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Fr\'echet Modules and Descent

We study several aspects of the study of Ind-Banach modules over Banach rings thereby synthesizing some aspects of homological algebra and functional analysis. This includes a study of nuclear modules and of modules which are flat with respect to the projective tensor product. We also study metrizable and Fr\'{e}chet Ind-Banach modules. We give explicit descriptions of projective limits of Banach rings as ind-objects. We study exactness properties of projective tensor product with respect to kernels and countable products. As applications, we describe a theory of quasi-coherent modules in Banach algebraic geometry. We prove descent theorems for quasi-coherent modules in various analytic and arithmetic contexts.

This article has no associated data in a data repository. The authors certify that they have no affiliations with or involvement in any organization or entity with any financial interest, or non-financial interest in the subject matter or materials discussed in this manuscript.

Introduction
The use of categorical and homological techniques in functional analysis has a long and complicated history which we can not adequately summarize here. This includes work of Helemskii [20], Meyer [25] [24], Cigler, Losert and Michor [14], Paugam [29], Taylor [38], Wengenroth [40] and others. We follow the approach of using the homological algebra of quasi-abelian categories of Prosmans and Schneiders [37], [34] generalized from the functional analysis of Banach and Ind-Banach spaces over complex numbers to general Banach rings.
Grothendieck developed the theory of nuclearity for topological vector spaces over C. In [34] these ideas are carried over to the closely related setting of ind-Banach spaces over C. We were able to prove analogues of these results in the setting of Ind-Banach modules over arbitrary Banach rings R. The definition of nuclearity we use is in Definition 4.9 and an equivalent characterization in Remark 4.15. Not having Hilbert space techniques available, we were unable to prove that subspaces and quotients of nuclear maps are nuclear. However, we can prove many other standard "permanence properties" of nuclearity. We discuss countable products and coproducts in Corollary 5.22 and a two out of three rule for strict short exact sequences in Lemma 4.11 and the projective tensor product of nuclear spaces in Lemma 4. 16. A different approach to nuclearity which works in both the Archimedean and non-Archimedean settings could be inspired by Schneider's notion (see [36]) of compact morphisms between Banach spaces. Corollary 5.9 proves that nuclearity also ensures an interesting interaction with products of dual spaces. Following work of Prosmans and Schneiders we prove that nuclear spaces can be written in certain canonical ways in Lemmas 4.17 and 4.18. We define metrizability in Definition 5.5. Important examples of metrizable modules are Banach or Fréchet modules. Notice that as nuclearity of an object implies it is flat for the projective tensor product (Lemma 4.19), one may ask what condition on an object might ensure that the projective tensor product with it commutes with countable products. This turns out to be a complete characterization of metrizability as proven in Lemmas 5. 18 and 5.19. Therefore, in combination the properties of nuclearty and metrizability for an object imply that the projective tensor product with it commutes with countable limits (Lemma 5.18). Banach algebraic geometry and its derived versions is an approach to analytic geometry which uses geometry relative to categories of Banach spaces (or modules) in the same way that usual algebraic geometry is based on categories of abelian groups. In particular, this philosophy applies to rigid analytic geometry [10], overconvergent rigid geometry [7] and Stein geometry ( [8], [31], [5]) and in these articles it was shown that the homotopy monomorphism topology specializes to conventional ones in special cases. There are also projects on derived analytic geometry [9] and analytic F 1 -geometry [11]. Most of the constructions in this article are based on an arbitrary Banach ring R. If R is a nonarchimedean Banach ring (see Definition 3.25), this entire article can be separately read in two different versions, depending on whether one considers the categories Ind(Ban R ) of all Banach modules or Ind(Ban na R ) of non-archimedean Banach modules. Therefore, in this case, notation such as symbols for limits, colimits, products and coproducts, can sometimes take on two different meanings. We have chosen to write everything with the default version being of the archimedean version. This has the appealing aspect of being completely the same for any R, archimedean or not. In the case that R is non-archimedean, the reader who wants to work in a non-archimedan context should replace all limits and colimits in the category Ind(Ban R ) by those in the category Ind(Ban na R ). All the proofs go through in a similar way. Given a union of subsets one often wants to describe modules on the union in terms of modules on the components together with gluing data. Our main descent results can be found in Theorem 7.9. To formulate this we introduce a generalization of a coherent module called a quasi-coherent module. This notion was also needed in [10,7,8] where some properties of quasi-coherent modules were studied, and in this article we extend that study. We relate our results to Tate's acyclicity theorem in Lemma 7.11 and modules on Stein covers.

Notation
We use the notation lim instead of lim ← and colim for lim → . The letter R denotes a general Banach ring, defined in Definition 3. 22. We denote categorical products by ∏ and categorical coproducts by ∐, it should be clear in what category these take place, usually it is sufficient to consider them in the category Ind(Ban Z ). Given an object A in Comm(Ind(Ban R )), we use spec(A) to just denote the same object in the opposite category. As usual, Z p denotes the p-adic integers, unless we are scaling the norm on Z with a real number in the sense of Definition 3.33, this should be clear from the context. For an abelian group A we use A × to denote A − {0}.

Some Category Theory and Its Uses in Functional Analysis and Geometry
3.1. Relative Algebra and Homological Algebra.
Definition 3.1. In an additive category with kernels and cokernels, a morphism f ∶ E → F is called strict if the induced morphism is an isomorphism. Here im(f ) is the kernel of the canonical map F → coker(f ), and coim(f ) is the cokernel of the canonical map ker(f ) → E. An object P is projective if for all strict epimorphisms E → F the associated map Hom(P, E) → Hom(P, F ) is onto. An object I is injective if for all strict monomorphisms E → F the associated map Hom(F, I) → Hom(E, I) is onto. If the category is equipped with a unital symmetric monoidal structure ⊗ then an object F is called flat if the functor (−)⊗F preserves strict monomorphisms.
