the transition maps are the truncation maps).
with g • f : X → X given by multiplication by a. We also write P ≤I ∞ = n≥0 P ≤I n ; informally, we can think of P ≤I ∞ as those objects which satisfy the defining property of objects in P "up to isogeny." In this note, we will be interested almost exclusively in the case where I ⊂ π 0 (R) is a finitely generated ideal. By contrast, when I = I 2 , this type of definition is frequently used in almost ring theory [23].
(1) When I = (1), then P ≤I = P. This follows because P is closed under retracts.
(2) For convenience, we have formulated the above for ∞-categories, but the above definition only depends on the underlying homotopy category Ho(C) (which is naturally enriched in π 0 (R)-modules). (3) Let F : C → D be an R-linear functor of R-linear additive ∞-categories. Suppose P ⊂ C, P ′ ⊂ D are full subcategories closed under finite direct sums and retracts. Suppose F carries P into P ′ . Then F carries P ≤I into P ′ ≤I . (4) Let J ⊂ π 0 (R) be another ideal. Then (P ≤I ) ≤J ⊂ P ≤IJ . (5) Let {I α , α ∈ A} by a filtered system of ideals and let I = I α . Then P ≤I = P ≤Iα .
It will be convenient to rephrase Definition 2.1 in the finitely generated case in terms of the notion of ≤ I-split surjection (resp. injection), as will follow from Proposition 2.5 below.

Definition 2.3 (≤ I-split surjections and injections).
(1) A map of (discrete) π 0 (R)-modules M → N is said to be ≤ I-surjective if its cokernel is annihilated by I.
This is equivalent to the statement that for each a ∈ I, there exists g a : Y → X such that f • g a : Y → Y is given by multiplication by a. (3) A map f : X → Y in C is said to be a ≤ I-split injection if for all Z ∈ C, the map of π 0 (R)-modules π 0 Hom C (Y, Z) → π 0 Hom C (X, Z) is ≤ I-surjective. This is equivalent to the statement that for each a ∈ I, there exists g a : Y → X such that g a • f : X → X is given by multiplication by a.
Proposition 2.5. Suppose I ⊂ π 0 (R) is a finitely generated ideal. Then the following are equivalent: (1) An object X ∈ C belongs to P ≤I .
Proof. It is easy to see from the definitions that (2) or (3) implies (1). Suppose (1). Let x 1 , . . . , x n ∈ I be a set of generators. We have objects X ′ 1 , . . . , X ′ n ∈ P and maps f i : X → X ′ i and g i : Since the x i generate I, it is not difficult to see that this map is a ≤ I-split injection. Similarly, the map i → X is a ≤ I-split surjection. Corollary 2.6. Suppose I is finitely generated and P is the filtered union of the full additive, idempotent-complete subcategories P α , α ∈ A. Then P ≤I = α∈A (P α ) ≤I .
We will use Definition 2.1 in the following instances.
(1) Suppose P is the subcategory of zero objects. Then an object X ∈ C belongs to P ≤I if and only if every element of I acts by zero on X. In this case, we will say that X is ≤ I-isogenous to zero.
(2) Let D = Fun(∆ 1 , C) denote the ∞-category of arrows X → X ′ in C, and let P be the subcategory of isomorphisms. An arrow in C belonging to P ≤I is said to be an ≤ I-isogeny. (3) Suppose C is a compactly generated R-linear additive ∞-category and P is the subcategory of compact objects. We say that an object is ≤ I-compact if it belongs to P ≤I .
For the next results, we note that there is a small subtlety that the construction of the arrow ∞category does not commute with taking homotopy categories, i.e., Ho(Fun(∆ 1 , C)) = Fun(∆ 1 , Ho(C)) (if so, the proofs could be shortened). Instead, we have a fiber sequence Suppose f 1 : X → X ′ is ≤ I-split surjective and f 2 is an equivalence. Then f is ≤ I-split surjective.
Similarly, suppose f 1 is an equivalence and f 2 is ≤ I-split injective. Then f is ≤ I-split injective.
Proof. Combine Lemma 2.10 and the fiber sequence (1). Proposition 2.12. Let P, Q ⊂ C and suppose I, J ⊂ π 0 (R) be ideals. Let Fun(∆ 1 , C) be the ∞category of arrows in C. Let R ⊂ Fun(∆ 1 , C) be the subcategory of arrows X → Y with X ∈ P, Y ∈ Q. Then an arrow X ′ → Y ′ in Fun(∆ 1 , C) with X ′ ∈ P ≤I and Y ′ ∈ Q ≤J belongs to R ≤IJ .
Proof. Without loss of generality, we can assume I, J are finitely generated. By assumption and Proposition 2.5, there is a ≤ I-split surjection W ′ → X ′ with W ′ ∈ P. We then get a map of which is a ≤ J-split injection via Proposition 2.11. However, (W ′ → Z ′ ) ∈ R, so we conclude.
Our main application of Proposition 2.12 is that the ≤ I-construction behaves well with respect to extensions. Corollary 2.13. Suppose C is an R-linear stable ∞-category, and let I, J ⊂ π 0 (R) be finitely generated ideals. Let P, Q, R be full additive, idempotent-complete subcategories. Suppose the cofiber of any map with source in P and target in Q belongs to R. Then the cofiber of any map with source in P ≤I and target in Q ≤J belongs to R ≤IJ .
Proof. This is a consequence of Proposition 2.12. We consider the cofiber functor cofib: Fun(∆ 1 , C) → C. Let W ⊂ Fun(∆ 1 , C) be the subcategory of arrows with source in P and target in Q. The cofiber functor carries W into R and therefore W ≤IJ into R ≤IJ . By Proposition 2.12, any arrow with source in P ≤I and target in Q ≤J belongs to W ≤IJ . Corollary 2.14. Suppose C is an R-linear stable ∞-category and P ⊂ C a thick subcategory. Then P ≤I ∞ ⊂ C is also a thick subcategory. Proposition 2.15 (Characterization of ≤ I-compact objects). Let C be a compactly generated R-linear additive ∞-category. Then the following are equivalent for X ∈ C: a) X is ≤ I-compact. b) For every filtered system Y i , i ∈ I in C, the map is an ≤ I-isogeny of π 0 R-modules. c) For every filtered system Y i , i ∈ I in C, the map is an ≤ I-isogeny of R-modules.
Proof. Suppose a). For every filtered system Y i , i ∈ I, we define a functor on C with values in arrows in R-modules, given by . This carries compact objects in C to isomorphisms, so it carries ≤ I-compact objects to ≤ I-isogenies. Thus, a) implies c), and c) clearly implies b). To see that b) implies a), we write X as a filtered colimit of compact objects and use that (2) is a ≤ I-isogeny to exhibit X as ≤ I-compact.

2.2.
Modules. We now specialize to the case where C is given by modules over a connective E ∞ring R (or some appropriate subcategory), and begin by reviewing some finiteness conditions. Definition 2.16 (≤ I-finitely generated modules). Suppose C = Mod ♥ (R) is the category of discrete R-modules, and P is the subcategory of finitely generated modules. Then we say that a discrete R-module M is ≤ I-finitely generated if it belongs to P ≤I . It is not difficult to see that this holds if and only if M is ≤ I-isogenous to a finitely generated module.