Consider a unital, closed symmetric monoidal category (C, ⊗, e = id C ) with finite limits and colimits (more details in [10]). The internal Hom in C will be denoted Hom as it should be clear from the context what category we are working in. We will always suppress the commutativity, unitality, and associativity natural transformations from the notation. It is easy to see the following lemma. Recall [39] the category Comm(C) of commutative unital rings with respect to (C, ⊗, e = id C ) and for any S ∈ Comm(C) the category Mod(S) of S-modules internal to (C, ⊗, e = id C ). If we further assume that (C, ⊗, e = id C ) has countable coproducts then an important construction is the symmetric ring construction which is a left adjoint to the forgetful functor Comm(C) → C 3.2. Quasi-abelian categories.
Definition 3.3. Let E be an additive category with kernels and cokernels. We say that E is quasi-abelian if it satisfies the following two conditions: • In a cartesian square if f is a strict epimorphism then f ′ is a strict epimorphism. • In a co-cartesian square f is a strict monomorphism then f ′ is a strict monomorphism.
A quasi-abelian category is a category where the strict monomorphisms and strict epimorphisms satisfy the conditions of a Quillen exact category. It may be useful to allow for more general Quillen exact structures (see [11]) for instance using short exact sequences that split over R but in this work we avoid this. Here a sequence in a quasi-abelian category is strictly exact when the image of the first map is isomorphic to the kernel of the second, and e ′ is strict.
Lemma 3.5. Let C and D be quasi-abelian categories. Let L ∶ C → D be any functor with a right adjoint R ∶ D → C. Then L preserves strict epimorphisms and R preserves strict monomorphisms.
Proof. Let f ∶ V → W be a strict epimorphism in C. Then of course L(f ) is an epimorphism. Because f is a strict epimorphism, we have W = coker(ker(f ) → V ). Therefore, since left adjoints preserve cokernels, L(f ) expresses L(W ) as the cokernel of the morphism L(ker(f )) → L(V ). The second statement is proven in a similar way.
Definition 3. 6. Let E be a quasi-abelian category. Let K(E) be its homotopy category. A morphism in K(E) is called a strict quasi-isomorphism if its mapping cone is strictly exact.
Definition 3.7. Let E be an additive category with kernels and cokernels. An object I is called injective if the functor E ↦ Hom(E, I) is exact, i.e., for any strict monomorphism u ∶ E → F , the induced map Hom(F, I) → Hom(E, I) is surjective. Dually, P is called projective if the functor E ↦ Hom(P, E) is exact, i.e., for any strict epimorphism u ∶ E → F , the associated map Hom(P, E) → Hom(P, F ) is surjective.
Definition 3.8. A quasi-abelian category E has enough projectives if for any object E there is a strict epimorphism P → E where P is projective. A quasi-abelian category E has enough injectives if for any object E there is a strict monomorphism E → I where I is injective.
Definition 3.9. Let E be an additive category. An object E is called: for any small family (F i ) i∈I of E whenever the coproduct on the left exists • tiny, if (3.3) Hom(E, colim for any filtering inductive system I → E whenever the colimit on the left exists. Definition 3.10. Let E be a quasi-abelian category. A strict generating set of E is a subset G of Ob(E) such that for any monomorphism with G ∈ G which does not factor through m.
Definition 3.11. A quasi-abelian category is quasi-elementary (resp. elementary) if it is cocomplete and has a small strictly generating set of small (resp. tiny) projective objects.
For abelian categories quasi-elementary is equivalent to elementary. We will freely use the following proposition which comes from Proposition 2.1.18 of [37] Proposition 3.12. Let C be a small, closed, symmetric monoidal, quasi-abelian category and R ∈ Comm(C). Then Mod(R) is elementary if C is, R⊗P is tiny in Mod(R) whenever P is tiny in C. If G is a strict generating set of C then {R⊗G G ∈ G} is a strict generating set of Mod(R).

3.3.
Ind-Categories and Ind-Categories of quasi-abelian categories. Recall that for any category C we can define its ind-completion. Definition 3.13. Let C be a category. An ind-completion of C is a category D with a functor i ∶ C → D, such that D is closed under filtered colimits, and the functor i is initial with respect to functors into categories closed under filtered colimits.
Lemma 3.14. Let C be a category. Its ind-completion exists and can be realized as the full subcategory of the category Pr(C) = Fun(C op , Set) whose objects are filtered colimits of representable functors (note that the category of presheaves is cocomplete).
We will denote the ind-completion of C by Ind(C). Given two presentations of objects E ≅ "colim i∈I "E i and F ≅ "colim j∈J "F j , we have a canonical isomorphism Remark 3.15. Therefore, a morphism can be represented by a functor α ∶ I → J and for each i ∈ I an element of Hom(E i , F α(i) ) giving a natural transformation E → F ○ α.
One way of getting elementary quasi-abelian categories is by looking at ind-completions of quasi-abelian categories (2.1.17 in [37]): Theorem 3. 16. Let E be a small quasi-abelian category with enough projective objects. Then, Ind(E) is an elementary quasi-abelian category.
The following is 2.1.19 in [37]: Proposition 3.17. Let E be a small, closed, symmetric monoidal, quasi-abelian category. The category Ind(E) has a canonical closed symmetric monoidal structure extending that on E. Hence, if E has enough projectives, Ind(E) is a closed symmetric monoidal elementary quasi-abelian category.
Proof. The extension is given as follows: where (G j ) j∈J is a small family of elements of G.
We assume the reader is familiar with the notions of a family of injective objects with respect to a functor between quasi-abelian categories [37]. In this section we recall how to derive the inverse limit functor in quasi-abelian categories.