Definition 2.17 (Perfect and almost perfect modules). Let Perf(R) ⊂ Mod(R) denote the ∞category of perfect R-modules, or equivalently the compact objects in Mod(R). We let APerf(R) ⊂ Mod(R) denote the subcategory of almost perfect (or pseudocoherent) R-modules (see [30,Sec. 7.2.4]; in the discrete case the definition is due to [1, Exp. I]). Given a qcqs spectral scheme X, we let QCoh(X) denote the ∞-category of quasi-coherent modules on X, and Perf(X), APerf(X) ⊂ QCoh(X) the associated subcategories of perfect and almost perfect objects.
Our goal is to study ≤ I ∞ -versions of perfectness and almost perfectness. For this, we will need to use the following intermediate property. We review the notion of being "perfect to order n," as in [31,Sec. 2.7] or [40,Tag 064N], and then formulate its ≤ I-analog. (1) For every filtered system N i , i ∈ I in Mod(R) ≤n , the natural map of R-module spectra has homotopy fiber in Mod(R) <0 : that is, it induces an isomorphism on π i for i > 0 and an injection on π 0 . (2) For every filtered system N i , i ∈ I in Mod(R) ≤n such that each transition map N i → N j induces an injection on π n , the natural map (4) is an isomorphism on connective covers. (3) For every filtered system N i , i ∈ I of discrete π 0 (R)-modules, the natural map is an isomorphism for i < n and an injection for i = n. The condition of being perfect to order n only depends on the truncation τ ≤n M ∈ Mod(R) ≤n , so we will often view being "perfect to order n" as a property of objects in Mod(R) ≤n . Note finally that a bounded-below R-module M is almost perfect if and only if it is perfect to each order n.   (1) M ′ is perfect to order n − 1.
(2) M ′′ is perfect to order n.
Proof. This follows from a diagram-chase. Let N i , i ∈ I be a filtered system in Mod(R) [0,n] . Let F be the functor which carries an R-module Q to fib(lim − → Hom R (Q, N i ) → Hom R (Q, lim − → N i )). By assumption, F (M ) ∈ Mod(R) <0 ; since we have a fiber sequence F (M ′′ ) → F (M ) → F (M ′ ), the result now follows easily.
Definition 2.22 (Modules ≤ I-perfect to order n). Let M ∈ Mod(R) ≤n . If P ⊂ Mod(R) ≤n is the subcategory of objects which are perfect to order n, then we will say that M is ≤ I-perfect to order n if M belongs to P ≤I . Given an arbitrary R-module M , we will say that M is ≤ I-perfect to order n if τ ≤n M is. An object is ≤ I ∞ -perfect to order n if it is ≤ I r -perfect to order n, for some r.
Proof. Note that this question depends only on τ ≤n−1 M . Our assumption implies that τ ≤n−1 M receives a ≤ I ∞ -split surjection from τ ≤n−1 of a perfect R-module P . Therefore, we find that is an isomorphism, we obtain the result for M as well.
Proposition 2.24. Let M ∈ Mod(R) ≤n be bounded-below. In order for M to be ≤ I-perfect to order n, it is necessary and sufficient that for every filtered system N i , i ∈ I in Mod(R) ≤n with injective transition maps on π n , the connective cover of the natural map (4) is an ≤ I-isogeny.
Proof. Without loss of generality, we can assume I finitely generated. Necessity is clear, since the connective cover of (4) is an equivalence for M perfect to order n under our assumptions. We can find a filtered system N i ∈ Mod(R) ≤n with injective transition maps on π n such that each N i is perfect to order n and such that lim − → N i ≃ M , e.g., using Example 2.20 and that M is a filtered colimit of perfect modules. Our assumption now shows that one of the N i maps to M via a ≤ I-split surjection, which is enough to imply the claim. (1) M ′ is ≤ I-perfect to order n − 1.
Proof. This follows from Proposition 2.21, in light of Corollary 2.13. Alternatively, one can argue directly using Proposition 2.24.

2.3.
Passage from the generic fiber. In this subsection, we will prove various instances of the following principle: under (derived) I-completeness assumptions, for a module to have a certain property outside I is equivalent to having it up to ≤ I ∞ -isogeny integrally, or equivalently up to ≤ I r modulo I n for every n (where r is independent of n).
To begin with, we consider the case of discrete modules over a commutative ring. Fix a commutative ring R and a finitely generated ideal I ⊂ R. Recall the notion of derived completeness; see [40,Tag 091N] or [31, Sec. 7.3] for accounts. A derived I-complete module M is generally not I-adically separated (i.e., one may have I n M = 0); this is the key difference between derived I-complete and classically I-complete modules. Nonetheless, given a derived complete module M ∈ Mod(R) ♥ , if M/IM = 0 then M = 0. In the future, we will need a slight refinement of this fact, as follows.
Lemma 2.28. Let R be a commutative ring with a fixed element t ∈ R. Let M, N, P be discrete R-modules and suppose P is derived t-complete. Let f : M → N and g : N → P be maps. Suppose that both f and g are t-divisible elements of Hom R (M, N ), Hom R (N, P ). Then g • f = 0.
Proof. We will use the following basic fact: among t-torsion-free objects, derived and classical completion coincide; this follows, e.g., from the formula below (Construction 2.32). Clearly g annihilates all t-power torsion elements of N , so it factors through a map (N/N tors ) → P . It thus suffices to show that the composite map M f → N → (N/N tors ) is zero. Indeed, this composite map is divisible by arbitrary powers of t, but (N/N tors ) (as the derived, and hence classical, t-completion of a t-torsion-free module) is t-adically separated, forcing the map to vanish.
Our starting point for this section is the following "uniform boundedness" result. See also [5, Lemma 2.2] for a special case (when t = p).
Theorem 2.29 (Bhatt [7]). Let R be a commutative ring containing a finitely generated ideal I ⊂ R. Let M be a derived I-complete discrete R-module. Then the following are equivalent: (1) M vanishes away from I: that is, M [1/t] = 0 for t ∈ I.
(2) There exists r such that M is ≤ I r -isogenous to zero.
(3) There exists r such that for all n, M/I n M is ≤ I r -isogenous to zero.
Proof. The equivalence of (1) and (2) appears in [7]. It remains only to show that (3) implies (2). Taking n = r + 1, we find that I r M = I r+1 M = I(I r M ). Since I r M is derived I-adically complete (as the image of a map from a finite direct sum of copies of M to M ), it follows that I r M = 0, as desired.
Proposition 2.30. Let R be a ring which is derived I-complete for some finitely generated ideal I ⊂ R. Let M be a derived I-complete discrete R-module. Then the following are equivalent: (1) M is finitely generated outside I (that is, for t ∈ I, M [1/t] is a finitely generated R[1/t]module). (2) There exists r such that M is ≤ I r -finitely generated.
(3) There exists r such that for all n, M/I n M is ≤ I r -finitely generated. In fact, the best possible r for (2) and (3) are the same.