Given a functor V ∶ I → C the Roos complex of V is of the form where the product is over all composable sequences of n morphisms in I. The differential Usual abstract nonsense arguments show the existence of a derived functor Rlim i∈I .
Definition 3.19. The lim i∈I -acyclic objects are objects V of C I satisfying Rlim object V of C I will be called Roos-acyclic if the differentials in the Roos complex are strict and the cohomology of the Roos complex is concentrated in degree zero.
Inverse limits have an explicit derived functor because of the following proposition of Prosmans [33].
Proposition 3.20. Let I be a small category and C a quasi-abelian category with exact products. Then the family of objects in C I which are Roos-acyclic form a lim i∈I -acyclic family. As a result, the functor lim i∈I ∶ C I → C is right derivable and for any object V ∈ C I , we have an isomorphism where the right hand side is the Roos complex of V . The family of lim i∈I -acyclic objects for the functor lim i∈I ∶ C I → C form a family of injectives relative to this functor (a concept appearing in [37]).
Because of the explicit formula of the Roos complex, notice that Corollary 3.21. Let I be a small category and C a quasi-abelian closed symmetric monoidal category with exact products. If W is flat in C and W ⊗(−) commutes with products in C then the natural morphism is an isomorphism. In particular, if V ∈ C I is lim i∈I -acyclic then so is W ⊗V and the canonical morphism is an isomorphism.
We will use this material again in Lemma 6.10.
3.4. Relative Geometry. Just as algebraic geometry is "built" from the theory of commutative rings and their modules, much work on other kinds of geometry and topology is based on commutative monoids and their modules internal to general symmetric monoidal categories (C, ⊗, e), for instance see [39]. In our approach we also ask that they be equipped with compatible Quillen exact structures. The category most important for us is the quasiabelian example of Ind-Banach modules over a Banach ring together with its projective tensor product and its applications to analytic and arithmetic geometry. An important class of morphisms between "affine schemes" in relative geometry are opposite to those morphisms Such a morphism will be called a homotopy epimorphism. We use that terminology because this notion actually is equivalent to the general model or infinity-category notion of a homotopy epimorphism as found in work of Toën and Vezzosi (as used in homotopical or derived algebraic geometry). However, other sources call this a stably flat morphism [26], an isocohomological morphism [25] or a homological epimorphism [16], [13]. It appears in homotopy theory [18], representation theory, and algebra under different names. Practically, the only way we know to prove that a morphism is a homotopy epimorphism is to resolve B by projective and flat A-modules in a clever way allowing for computation of the derived projective tensor product. In particular, we must prove that the resolution remains a resolution after applying the projective tensor product with B. A particular example of a homotopy epimorphism is a flat epimorphism, however there are other examples which will be shown in the next section.
3.5. Banach Rings and Banach Modules. Definition 3.22. By a complete normed (or Banach) ring we mean a commutative ring with identity R equipped with a function, The category of Banach rings has as morphisms those ring homomorphisms R → S such that there exists a constant C > 0 such that φ(a) S ≤ C a R for all a ∈ R, in other words bounded ring homomorphisms.
The initial Banach ring is the integers Z equipped with the standard absolute value as norm.
Definition 3.23. Let (R, ⋅ R ) be a Banach ring. A Banach module over R is an R-module M equipped with a function ⋅ M ∶ M → R ≥0 such that for any m, n ∈ M and a ∈ R: Example 3.24. Any abelian group or ring can be considered a Banach ring by equipping it with the trivial norm which assigns 0 to the zero element and 1 for each non-zero element. We use notation such as for example Z triv for the integers equipped with the trivial norm. If M is a module over a Banach ring R, we can make M into a Banach module by equipping it with the trivial norm.
Definition 3.25. A Banach ring or a Banach module over a Banach ring is called nonarchimedean if its semi-norm obeys the strong triangle inequality: for any two elements v, w we have v + w ≤ max{ v , w }. The category of Banach modules with bounded morphisms is denoted by Ban R . If R is non-archimedean Ban na R denotes the category of non-archimedean Banach modules with bounded morphisms.
Lemma 3.28. For any Banach ring R, R is projective as a Banach R-module. Lemma 3.29. For any projective R-module, P , and any real number r > 0, P r is also projective.
Definition 3.30. Given M, N ∈ Ban R we define M⊗ R N as the (separated) completion of M ⊗ R N with respect to the semi-norm Similarly, if R is non-archimedean, given M, N ∈ Ban na R we define M⊗ na R N as the (separated) completion of M ⊗ R N with respect to the semi-norm The internal Hom in these categories is denoted by Hom R (V, W ) and given by the Banach space whose underlying vector space is just the bounded R-linear maps

The categories
Ban R and Ban na R are both closed symmetric monoidal when equipped with these projective tensor product with unit object given by R.
Definition 3.31. The category Ban ≤1 R is defined to have the same objects as Ban R . The morphisms are the linear maps with norm less than or equal to one (these are called nonexpanding or sometimes just contracting).
This defines a closed symmetric monoidal category with the same internal hom and tensor product and as before it has two versions (one of which exists only when R is nonarchimedean). Infinite products and coproducts in Ban ≤1 R exist even though they do not exist in Ban R . In the archimedean case (see page 63 of [20] If R is non-archimedean and we choose to work in the non-archimedean case, they can be computed as in [17]: the product ∏ ≤1 i∈I V i of a collection {V i } i∈I in Ban ≤1 R is given by General limits (resp. colimits) are constructed out of kernels and products (resp. cokernels and coproducts) in the usual way. Finite limits (resp. finite colimits) in Ban ≤1 R agree with those in Ban R .