Proof. Suppose (1). Then there exists a finitely generated free R-module F and a map f : F → M which induces a surjection after inverting any t ∈ I (it suffices to check this as t runs over a system of generators). By Theorem 2.29, we have I r coker(f ) = 0 for some r. Replacing F with im(f ), we find that M is ≤ I r -finitely generated, proving (2). Clearly (2) implies (1), so it suffices to show that (3) implies (2). Taking n = r + 1, we have that M/I r+1 M is ≤ I r -finitely generated. Therefore, there exists a finitely generated free R-module F ′ and a map g : F ′ → M such that coker(g)/I r+1 coker(g) is annihilated by I r , and therefore I r coker(g) = 0 by derived completeness. Thus im(g) ⊂ M is a ≤ I r -isogeny and the result follows. Now we switch to the non-discrete setting: let R be a connective E ∞ -ring, and let I ⊂ π 0 (R) be a finitely generated ideal. We briefly recall the theory of I-complete and I-torsion objects in Mod(R), from [19]; see also [31,Sec. 7.3] for a detailed account. (1) Given M ∈ Mod(R), M is said to be I-torsion if all the homotopy groups are I-power torsion (i.e., every element in π * (M ) is annihilated by a power of I). ( This holds if and only if each homotopy group π i (M ) is derived I-complete as a discrete π 0 (R)-module.
Note that this tower {R n } is not canonical (it relies on a choice of generators of I) and it does not generally form a tower of E ∞ -R-algebras, but only of R-modules. In the future, we will fix an explicit choice of generators of I and thus of a tower {R n }.
Throughout, we will need to use the following fundamental equivalence between complete and torsion modules, which goes back to [ Theorem 2.33. Let R be a connective E ∞ -ring, and let I ⊂ R be a finitely generated ideal as above. The functor of I-completion also induces an equivalence between the subcategories of Itorsion modules and I-complete modules inside Mod(R).
Next, we prove the analogs of Proposition 2.30 in the context of module spectra. The main results are similar: perfectness to any order on the generic fiber can be checked integrally (or modulo any power of the ideal in an appropriate sense) up to bounded isogeny. Proposition 2.34. Let R be a connective E ∞ -ring and I = (x 1 , . . . , x p ) ⊂ π 0 (R) a finitely generated ideal; fix a tower {R n } as in Construction 2.32. Let M ∈ Mod(R) be I-complete. Then the following are equivalent: (1) For each m < 0, π m (M ) is ≤ I ∞ -isogenous to zero.
is annihilated by a fixed power of I uniformly in n, which forces (by Theorem 2.29) π −1 (M ) to be ≤ I ∞ -isogenous to zero. Proposition 2.35. Let R be a connective E ∞ -ring which is I-complete for I = (x 1 , . . . , x p ) ⊂ π 0 (R) a finitely generated ideal; fix {R n } as in Construction 2.32. Let M be an object of Mod(R) which is bounded-below and I-complete. Given m ≥ 0, the following are equivalent: Proof. Without loss of generality (i.e., by shifting), we may assume M connective. We use induction on m. In the case m = 0, all three conditions are equivalent to π 0 (M ) being ≤ I ∞ -finitely generated (via Proposition 2.30). Now suppose m > 0. Since all three conditions are isogeny invariant, we can assume without loss of generality that π 0 (M ) is finitely generated. Choose a finitely generated free R-module P and a map P → M inducing a surjection π 0 (P ) ։ π 0 (M ), and write F = fib(P → M ). Thanks to Proposition 2.26, conditions (1)

The construction M(R)
Throughout this section, we fix a E ∞ -ring R equipped with a finitely generated ideal I ⊂ π 0 (R). We will construct a stable ∞-category M(R), which we should regard as associated to the "generic fiber" of the formal spectrum Spf(R) (we will not need a precise notion here). Our main result is that APerf(Spec(R) \ V (I)) is naturally a full subcategory of M(R).

Definition of M(R).
To begin with, we give the abelian version of the construction M(R); in fact, there are two natural candidates, involving complete and torsion modules respectively. In the derived setting, the distinction between the two goes away.
Definition 3.1 (The construction A(R)). We let A(R) denote the abelian category obtained as the Serre quotient of the category of I-power torsion discrete π 0 (R)-modules by the subcategory of those modules which are ≤ I ∞ -isogenous to zero. Given a map R → R ′ , base-change gives a right exact functor A(R) → A(R ′ ) with a right adjoint (given by restriction of scalars) which is exact.
Definition 3.2 (The construction B(R)). We let B(R) denote the abelian category obtained as the Serre quotient of the category of derived I-complete discrete π 0 (R)-modules by the subcategory of those modules which are ≤ I ∞ -isogenous to zero.
We can regard B(R) as an abelian version of the category of Banachian spaces studied in [17]. Next, we need the stable versions. Construction 3.3 (t-structures on Verdier quotients). Let E be a stable ∞-category equipped with a t-structure with heart E ♥ . Given a Serre subcategory B 0 ⊂ E ♥ , we define a thick subcategory E ′ ⊂ E consisting of those objects all of whose homotopy groups lie in B 0 . Then E ′ also inherits a t-structure, as does the Verdier quotient E/E ′ . The heart E ′ is given by B 0 and the heart of of E/E ′ is given by the Serre quotient E ♥ /B 0 . (1) Let Mod(R) cpl >−∞ ⊂ Mod(R) denote the subcategory consisting of R-modules which are bounded-below and I-complete. Let (2) We have an equivalence of ∞-categories between Mod(R) cpl >−∞ and the ∞-category of Rmodules which are bounded-below and I-torsion (Theorem 2.33, since both functors involved have bounded amplitude). Therefore, Mod(R) cpl >−∞ is equipped with a t-structure, which we refer to as the I-torsion t-structure, whose heart is the abelian category of I-power torsion discrete π 0 (R)-modules. This restricts to a t-structure on Mod(R) nil >−∞ whose heart is the abelian category of π 0 (R)-modules which are ≤ I ∞ -isogenous to zero.
as an R-linear stable ∞-category; it also inherits a right-bounded (i.e., every object is bounded below) t-structure whose heart is given by the abelian category A(R) (Construction 3.3). It is linear over Perf(R) and the action annihilates the I-torsion objects in Perf(R), so it becomes linear over Perf(Spec(R) \ V (I)). We where the transition maps are the truncation functors. Similarly, M(R) is a stable ∞-category (linear over Perf(Spec(R) \ V (I))) with a t-structure, which we refer to as the I-torsion t-structure; for each n, we have M(R) ≤n = M 0 (R) ≤n .

Remark 3.5.
There is no distinction between the idempotent-complete and the non-idempotent complete Verdier quotient in the definition of M 0 (R). Indeed, via [41], this follows from the localization theorem in K-theory. It suffices to show that K 0 (Mod(R) nil >−∞ ) = 0. This in turn follows because for any object  also admits a t-structure obtained by restriction from the usual t-structure on all R-modules, whose heart is given by derived I-complete discrete π 0 (R)-modules; we refer to this as the I-complete tstructure. Similarly, using Construction 3.3, this t-structure descends to M 0 (R). We observe that it also descends to M(R) with heart B(R). This follows because the I-torsion and I-complete t-structures on M 0 (R) (and on Mod(R) cpl >−∞ ) differ by a bounded amplitude. In particular, M(R) is also the left completion of M 0 (R) with respect to the I-complete t-structure. Construction 3.8 (Functoriality of M(R)). The construction R → M(R) defines a functor in the connective E ∞ -ring R (with choice of ideal I ⊂ π 0 (R)) to the ∞-category of right-bounded, leftcomplete stable ∞-categories with t-structures (for either choice of t-structure), and right t-exact functors between them; on hearts it is the base-change functor on abelian categories A(R), B(R). Finally, given any map R → R ′ of connective E ∞ -rings, we get a right adjoint M(R ′ ) → M(R) to the base-change functor (given by restriction of scalars) which is t-exact.

3.2.
Almost perfect complexes. Throughout this subsection, we use the I-complete t-structure and we assume R is itself I-adically complete (no loss of generality by Remark 3.6). It will be necessary to compute some mapping spaces in M(R); for this, we use the following construction. Proposition 3.10. Suppose R is I-complete. Suppose X, Y ∈ Mod(R) cpl >−∞ . Suppose X is ≤ I ∞perfect to order n + 1 and Y is n-truncated in the I-complete t-structure on Mod(R) cpl >−∞ . Then the functor j * induces an equivalence of connective spectra τ ≥0 Hom M0(R) (X, Y ) ≃ τ ≥0 Hom QCoh(Spec(R)\V (I)) (j * X, j * Y ).
Note that the R-module Y is n-truncated (with respect to the Postnikov t-structure). Therefore, is an isomorphism by Proposition 2.23, since X is ≤ I ∞ -perfect to order n + 1. To complete the proof, it suffices to show that for any X ′ → X in Mod(R) cpl >−∞ whose cofiber belongs to Mod(R) nil >−∞ , then the map induces an equivalence after inverting any t ∈ I. However, since Y is n-truncated, we can identify this with the map τ ≥0 Hom Mod(R) cpl Our goal is to identify a subcategory of "weakly almost perfect objects" in M(R) and then to show that this subcategory is equivalent (via j * ) to the ∞-category APerf(Spec(R) \ V (I)). Definition 3.12 (Objects weakly perfect to order n). We will say that an object of M 0 (R) is weakly perfect to order n if for any representative X ∈ Mod(R) cpl >−∞ , X is ≤ I ∞ -perfect to order n. This definition is clearly independent of the choice of representative X, since any two are related by a zig-zag of quasi-isogenies. We will say that an object is weakly almost perfect if it is weakly perfect to any order.
The condition that an object in M 0 (R) should be weakly perfect to order n depends only on its n-truncation with respect to the I-complete t-structure. Therefore, we also obtain analogous definitions of weakly almost perfect (resp. weakly perfect to order n) for objects of M(R).
(1) X is weakly perfect to order zero if and only if there exists a map R r → X (for some r ≥ 0) whose cofiber C belongs to M(R) ≥1 .
(2) Suppose n > 0 and X is weakly perfect to order n − 1. Then X is weakly perfect to order n if and only if, for any (or every) map R r → X with cofiber C ∈ M(R) ≥1 , we have that C[−1] is weakly perfect to order n − 1.
Proof. The condition that X should be weakly perfect to order n only depends on the n-truncation of X; moreover, maps R r → X only depend on τ ≤g X (Corollary 3.11), if I is generated by g elements. Therefore, without loss of generality we may assume that X ∈ M 0 (R) <∞ and can be represented as the image in M 0 (R) of some Y ∈ Mod(R) cpl >−∞ , which without loss of generality we can take to be connective. Then X is weakly perfect to order zero if and only if there exists a map R r → Y whose cofiber in Mod(R) has π 0 which is ≤ I ∞ -isogenous to zero, cf. Example 2.25. Assertion (1) now follows. Since any map of objects in M 0 (R) can be represented by a map in Mod(R) cpl >−∞ (subject to working up to quasiisogenies), assertion (2) follows by Proposition 2.26.
(1) The collection of objects in M(R) ≥0 which are weakly almost perfect is closed under finite colimits and geometric realizations.
(2) An object of M(R) ≥0 is weakly almost perfect if and only if it can be written as a geometric realization of a simplicial object in M(R) ≥0 each of whose terms is a finite direct sum of copies of R.
Proof. Part (1) follows from the fact that the condition of being ≤ I ∞ -perfect to some order n is closed under finite colimits and depends only on the n-truncation. Part (2) follows by iteration. Given a weakly almost perfect X ∈ M(R) ≥0 , one produces (using Proposition 3.13) a filtered object 0 = P −1 → P 0 → P 1 → . . . such that P i /P i−1 is equivalent to a finite direct sum of copies of R[i] and whose colimit (which exists for connectivity reasons) is X. Converting this into a simplicial object via the Dold-Kan correspondence [31, Sec. C.1.4], we see that (2) follows.
Proposition 3.15. Let R be I-complete. Suppose X, Y ∈ M(R) and X is weakly almost perfect. Then j * X ∈ QCoh(Spec(R) \ V (I)) is almost perfect. Furthermore, the natural map induces an equivalence of R-module spectra Hom M(R) (X, Y ) ≃ Hom QCoh(Spec(R)\V (I)) (j * X, j * Y ).
Proof. Any object in Mod(R) cpl >−∞ which is ≤ I ∞ -perfect to order n restricts to an object of QCoh(Spec(R) \ V (I)) which is perfect to order n, so the first assertion follows. Since M(R) is left-complete, it suffices to show for each n that τ ≥0 Hom M(R) (τ ≤n+1 X, τ ≤n Y ) ∼ − → τ ≥0 Hom QCoh(Spec(R)\V (I)) (j * τ ≤n+1 X, j * τ ≤n Y ); these (connective) mapping spectra can be computed in M 0 (R) ≤n . Thus, the result follows from Proposition 3.10 since we can represent τ ≤n+1 X by an I-complete R-module which is perfect to order n + 1. Proof. Full faithfulness is a consequence of Proposition 3.15. For essential surjectivity, it suffices to show that any object of APerf(Spec(R) \ V (I)) ≥0 can be written as a geometric realization of a simplicial object which at each level is a direct sum of copies of the unit. This holds more generally for any quasi-affine spectral scheme (modeled on the spectrum of a connective E ∞ -ring).
Indeed, let F ∈ APerf(Spec(R) \ V (I)) ≥0 . Then by [31, Prop. 9.6.6.1], there is a map from a direct sum of copies of the unit to F inducing a surjection on π 0 -sheaves. Since F is almost perfect and connective, we can assume that this is a finite direct sum. Continuing inductively as in the proof of Proposition 3.14, we can now write F as a geometric realization as desired. The assertions for M(R) follow from those for M(R) ≥0 .

Review of monadicity and descent
Our descent results take the form of an expression for a stable ∞-category as a homotopy limit of a cosimplicial stable ∞-category. In this section, we review some general results from [30,Sec. 4.7.5] for identifying such homotopy limits; these are closely related to the Barr-Beck-Lurie monadicity theorem.