R . Then observe that the natural morphism   Proof. There is a canonical strict epimorphism ∐  Proof. Using Lemma 3.34, we can complement P and Q by modules P ′ and Q ′ in order to conclude that there is a module and so the lemma follows from another application of Lemma 3.34. Proof. Let P be a projective in Ban R . There is a canonical strict epimorphism ∐ p∈P × ≤1 R p → P discussed in [10]. As usual, it splits and so ∐ p∈P × ≤1 R p is coproduct (in Ban R ) of the kernel and P . Hence P is flat by Lemma 3.2.
The proof of the following is obvious from the definitions.
Lemma 3.37. Any finite coproduct of projective objects in Ban R is projective in Ban R .
Lemma 3.38. [17] A filtered colimit of strict, short exact sequences in Ban ≤1 R is a strict short exact sequence.
Proof. See Proposition 1 on page 69 of [17]. Lemma 3.39. If V → W is a strict epimorphism and P is projective then the corresponding morphism Hom(P, V ) → Hom(P, W ) is a strict epimorphism.
Proof. By Proposition 1.3.23 of [37] it is enough to show that for any projective Q, that Hom(Q, Hom(P, V )) → Hom(Q, Hom(P, W )) is surjective. This follows immediately from adjunction and from Lemma 3.35.  Proof. Let π ∶ V → W be a strict epimorphism and let f ∶ P → W be any morphism. Fix ǫ > 0. By Lemma 3.39, for each projective P s we get a strict epimorphism Hom(P s , π) ∶ Hom(P s , V ) → Hom(P s , W ).
Let f s be the restriction of f to P s so f s ≤ f . Using the strict epimorphism property (see the characterizations in the appendix of [10]), choose for each s a liftf s ∈ Hom(P s , V ) of f s such that f s ≤ f s + ǫ ≤ f + ǫ. As their norms are bounded independent of s, thef s assemble into a morphismf ∶ P → V inducing f . Proof. The proof is exactly as in [10].
Proof. It is enough to show that it induces an isomorphism of sets for any real number r ≥ 1. The canonical morphism identifies the left hand side with which agrees with the right hand side.
Proof. This is automatic from the definitions and Lemma 3.45.  Remark 3.48. If R is non-archimedean, all of this subsection goes through for Ban na R in place of Ban R . Just as finitely presentable rings play an important role in algebraic geometry, in Banach algebraic geometry over R, the nice objects of study are quotients of the above disk algebras by ideals, equipped with the quotient Banach structure. As the category of these affinoid algebras is not closed under filtered limits or colimits, it is natural to introduce also Stein and dagger algebras in Section 6 and perhaps even more general limits and colimits like quasi-Stein, Stein-dagger, and quasi-Stein-dagger, etc. Remark 3.49. Let R be a non-zero Banach ring with multiplicative norm. It is automatically an integral domain. Let S be a multiplicative subset, and equip the localization S −1 R of R with the completion of the norm r s = r s . Then for any Banach ring T , the map R → S −1 R identifies Hom( S −1 R, T ) with the bounded ring morphisms R → T sending S to invertible elements and so categorically, the map R → S −1 R is an epimorphism in the category of Banach rings, equivalently S −1 R⊗ R S −1 R ≅ S −1 R.
3.6. Geometric and Arithmetic Examples of Homotopy Epimorphisms. In this subsection we discuss several examples of homotopy epimorphisms and derived projective tensor products. These all have a geometric meaning in terms of the Berkovich or Huber spectrum of Z. As we are working over Z in this subsection, it takes place entirely in the archimedean context. We will return to looking at these in terms of covers of spec(Z) and descent in future work.
Observation 3.50. We have Q p⊗Z R = {0} = Z p⊗Z R for any prime p and for distinct primes p and q, we have Q p⊗Z Q q = {0} = Z p⊗Z Z q . As a consequence, Q p and Z p are not flat with respect to the completed tensor product over Z.
Proof. In Z p⊗Z R the element 1⊗1 can be written as p n⊗ p −n which has norm p −2n for any n. In Z p⊗Z Z q choose for each n, integers a n and b n with a n p n + b n q n = 1. Then 1⊗1 can be written as a n p n⊗ 1 + 1⊗b n q n which has norm less than or equal to p −n + q −n . Letting n go to infinity we see that in both Banach rings, 1 = 1⊗1 has norm zero and hence vanishes and so these rings are the zero ring. Applying the functor Z p⊗Z (−) to the strict short exact sequence Z → R → S 1 gives Z p → {0} → Z p⊗Z S 1 = {0} and so Z p is not flat. The proofs for the fraction fields with their obvious Banach structures are similar.
Using the resolutions we develop later, its easy to see that these rings are also orthogonal on the derived level. The lack of flatness with respect to the projective tensor product is similar to the known problem in analytic geometry that certain morphisms A → B of various Banach, Fréchet, or bornological algebras corresponding to the restriction of spaces of functions over various "open" sets do not exhibit B as a flat module with respect to the completed tensor product over A [10,11].
Example 3.51. Consider the usual Tate algebra A = Q p ⟨x⟩ of non-archimedean geometry. We can think of it as a Banach-algebraic version of the . However, A → A V is a homotopy epimorphism as proven in [10].
This explains our preference for using homotopy epimorphisms instead of flat epimorphisms. The analogous issue does not arise in finite dimensional algebraic or differential geometry in the standard topologies. Let R be a Banach ring. Let V be a finite rank, free Banach module over R. Let S R (V ) be the symmetric algebra of V in the category Ind(Ban R ), a free object in Comm(Ind(Ban R )). Note that as a bornological ring, A is a polynomial algebra over R with number of generators equal to the rank of V . Consider the algebra S ≤1 R (V ), a free object in Comm(Ban ≤1 R ). This is a Banach ring which can be explicitly described as the subring R{x 1 , . . . , that even when R is a non-archimedean ring or field, we can and will use this definition, because we are not restricting our attention to non-archimedean modules. The idea of writing Z p in terms of disk algebras over Z goes back to F. Paugam [30]. We use his idea in the following lemma which uses the disk algebras (Definition 3.47). We show here that one can think of Z p as a sort of archimedean type rational localization of Z. The symmetric ring construction in the contracting category [10] works equally well to define infinite dimensional disk algebras.