Definition 4.1 (The Beck-Chevalley condition). Let C • be an augmented cosimplicial ∞-category. We will say that C • satisfies the adjointability condition if for each α : [m] → [n] in ∆ + , the square is right adjointable: that is, the horizontal arrows admit right adjoints (denoted d 0 * ), and the induced Let R → R ′ be a map of E ∞ -rings, and let R ′• be the associatedČech nerve (an augmented cosimplicial E ∞ -ring). Then the diagram of ∞-categories Mod(R ′• ) satisfies the adjointability condition. The reason is that if we have a pushout square of E ∞ -rings then the induced square after applying Mod(·) is right adjointable.
For the next result, we let Cat ∞ denote the ∞-category of ∞-categories. (1) The (co)augmentation coaug : C −1 → C 0 is conservative and has the following property: totalizations of coaug-split cosimplicial objects in C −1 exist and are preserved by coaug.
(2) C • satisfies the adjointability condition. Then C • is a limit diagram.
In practice, the adjointability condition will be automatic (from e.g., Example 4.2), so to verify that certain diagrams are limit diagrams, it will be necessary to verify condition (1) of Theorem 4.3. There will be two basic tools: universal descent maps (for which condition (1) will hold for essentially diagrammatic reasons) and situations where one has a t-structure.
We begin with the universal descent case. The following definition is essentially from [32, Sec. 3] and [31, Sec. D.3]. The main example is given by base-change along a universal descent morphism of E ∞ -rings.

Definition 4.4 (Universal descent functors). Let C, D be idempotent-complete, stable ∞-categories.
We say that an exact functor f : C → D is of universal descent if there exists an exact functor ϕ : C → C with the following properties: (1) The identity functor id C is a retract of ϕ.
In the following, we denote by Cat perf ∞ the ∞-category of idempotent-complete, stable ∞categories and exact functors between them.
Proof. It remains to verify condition (1) of Theorem 4.3. First, the filtration (6) implies that any object in the kernel of coaug vanishes, so coaug is conservative. Next, consider the collection V of cosimplicial objects Y • ∈ Fun(∆, C −1 ) which admit a totalization in C −1 and which is preserved under coaug. Clearly V is a thick subcategory of Fun(∆, C −1 ), and it contains those cosimplicial objects which admit splittings. If X • ∈ Fun(∆, C −1 ) is such that coaug(X • ) admits a splitting, then our assumption implies that X • belongs to V, whence the claim.
Next, we prove a descent criterion in the case of t-structures. This is essentially a version of the argument used for faithfully flat descent in [31, Sec. D.6].
Proposition 4.6 (Descent criterion for left-complete stable ∞-categories). Let C • be an augmented cosimplicial object of Cat perf ∞ . Suppose that each C i , i ≥ −1 is equipped with a right-bounded, leftcomplete t-structure, and the following conditions are satisfied.
(2) Each cosimplicial structure map is right t-exact, and has a t-exact right adjoint. preserves totalizations and is conservative. Therefore, by Theorem 4.3, we find that C • [m,n] is a limit diagram. Taking the limit over n and the colimit over m, we obtain that C • is a limit diagram. A similar argument works for C • ≥0 .
. . . This is the derived version of faithfully flat descent from [31,Cor. D.6.3.3] (at least in the boundedbelow case), and the above argument is that of loc. cit.

The universal descent case
In this section, we show (Theorem 5.8) that almost perfect complexes on Spec( R I ) \ V (IR) form a sheaf with respect to the universal descent topology (Definition 5.1). As an application, we prove arc-descent results for the category of finiteétale covers, extending results of [8].  This class of morphisms was studied in [32] and [31,Sec. D.3], to which we refer for more details; see also [33] for a survey. In particular, one has the following basic result (a special case of Proposition 4.5): is an equivalence. In other words, Mod(·) is a sheaf for the universal descent topology.
In the case of finitely presented morphisms of discrete rings, one can give a concrete geometric criterion for a map to be universal descent, due to Bhatt-Scholze [10,Prop. 11.25] in the noetherian case; here we observe that it holds generally.
Theorem 5.5. Let f : R → R ′ be a finitely presented map of discrete rings. Suppose f is a v-cover ( [37] and [10,Sec. 2]). Then f is a universal descent morphism.
Proof. By [37,Theorem 6.4], it follows that f is obtained as the (underived) base-change of a v-cover of finitely presented Z-algebras, R 0 → R ′ 0 along a map R 0 → R. In particular, [10,Prop. 11.25]. Now the map (of E ∞ -rings) R → R ⊗ R0 R ′ 0 is universal descent by base-change. By Lemma 5.6, it follows that R → R ′ is a universal descent morphism.
Lemma 5.6. Let f : R → S be a map of connective E ∞ -rings which is universal descent. Suppose R is discrete. Then the composite map R → S → π 0 (S) is universal descent.
Proof. We use throughout the following basic observation, valid since R is discrete. Let R → M be a map of R-modules; then this map admits a section if and only if the composite R → M → τ ≤0 M admits a section.
Let S • beČech nerve of R → S. Then for some N , R is a retract of Tot N (S • ); equivalently, the map R → Tot N (S • ) → τ ≤0 Tot N (S • ) admits a section. Now the map τ ≤0 Tot N (S • ) → τ ≤0 Tot N (τ ≤N +1 (S • )) is an equivalence. Therefore, the map R → τ ≤0 (Tot N (τ ≤N +1 (S • ))) admits a section, and hence the map R → Tot N (τ ≤N +1 (S • )) admits a section too. It follows that R is a retract of a finite limit of a diagram of R-modules, each of which admits the structure of a τ ≤N +1 Smodule. Consequently R → τ ≤N +1 S is a universal descent morphism. Via the Postnikov tower, we see that τ ≤N +1 S → π 0 (S) is a universal descent morphism. Composing, the claim follows.
Proposition 5.7. Let R → R ′ be a universal descent map of E ∞ -rings, and let I ⊂ π 0 (R) be a finitely generated ideal. ThenR →R ′ is universal descent.
Proof. This follows because we can write R as a retract of a finite limit of a diagram of R-modules admitting the structure of R ′ -module, and then complete everywhere.
The main descent theorem that we prove in this section is the following. Proof. For any filtered category I and functor I → Mod(R) ≤n−e−1 , j → N j , we will verify that the natural map is an ≤ I re -isogeny (compare Proposition 2.24). More generally, for each functor f : I → Mod(R), we consider the map ω = ω(f ) of (7). Our hypothesis implies that if f lifts to a functor with values in Mod(R ′ ) ≤n−1 , then ω(f ) is a ≤ I r -isogeny. Let R ′• be theČech nerve of R → R ′ , considered as a cosimplicial R-module. For any R-module N , we can write N as a functorial retract of Tot e (N ⊗ R R ′• ). It follows that if N ∈ Mod(R) ≤n−e−1 , then we can write N as a functorial retract of Tot e (τ ≤n−1 (N ⊗ R R ′• )). Now returning to (7) in the case where f takes values in ≤ n − e − 1-truncated modules, we find that ω(f ) is a retract of a partial totalization Tot e of ω(τ ≤n−1 (f ⊗ R R ′• )). From the above, it follows that ω(f ) is a retract of a partial totalization Tot e of a diagram of ≤ I r -isogenies, whence the claim.
Proposition 5.11. Let R → R ′ be a universal descent map of I-complete connective E ∞ -rings of exponent e. Let M ∈ Mod(R) be I-complete and bounded-below. Suppose R ′ ⊗ R M ∈ Mod(R ′ ) is ≤ I ∞ -perfect to order n. Then M is ≤ I ∞ -perfect to order n − e − 1.
Proof. Fix a tower {R m } as in Construction 2.32. By Proposition 2.35, it suffices to show that there exists r such that M ⊗ R R m is ≤ I r -perfect to order n − e − 1 for all m. But by assumption, there exists r ′ such that (M ⊗ R R m ) ⊗ R R ′ is ≤ I r ′ -perfect to order n for all m. Now apply Proposition 5.10 (with r = r ′ e) to conclude. Proof. This follows from Proposition 5.11, since we can (up to truncating homotopy groups in high enough degrees) assume that M is represented by an object of Mod(R) cpl >−∞ .
Proof of Theorem 5.8. We first prove the results for Perf, APerf. Since Perf is the subcategory of dualizable objects in APerf, it suffices to prove the result for APerf. We have seen that the construction R ′ → M(R ′ ) is a sheaf for the universal descent topology (Proposition 5.9), and there is a natural fully faithful embedding APerf(Spec(R ′ ) \ V (I)) ⊂ M(R ′ ) (Proposition 3.16) with image the weakly almost perfect objects. It suffices now to show that the property of belonging to the image of this embedding (i.e., being weakly almost perfect) is local in the universal descent topology. But this follows from Corollary 5.12. Note that it suffices to work everywhere with I-complete E ∞ -rings (Proposition 5.7).
To obtain the results for APerf ≥0 (and hence Perf ≥0 ), it suffices to show that for a universal descent map R ′ → S ′ of E ∞ -R-algebras and an object M ∈ APerf(Spec(R ′ \ V (I))) whose pullback to APerf(Spec(Ŝ ′ \ V (I)) is connective, then M is connective. This is a local question, andR ′ →Ŝ ′ is of universal descent. Thus, it suffices to verify the following: if A → B is a universal descent map of connective E ∞ -algebras and M ∈ Mod(A) is almost perfect, then M is connective if and only if M ⊗ A B is connective. By induction, we can assume M is (−1)-connective. But then π −1 (M ) is a finitely presented π 0 (A)-module and π −1 (M ) ⊗ π0(A) π 0 (B) = 0. Since π 0 (A) → π 0 (B) is surjective on spectra, this forces (by Nakayama) π −1 (M ) = 0 as desired. Remark 5.13 (Derived completions versus completions). In (8), as usual, the notation R (x1,...,xn) refers to the derived completion (which is a connective E ∞ -ring). One can replace the derived completion with the classical completion. For any R, there is a map from the derived completion R (x1,...,xn) to the classical completionR cl (x1,...,xn) . The induced map on π 0 is surjective, and the kernel squares to zero; this follows from the lim ← − -spectral sequence (i.e., the Milnor exact sequence in this case). Alternatively, this fact follows directly from Lemma 2.28. Consequently, finiteétale schemes are the same whether one uses the classical or derived completion. Similarly, in this section, there is no extra generality gained by working with connective E ∞ -rings rather than discrete rings.
We use the following fundamental algebrization result, which allows one to replace the completion with the henselization. For a qcqs scheme X, we let FEt(X) be the category of finiteétale schemes over X.  (x 1 , . . . , x n ), the natural functor is an equivalence of categories.
Proof. This follows from Theorem 5.14, since henselization commutes with filtered colimits.
We now use the following definition, from [8] and due independently to Rydh. It is a refinement of the v-topology [37,10], which in turn is the non-noetherian version of Voevodsky's h-topology. We also note a slight variant of it, as in [8, Sec. 6.2].
Definition 5.16 (arc-covers). A map of commutative rings R → R ′ is said to be an arc-cover if for every rank ≤ 1-valuation ring V and map R → V , there is an extension of rank ≤ 1-valuation rings V → V ′ and a commutative diagram This defines the arc-topology on the category of affine schemes. We refer to [8,Ex. 6.15] for a discussion of the relationship between the arc x -topology (so n = 1 in the above) and surjectivity at the level of adic spectra.
Informally, Theorem 5.19 is an expression of a (well-known) principle that the theory of (purely algebraically defined) finiteétale schemes behaves well in analytic geometry, and satisfies very strong descent results; many cases of this result are already in the literature. For instance, for perfectoid spaces, v-descent of finiteétale schemes appears as [39,Prop. 9.7]. In the case of abelianétale covers, this result appears (and more generally for the higher cohomology) in [8,Cor. 6.17] (with the restriction to a principal ideal, but this is not necessary). Theorem 5.19 as stated can also be proved using Gabber's rigidity results for nonabelian cohomology, cf. [22] and [27,Exp. XX].
We explain here a quick proof of Theorem 5.19 using the theory of finiteétale algebra objects in a symmetric monoidal stable ∞-category ( [36,32]) and Theorem 5.8.
Construction 5.20 (Finiteétale algebra objects). For a small, idempotent-complete stably symmetric monoidal ∞-category C, one extracts [32, Def. 6.1] a category FEt(C) of "finiteétale algebra objects" of C, a full subcategory of the ∞-category CAlg(C) of commutative algebra objects of C. 2 In the case where C = Perf(X) for X a qcqs spectral scheme (modeled on the spectra of connective E ∞ -rings), then FEt(C) recovers precisely the opposite to the category of finiteétale schemes over X (or equivalently of the underlying scheme π 0 X).
Example 5.21 (Torsors in FEt(C)). Let G be a finite group. A G-torsor in FEt(C) is given by an object A ∈ Fun(BG, CAlg(C) (i.e., a commutative algebra equipped with a G-action) such that: (1) As an object of Fun(BG, C), A belongs to the thick subcategory generated by the induced Gobjects (i.e., the G-action is nilpotent; compare the discussion in [33,Sec. 4]). In particular, A hG exists in C.
(2) The natural map 1 → A hG is an equivalence.
(3) The shearing map A ⊗ A → G A is an equivalence in C.
These conditions are due to Rognes [36], who introduces the notion of a Galois extension of an E ∞ -ring.
Remark 5.22 (FEt(·) preserves some limits). In general, the construction C → FEt(C) need not preserve limits: the construction C → CAlg(C) does, but condition (1) involved may not. However, suppose we have an augmented cosimplicial stably symmetric monoidal ∞-category C • which is a limit diagram; suppose moreover that C −1 → C 0 is a universal descent morphism. Then ). This follows by considering G-torsors for each finite group G and [32,Cor. 5.40].
Proof of Theorem 5.19. By Remark 5.18, it is sufficient to verify that F is an arc-sheaf. We will apply the criterion of [8,Theorem 4.1] to the functor F . First, note that the functor F is finitary thanks to Corollary 5.15.