Definition 3.52. Let Z ǫ p be the completion of Z with respect to the norm ap n ǫ p = p −nǫ for p not dividing a ∈ Z, n ≥ 0 and 0 < ǫ < ∞ and r = p −ǫ .
Observation 3.53. For distinct primes p ≠ q and 0 < ǫ, δ < ∞ we have Z ǫ p⊗Z Z δ q = {1} similarly to Observation 3.50. On the other hand the tensor seminorm of Z = Z ⊗ Z Z ⊂ Z ǫ p ⊗ Z Z δ p is just the norm ap n = min{p −nǫ , p −nδ } for p not dividing a ∈ Z. Therefore, Z ǫ p⊗Z Z δ p is the completion of Z with respect to the norm ap n ǫ p = p −nζ for p not dividing a ∈ Z and ζ = max{ǫ, δ}.
Lemma 3.54. For 0 < r < 1, there is a strict short exact sequence i=0 a i p i with a i between 0 and p − 1 has norm ∑ ∞ i=0 a i r i . This is the usual Z p as an abstract ring. In particular for r = p −1 this gives the usual p-adic norm on Z 1 p = Z p . Proof. In order to see that the multiplication by and in fact we can solve for For every prime p there are isomorphisms of normed rings where x = r and the right hand side has the ǫ p norm. In order to explain this, given a polynomial Therefore the morphism Z{ x r } → Z ǫ p is strict. By computing order by order modulo powers of p with any polynomial ∑ m i=n a i x i where ∑ m i=n a i p i = 0, one finds that this element must be in the ideal (x − p). If the polynomial f maps to bp s where p does not divide b then bp s = ∑ m i=n a i p i and so bp s ǫ p ≤ f . We apply the completion functor to get the desired represents the subset of Berkovich points which send p to [0, r]. Example 3.55. We can compute S 1⊗ L Z Z p where S 1 = R Z using the above projective resolution of Z p . It is a non-zero Banach abelian group K sitting in degree −1 including for example the element 1 p ∑ ∞ i=0 (x p) i which has norm 1 p−1 and is in the kernel K of the strict epimorphism S 1 {px} Remark 3.56. This geometric perspective can be useful, for instance, one could define the p-adic completion of a Banach ring R as R{px} (x − p). There are also interesting new rings to define such as the following Fréchet version of the p-adic integers. For instance we can consider the functions on the open disk of radius 1 p over spec(Z), lim r<1 p Z{ x r } in place of Z{px} in the role it plays in Lemma 3.54.
There are bounded morphisms Z triv → Z p and homotopy epimorphisms Z → Z † p → Z p . As far as the Fréchet version, this will appear in future work but for now we show: Lemma 3.58. The natural map Z → Z † p is a homotopy epimorphism. Proof. Let p be a prime and let R be any Banach ring without p-torsion and with multiplicative norm and let s be any real with 1 < s < p. Further assume that p R s > 1. Consider the bounded morphism R{sx} and in fact we can solve for Therefore, Therefore, for 1 < u < s < p we have the strict short exact sequence and it becomes a strict short exact sequence after applying the functor (−)⊗ Z Z log p u p . Passing to colimits over u < s < p we find that we have a strict short exact sequence which is a projective resolution. Furthermore, after applying the functor (−)⊗ Z Z † p the result is the strict short exact sequence given by the colimits over u < s < p of we find a strict short exact sequence

3.7.
Bornological and Ind-Banach Modules. Let R be a Banach ring as defined in Definition 3.22. A bornological module over R is a pair consisting of an R-module M together with a bornology on the set M as in Definition 3.36 of [7] such that the structure morphisms for addition and action of R are bounded. The morphisms of bornological modules are bounded R-linear maps. The homological properties of bornological spaces over C were discussed in [32]. We define the full subcategory CBorn R as those bornological modules for which there is an increasing union of subsets, each of which has the structure of an object of Ban R and the inclusion of the subsets in M and in one another are all bounded morphisms. This category is equivalent to the subcategory of essentially monomorphic objects in Ind(Ban R ).
For more on this category see [7] and [8]. The importance of bornological spaces in complex geometry was studied by Houzel [19]. Given an object A ∈ Comm(Ind(Ban R )), the category Mod(A) shares all the nice properties of the category Ind(Ban R ). We just remark here that the completed projective tensor product is defined by Lemma 3.59. The category CBorn R is closed, symmetric monoidal, quasi-abelian, complete and co-complete. It has enough flat projectives.
Lemma 3.60. Direct products in CBorn R commute with cokernels.
Since CBorn R has enough projectives, it has exact products by Proposition 1.4.5 of [37]. Therefore ∏ W i → ∏ C i is a strict epimorphism as it is the product of strict epimorphisms. The kernel is ∏ V i so we are done.
By Proposition 3.17 we have: Lemma 3.61. If R is a Banach ring the categories Ind(Ban R ) (or Ind(Ban na R ) for R nonarchimedean) is a closed, symmetric monoidal, complete and co-complete elementary quasiabelian category.
Definition 3.62. A Fréchet module over R is an object of Ind(Ban R ) which is a countable limit of a diagram in Ban R . We consider Fréchet modules as a full subcategory of CBorn R .