We show that F satisfies h-descent. Let R → R ′ be an h-cover, i.e., a finitely presented v-cover. It suffices to show that in this case, Indeed, by Theorem 5.5, R → R ′ is a universal descent morphism, so by Theorem 5.8 we have a limit diagram (involving the iterated derived tensor products of R ′ over R). Applying Construction 5.20, and noting that the higher homotopy groups in the derived tensor product do not affect finiteétale schemes, (10) follows (also via Remark 5.22).
To check the criterion of [8,Theorem 4.1], it suffices to check the "aic-v-excision" condition of loc. cit. Suppose V is an absolutely integrally closed valuation ring and p ⊂ V a prime ideal; we need to show that the square (11) F (V ) is a pullback. To see this, we consider the map V → V /p × V p , which is of universal descent because of the pullback description V ≃ V /p × (V /p)p V p . It follows as in the previous paragraph that F carries theČech nerve of this map to a limit diagram; unwinding theČech nerve now shows that (11) is a pullback square as desired.
6. Example: flat descent on classical rigid spaces Throughout this section, we fix a complete nonarchimedean field K with ring of integers O K ⊂ K, and a nonzero nonunit π ∈ O K . We will work in the setting of classical rigid geometry and K-affinoid algebras. 3 In this section, we give a proof of Theorem 1.4 from the introduction (as Theorem 6.16 below), i.e., of flat descent of almost perfect complexes for K-affinoid algebras. This result can also be deduced from the case of coherent sheaves [13], although our argument will be independent; cf. also [25] for the case ofétale descent.
We freely use the flattening results of Bosch-Lütkebohmert [14] and various finiteness properties of topologically finitely presented O K -algebras, but otherwise the methods make no reference to results of rigid geometry (e.g., Kiehl's theorem that coherent modules satisfy descent in the analytic topology; these methods recover Kiehl's theorem). Our main observation (Proposition 6.14) is that if A → B is a faithfully flat map of affinoid K-algebras, then the map A 0 → B 0 of appropriate rings of definition is a universal descent morphism; furthermore, the iterated derived tensor products of B 0 over A 0 have higher homotopy which is bounded π-power torsion. The result will then follow from Theorem 5.8. 6.1. Review of coherent rings. In the following it will be necessary to work with coherent (and especially stably coherent) rings; these are rings for which many of the convenient module-theoretic finiteness properties of noetherian rings still hold, provided one restricts to finitely presented modules. See [24] for a textbook reference. Definition 6.1 (Coherent rings). A ring R is called coherent if the category of finitely presented R-modules is abelian (equivalently, stable under kernels). This holds if and only if each finitely generated ideal I ⊂ R is finitely presented as an R-module. A commutative ring R is called stably coherent if every finitely presented R-algebra is coherent. Definition 6.2 (Coherent E ∞ -rings). Let R be a connective E ∞ -ring. We say that R is coherent (resp. stably coherent ) if π 0 (R) is coherent (resp. stably coherent) as a commutative ring and each π i (R), i ≥ 0 is finitely presented as a π 0 (R)-module. Remark 6.3 (Characterization of almost perfect complexes). Suppose R is a coherent E ∞ -ring. In this case, a bounded-below R-module spectrum M ∈ Mod(R) is almost perfect if and only if the homotopy groups π i (M ) are finitely presented R-modules, cf. [30,Prop. 7.2.4.17]. In particular, APerf(R) acquires a t-structure (by restriction from Mod(R)) whose heart is the category of finitely presented discrete π 0 (R)-modules; this t-structure is right-bounded and left-complete.
Let R be a coherent E ∞ -ring. There is similarly a characterization of almost finitely presented E ∞ -R-algebras. Given any connective E ∞ -ring R, recall [30,Sec. 7.2.4] that a connective E ∞ -R-algebra R ′ is said to be almost finitely presented if, for each n, the truncation τ ≤n R ′ defines a compact object of the ∞-category of connective, n-truncated E ∞ -R-algebras. Proposition 6.4 (Characterization of almost finitely presented algebras). Let R be a stably coherent E ∞ -ring. Let R ′ be a connective E ∞ -R-algebra. Then the following are equivalent: (1) R ′ is almost finitely presented.
(2) The ring π 0 (R ′ ) is finitely presented as a π 0 (R)-algebra. Moreover, R ′ is coherent as an E ∞ -ring. In this case, R ′ is stably coherent as well.
Remark 6.5. Coherence will be useful for us in the following situation. Let R be an E ∞ -O Kalgebra which is coherent. Suppose that R[1/π] is discrete. Then for each i > 0, the homotopy group π i (R) is ≤ π ∞ -isogenous to zero. This follows because, for i > 0, π i (R) is a finitely presented module over π 0 (R) and π i (R)[1/π] = 0 by assumption. We will need the following strengthening of the above example, which appears in work of Fujiwara-Gabber-Kato [ [28] for a survey.
Theorem 6.7. Let K be a complete nonarchimedean field. Let R be a finitely presented algebra over the π-completed polynomial ring O K T 1 , . . . , T n . Then: (1) R is coherent.
(2) Given any finitely generated R-module M , the π-power torsion submodule is finitely generated. Any π-torsion-free finitely generated module is finitely presented. (3) The map from R to its π-adic completion (which is also the derived π-adic completion by (2)), R →R, is flat.
In particular, the result implies that for finitely presented algebras over O K T 1 , . . . , T n and finitely generated modules, there is no distinction between classical and derived completion. (1) An O K -algebra A 0 is topologically of finite presentation if A 0 is a quotient of some O K T 1 , . . . , T n by a finitely generated ideal; if A 0 is torsion-free, it suffices that A 0 is a quotient of some O K T 1 , . . . , T n . Let Alg tfp OK denote the category of O K -algebras which are topologically of finite presentation. We let Alg tfp,♭ OK ⊂ Alg tfp OK be the subcategory of those algebras which are flat (i.e., torsion-free) over O K .
(2) An K-affinoid algebra is a K-algebra which is a quotient of the Tate algebra T n = K T 1 , . . . , T n def = (O K T 1 , . . . , T n )[1/π]. Let Affinoid K denote the category of K-affinoid algebras. We have an essentially surjective functor Alg tfp OK → Affinoid K given by tensoring with K.
The categories Alg tfp OK , Affinoid K admit finite colimits via the completed tensor products; the natural functor Alg tfp OK → Affinoid K preserves them. Definition 6.9 (Flat morphisms in Alg tfp,♭ OK , Affinoid K ). We define a morphism in Alg tfp,♭ OK , Affinoid K to be flat (resp. faithfully flat ) if it is flat (resp. faithfully flat) as a map of ordinary rings. This condition is stable under (completed) base-change: (1) In Alg tfp,♭ OK , a map A 0 → B 0 is flat (resp. faithfully flat) if and only if it is flat (resp. faithfully flat) modulo each power of π. In the case of flatness, this follows because the flatness condition can be tested on coherent (discrete) A 0 -modules, and these are automatically π-complete. Faithful flatness is implied by flatness and universal descent, so the claim for faithful flatness follows from Lemma 6.15 below.
(2) The statement that flat (resp. faithfully flat) maps in Affinoid K are stable under basechange follows from the existence of flat (resp. faithfully flat) formal models (Theorem 6.11 below).
One can ask when a map in Alg tfp,♭ OK has flat generic fiber. It turns out that the above two are the essential cases. To see this, we will use the flattening results of [14], after the work of Raynaud-Gruson [35] in the case of schemes. See also [3,Sec. 5.8] for an account. The last assertion (of faithful flatness rather than flatness) follows from [3, Prop. 5.5.10].
/ / X such that: (1) X ′ → X is a formal blow-up along an open, finitely generated ideal We can restate the above result purely in terms of schemes (rather than formal schemes), and in the faithfully flat case, as follows. Proposition 6.12. Let A 0 → B 0 be a map in Alg tfp,♭ OK such that A 0 [1/π] → B 0 [1/π] is faithfully flat. Then there exists a finitely presented A 0 -algebra A ′ 0 , flat over O K , such that: This follows from Theorem 6.11: instead of taking the formal blow-up along the relevant (finitely generated) ideal I ⊂ A 0 , we take the actual scheme-theoretic blowup X ′ → X = Spec(A 0 ); note that X ′ [1/π] → X[1/π] is an isomorphism. Since I is finitely generated, X ′ → X is a finite type map of schemes; since everything is flat over O K , X ′ → X is finitely presented (Theorem 6.7). If we let Y ′ be the strict transform of Y = Spec(B 0 ), then the previous result states that Y ′ → X ′ defines a map of O K -schemes which is flat modulo π. We can then take A ′ 0 to be an affine cover of X ′ .
Note that A 0 [1/π] → A ′ 0 [1/π] is a Zariski cover, verifying (2). Letting B ′♭ 0 = B ′ 0 / {π ∞ − torsion}, the assertions of Theorem 6.11 give that A ′ 0 → B ′♭ 0 becomes faithfully flat modulo any power of π. Moreover, it is faithfully flat after inverting π by base-change, by our assumptions. This implies that A ′ 0 → B ′♭ 0 is faithfully flat. Finally, to see that A 0 → A ′ 0 is a universal descent morphism, we argue via Theorem 5.5: indeed, X ′ → X is a proper, finitely presented and surjective morphism and hence a v-cover. It follows that A 0 → A ′ 0 is a finitely presented v-cover and hence a universal descent morphism. Lemma 6.13 (Discreteness up to quasi-isogeny criterion). Let A 0 ∈ Alg tfp OK . Let A ′ 0 be a finitely presented A 0 -algebra such that: (1) A 0 → A ′ 0 is a universal descent morphism.
is quasi-isogenous to a flat, discrete A ′ 0 -module. Then each (derived) tensor power M ⊗n ∈ Mod(A 0 ), n ≥ 0 is quasiisogenous to a discrete module.
Proof. By Proposition 6.12, there exists a finitely presented, universal descent, discrete A 0 -algebra A ′ 0 such that A 0 [1/π] → A ′ 0 [1/π] is flat and such that B ′♭ 0 def = π 0 (B 0 ⊗ A0 A ′ 0 )/ {π ∞ − torsion} is faithfully flat over A ′ 0 . Then the (derived) tensor product B 0 ⊗ A0 A ′ 0 has the structure of an almost finitely presented E ∞ -B 0 -algebra which becomes discrete after inverting π. By coherence (Theorem 6.7 and Remark 6.5), B 0 ⊗ A0 A ′ 0 is quasi-isogenous to the flat, discrete A ′ 0 -module B ′♭ 0 : in particular, the higher homotopy groups and the torsion in π 0 are bounded torsion. The claim (2) now follows from Lemma 6.13. Finally, we need to verify that A 0 → B 0 is universal descent. By assumption, A 0 → A ′ 0 is a universal descent morphism. Since the source is π-complete, it follows that A 0 → A ′ 0 is a universal descent morphism (Lemma 6.15). Therefore, it suffices to show (by two-out-of-three as in Remark 5.2) that A ′ 0 → B ′♭ 0 is a universal descent morphism: indeed, then A 0 → B ′♭ 0 is universal descent and hence so is A 0 → B 0 . But A ′ 0 → B ′♭ 0 is faithfully flat and finitely presented modulo π, so the claim again follows from Lemma 6.15. Lemma 6.15. Let R → S be a map of π-torsion-free O K -algebras. Suppose R is π-adically complete. Then: (1) R → S is a universal descent morphism if and only if R → S is a universal descent morphism. (2) Suppose R/π → S/π is faithfully flat and finitely presented. Then R → S is a universal descent morphism.
Proof. Let I = fib(R → S) in Mod(R), so we have a canonical map f : I → R. Then R → S is a universal descent morphism if and only if f ⊗n : I ⊗n → R is nullhomotopic (in Mod(R)) for n ≫ 0. Similarly, R → S is universal descent if and only if ( I) ⊗n → R is nullhomotopic for n ≫ 0. These conditions are equivalent because I ⊗n , ( I) ⊗n have the same π-completion and R is π-complete. This proves (1). Now suppose R/π → S/π is faithfully flat and finitely presented. By [31,Lemma D.3.3.7], the map f ⊗n vanishes after base-change to any quotient O K /π k for n ≥ 2; indeed, I ⊗n /π k is the (−n)-shift of a flat discrete R/π k -module which is at most countably presented, whence π 0 Hom R/π k (I ⊗n /π k , R/π k ) = 0 by loc. cit.. Therefore, the map f ⊗2 : I ⊗2 → R is divisible by any power of π. Consider the composable maps π 0 Hom R ( I ⊗4 , I ⊗4 ) f ⊗2 •· − −−− → π 0 Hom R ( I ⊗4 , I ⊗2 ) f ⊗2 •· − −−− → π 0 Hom R ( I ⊗4 , R), each of which is divisible by any power of π; it follows from Lemma 2.28 that the composite is zero. Therefore, f ⊗4 is nullhomotopic as desired. OK such that A 0 is carried to B 0 . ThenČech descent of APerf(·) along A → B follows from Proposition 6.14 and Theorem 5.8. From this, it follows by flatness that the subcategories APerf(·) [a,b] form sheaves on on Affinoid K for each a ≤ b; these sheaves are necessarily hypercomplete since they take values in truncated ∞-categories. Taking the limit in a, b, we find that APerf(·) is a hypercomplete sheaf.

Flat descent
In this section, we prove our main faithfully flat descent theorem (Theorem 7.8). Let R be a connective E ∞ -ring with a finitely generated ideal I ⊂ π 0 (R). We need the following definition, which generalizes Definition 1.5. For an E ∞ -R-algebra S, we denote by Mod(S) I−tors the full subcategory of Mod(S) spanned by the I-torsion modules.
for Perf ≥0 now follows by taking the intersection of APerf ≥0 and Perf. Finally, Perf [a,b] is given by those objects of Perf which belong to Perf ≥a and whose dual belongs to Perf ≥−b , so this is also a local condition.