Note that many function spaces in complex analytic geometry carry natural Fréchet structures or more generally locally convex structures. We would like to relate these to the category Ind(Ban C ). Let Tc be the category of locally convex topological vector spaces over C and Fr the sub-category of Fréchet spaces. Note that both of these categories are quasiabelian but they don't share all of the nice properties of Ind(Ban C ) such as having enough projectives and having a closed symmetric monoidal structure. The following definition is [34] Note that if E is a Banach space then IB(E) = E. The functor IB is in many cases fully faithful ([34] proposition 1.5): We also have that IB has a left adjoint when restricted to the category of complete locally convex vector spaces ([34] proposition 1.6).

Nuclear Modules
4.1. Nuclear Banach Modules. Let C be a closed symmetric monoidal category with monoidal structure ⊗ and unit e. This subsection will be applied only in the case that C is one of four categories. For a general Banach ring we consider Ban R and Ban ≤1 R . Also, if R is non-archimedean, we could still consider those but also in this case we could consider Ban na R and Ban na,≤1 R . These definitions and results will not be applied to the Ind-categories we consider, on the other hand, in sub-section 4.2 we will separately define nulclear objects of Ind-categories and work with objects in Ind-categories in a way that extends the definitions given in this subsection. An object is called nuclear if the identity morphism of this object is nuclear.

From Higgs and Rowe [21]
Lemma 4.2. If a morphism is nuclear then so is its dual. If a morphism is nuclear then so is any pre or post composition with it. The monoidal product of two nuclear morphisms is nuclear.
Proof. The first two statements can be found in Proposition 2.2 of [21]. The statement about the monoidal product can be found in Proposition 2.3 of [21] Lemma 4.3. The following are equivalent [21]: (1) The object V is nuclear.
Lemma 4.4. [21] If an object V is nuclear then its dual is also nuclear. Any nuclear object is reflexive.
preserves strict epimorphisms. If in addition, e is projective, then V is projective.
(1) Since V is nuclear, we can consider the naturally isomorphic functor By Lemma 3.5, this functor preserves strict epimorphisms so we are done.
. Therefore, we can consider the naturally isomorphic functor This preserves strict monomorphisms by Lemma 3.5.
The following Lemma was shown over C in [34]. The following is our version over a Banach ring R. Lemma 4.6. Let V and W be two Banach modules over a Banach ring R and let f ∶ V → W be a nuclear morphism in Ban R . Then there exists a countable set S and a map m ∶ S → R >0 , a nuclear morphism The morphism p is actually nuclear as it can be written as ∑ We have Lemma 4.7. Let V and W be two Banach modules over a Banach ring R and let f ∶ V → W be a nuclear morphism in Ban R . Then there exists a countable set S and a map m Remark 4.8. If R is non-archimedean, all of this subsection goes through for Ban na R in place of Ban R .

Nuclear Ind-Banach Modules.
This subsection is about nuclear objects in Ind(Ban R ) or if R is non-archimedean, about nuclear objects in Ind(Ban na R ). For readability, we suppress the non-archimedean versions, all the statements and proofs in the non-archimedean case are the same, up to the obvious substitutions. Of course in the non-archimedean version, all categorical constructions in Ban ≤1 R are replaced by those in Ban ≤1,na R and⊗ R is replaced bŷ ⊗ na R . As the beginning is more general, we work with a closed symmetric monoidal category C with monoidal structure ⊗ and unit e with finite limits and colimits, but the reader is invited to take C = Ban R , ⊗ =⊗ R and e = R or if R is non-archimedean there is also the option C = Ban na R , ⊗ =⊗ na R and e = R. Definition 4.9. An object V in Ind(C) is called nuclear if for every object W of C the natural morphism is an isomorphism.
Remark 4.10. In general, an object which is nuclear under this definition is not nuclear in the sense of subsection 4.1 applied to the category Ind(C). If we consider an object V of Ind(C) which happens to be in C ⊂ Ind(C) itself then the definitions agree by Lemma 4.3. Later, we find another situation when the two definitions agree in Lemma 5.8.
and V 3 are nuclear then V 2 is as well.
Proof. This follows immediately from the fact that the assumption allows us to identify the strict sequences . Hence both can be completed to strict short exact sequences. Now because the outer terms are identified by assumption, the needed isomorphism also holds for V 2 and hence it is nuclear.  On the other hand, by definition Hom(e, Hom(W j , V i )).
Therefore, the canonical map is an isomorphism and so for any element φ in Hom(W, V ) and for any j ∈ J there exists i ∈ I and a nuclear map φ i,j ∶ W j → V i assigned to it. The collection of the φ i,j define a morphism of inductive systems representing φ.
Lemma 4.13. Given a filtered inductive system W i of C where all the maps in the system are nuclear, then the object of Ind(C) given by "colim i∈I "W i is nuclear in the sense of Definition 4.9.
Proof. Let V be any object of C. Consider the canonical morphism Since V is in C (hence a compact object in Ind(C)) we can equivalently write this as the colimit of the morphisms W i ⊗V ∨ → Hom(V, W i ). Consider any W i → W j in the system corresponding to a non-identity arrow i → j. Since they are nuclear, the precomposition V → W i → W j is also nuclear by Lemma 4.2. Therefore it lies in the image of Hom(e, V ∨ ⊗W j ). This constructs a two-sided inverse Lemma 4.14. If an object W is nuclear in Ind(C) and presented as "colim i∈I "W i for I a filtering ordered set with transition maps φ ij ∶ W i → W j , then for each i ∈ I there exists a j ∈ I with j > i such that morphism φ ij is nuclear.
Proof. Consider Lemma 4.12 in the case V = W = "colim i∈I "W i in the case of identical inductive systems applied to the element id ∈ Hom(W, W ). A representative of the identity is given by a cofinal choice of transition maps. The lemma provides the nuclear maps φ ij representing the identity, which are therefore transition maps in the given presentation of W .  Proof. If we consider Remark 4.15 and choose presentations of X and Y with nuclear transition maps, then the induced monoidal structure presentation of X⊗Y also has nuclear transition maps by Lemma 4.2.
Lemma 4.17. Given I, an infinite filtering ordered set, and a functor I → Ban R such that the corresponding object W = "colim i∈I "W i is nuclear in Ind(Ban R ), there is a filtered category K with the same cardinality of objects and morphisms as I, a functor K → Ban R with corresponding object P = "colim k∈K "P k and an isomorphism W ≅ P such that each Banach space P k is a countable coproduct in Ban ≤1 R of weighted copies of R and the transition functions are nuclear.
Proof. Using Lemma 4.14 we may assume that for each i there is a j > i such that φ ij ∶ W i → W j is nuclear. Define Using Lemma 4.6 we can decompose each such morphism φ ij into c ij ○ p ij ∶ W i → P ij → W j where P ij is a countable non-expanding coproduct of weighted copies of R and p ij is nuclear. Given two pairs k = (i, j) with j ≥ i and k ′ = (i ′ , j ′ ) with j ′ ≥ i ′ of K we define the nuclear (see Lemma 4.2) morphism n kk ′ ∶ P k → P k ′ by p k ′ ○ c k for any pair k, k ′ such that j = i ′ . As in the proof of Lemma 2.3 of [34] we have a filtering inductive system (K, {P k }, {n kk ′ }) which defines an object P of Ind(Ban R ) isomorphic to W which has the desired properties.
Lemma 4.18. Given I, an infinite filtering ordered set and a functor I → Ban R such that the corresponding object W = "colim i∈I "W i is nuclear in Ind(Ban R ), there is a filtered category K with the same cardinality of objects and morphisms as I, a functor K → Ban R with corresponding object L = "colim k∈K "L k and an isomorphism W ≅ L such that each Banach space L k is a countable product in Ban ≤1 R of weighted copies of R and the transition functions are nuclear.
Proof. Using Lemma 4.14 we may assume that for each i there is a j > i such that where L ij is a countable non-expanding product of weighted copies of R and p ij is nuclear. Given two pairs k = (i, j) with j ≥ i and k ′ = (i ′ , j ′ ) with j ′ ≥ i ′ of K we define the nuclear (see Lemma 4.2) morphism n kk ′ ∶ L k → L k ′ by c k ′ ○ p k for any pair k, k ′ such that j = i ′ . As in the proof of Lemma 2.3 of [34] we have a filtering inductive system (K, {L k }, {n kk ′ }) which defines an object L of Ind(Ban R ) isomorphic to W which has the desired properties. Proof. Using Lemma 4.17 we can write a nuclear object in a certain nice form as the formal filtered colimit of countable coproducts in Ban ≤1 R of weighted copies of R. Each weighted copy of R is projective by Lemma 3.44. Therefore their coproduct in Ban ≤1 R is projective by Lemma 3.37 and hence flat by Lemma 3.36. By Lemma 3.37 this colimit of flat objects is flat.
Remark 4.20. If R is non-archimedean, all of this subsection goes through for Ban na R in place of Ban R .

The interaction of products and tensor products
Just as flatness in our context is about commuting kernels and completed tensor products, we need to investigate the interaction of the other type of limit (products) with the completed tensor product. Many of the results in this section are either taken from or inspired by the book [24] by R. Meyer and discussions with him. In this section we define metrizable modules and examine how the tensor product with them interacts with products. We work in a general context and as usual, for simplicity, we look at the archimedean setting, even if R is non-archimedean. In that setting, all the proofs in this section go through with the obvious modifications for Ban na R in place of Ban R . Definition 5.1. Let λ be a cardinal. A poset J is called λ-filtered if any subset S of J with S < λ has an upper bound.
Remark 5.2. Let λ be a cardinal. A finite product of λ-filtered posets is λ-filtered. Lemma 5.3. Let λ be a cardinal. Suppose that I is a poset and we are given a functor F ∶ I × J → Set where J has cardinality less than λ and I is λ-filtered. Then the natural morphism colim is an isomorphism.
Proof. This is well known in set theory. For example when λ = ℵ 0 , one can consider sets as can be shown by hand.
By considering objects in Ind(C) as functors from C op to sets, Lemma 5.3 immediately implies the following.
Lemma 5.4. Let λ be a cardinal. Suppose that I is a poset and we are given a functor F ∶ I × J → Ind(C) where J has cardinality less than λ and I is λ-filtered. Then the natural morphism colim is an isomorphism.
Suppose now that C is a closed symmetric monoidal category.
Definition 5.5. An object V of Ind(C) will be called metrizable if the category whose objects consist of objects of C along with morphisms to V and whose morphisms are commuting triangles is ℵ 1 -filtered. Proof. If there exists a functor as in the statement of the lemma, any morphism W → V would factor via F (i) for some object i ∈ I.
where V i ∈ C and I has cardinality less than λ.
Then there is an isomorphism Proof. By Lemma 5.4 we have Lemma 5.8. Let V = "colim i∈I "V i ∈ Ind(C) where V i ∈ C and I has cardinality less than λ. Let W = "colim j∈J "W j where J is λ-filtered. Assume that W is nuclear in Ind(C), then Proof. Using Lemma 4.14, without loss of generality we can assume that W is presented by a system where all the structure morphisms are nuclear. Consider the morphism ("colim i∈I "W i )⊗V ∨ → Hom(V, "colim i∈I "W i ).
By Corollary 5.7 we can equivalently write this as the colimit of the morphisms Consider any W i → W j in the system corresponding to a non-identity arrow i → j. Since they are nuclear, the precomposition V → W i → W j is also nuclear by Lemma 4.2. Therefore, it lies in the image of Hom(e, V ∨ ⊗W j ). This constructs an inverse ("colim i∈I "W i )⊗V ∨ ← Hom(V, "colim i∈I "W i ).
where V i ∈ C and I has cardinality less than λ.
is an isomorphism.
Proof. The left hand side is V ∨ ⊗W, which is isomorphic to where we have used Lemma 5.8. For the second statement let I = Z >0 and just consider the system At this point in the subsection, we need to reduce the generality and take C = Ban R for a Banach ring R. Of course, as usual, if R is non-archimedean, one can use C = Ban na R instead with the obvious modifications, which we suppress to save space.
Lemma 5.11. Suppose we are given a set V i in Ban R (n.b. not in Ind(Ban R )) indexed by a set I. Then the natural morphism in Ind(Ban R ) is an isomorphism in Ind(Ban R ) where the product on the right is taken in Ind(Ban R ), the product on the left is taken in Ban ≤1 R and the notation (V i ) ψ(i) −1 uses Definition 3.33.
Proof. It is enough to show that the morphisms are isomorphisms of abelian groups for any M ∈ Ban R . We have Notice here that in order to pass from colimits over Ψ to colimits over Z >0 , in the isomorphism colim ψ∈Ψ i∈I we have used that in the category of sets, products and filtered colimits distribute (not commute!). This means that for a set indexed by I of filtered sets See for instance [1] and [2], where this is explained. ◻ Lemma 5.12. Suppose we are given a countable set V i in Ban R (n.b. not in Ind(Ban R )) indexed by a set I. Then ∏ i∈I V i is metrizable in Ind(Ban R ).
Proof. Notice that Ψ is a non-full subcategory of Υ but they have the same objects. The functor Ψ → Ban R which sends ψ to ∏ i∈I ≤1 (V i ) ψ(i) −1 can be extended to a functor Υ → Ban R .
Indeed suppose that ψ 1 (i) < ψ 2 (i) for all i ∈ I − J where J is a finite subset of I. Then we have The inclusion Ψ → Υ is a final functor. This is because if ψ 1 (i) < ψ 2 (i) for all i in I − J then by letting ψ 3 = ψ 1 + ψ 2 we have ψ 1 (i) < ψ 3 (i) and ψ 2 (i) < ψ 3 (i) for all i ∈ I. The category Υ is ℵ 1 -filtered: Given a countable collection ψ 1 , ψ 2 , . . . of objects of Υ, define α ∈ Υ by α(i) = 1 + ∑ i k=0 ψ k (i). Then clearly for all i ≥ j we have ψ j (i) < α(i) and so in Υ we have that ψ < α. Hence the inclusion induces an isomorphism Combining this with the isomorphism of Lemma 5.11 we see that ∏ i∈I V i is metrizable.
Corollary 5.13. Suppose we are given a system in Ban R (n.b. not in Ind(Ban R )) such that all morphisms in the system are in Ban ≤1 R . Let Ψ be the poset consisting of non-decreasing functions ψ ∶ Z ≥1 → Z ≥1 with the order ψ 1 ≤ ψ 2 if ψ 1 (i) ≤ ψ 2 (i) for all i ∈ Z ≥1 . Then the natural morphism in Ind(Ban R ) V i is metrizable and so any Fréchet module is metrizable.
Proof. First notice that lim i∈Z ≥1

Using Lemma 5.11 we can write
This happens as long as sup i∈Z ≥1 v i i φ(i − 1) −1 < ∞ since the maps are non-expanding. For φ = ψ, ψ(i − 1) −1 ≥ ψ(i) −1 so there is no reason why this should be true. However, if we define φ = sψ by (sψ)(i) = ψ(i + 1) then we do have the map induced by the obvious maps are non-expanding because of the inequalities Therefore, we can rewrite (5.2) as Because the functor "colim ψ∈Ψ " is exact, we are done. As in the proof of Lemma 5.12 we can replace Ψ with the ℵ 1 -filtered category Υ and conclude that Remark 5.14. It is completely fine to take some or all of the maps V i+1 → V i to be the identity. In particular, taking them all to be the identity we see that any Banach module is metrizable.
Lemma 5.15. For each k ∈ K suppose we are given an inductive system I k → Ban R given by the system of Banach modules W (k) i . Assume that for i 1 < i 2 the morphisms W (k) are non-expanding. Let Φ be the poset whose objects are pairs (φ 1 , φ 2 ) where φ 1 ∶ K → ∐ k∈K I k such that φ 1 (k) ∈ I k for all k and φ 2 ∶ K → N ≥1 . This has a partial order defined by is an isomorphism. To explain the structure maps in the formal filtered colimit, for is the non-expanding product over k ∈ K of the obvious morphisms (W (k) Proof. It is enough to show that the morphisms are isomorphisms of abelian groups for any M ∈ Ban R . We have Hom(M, " colim that in the category of sets, products and filtered colimits distribute (not commmute!) as explained in [1]. Let us now assume that each I k is ℵ 1 -filtered and K is countable (so we can assume that K = N). Let Λ be the set whose objects are pairs λ = (φ 1 , φ 2 ) where φ 1 ∶ K → ∐ k∈K I k such that φ 1 (k) ∈ I k for all k and φ 2 ∶ K → N ≥1 . This has a partial order defined for all but a finite number of k ∈ K. Say that we are given a collection λ (1)    k. It has a partial order defined by φ 1 < φ ′ 1 when φ 1 (k) < φ ′ 1 (k) for all but a finite number of k ∈ K. Denote this poset by Φ 1 . The natural morphism is an isomorphism.
Lemma 5.17. Let K be a countable set. For each k ∈ K suppose we are given an inductive system I k → Ban R given by the system of Banach modules W (k) Ind(Ban R ) for each k. Then for any U ∈ Ban R the natural morphism is an isomorphism. If U is flat over R and K → Ind(Ban R ) is any functor written as k ↦ W (k) then the natural morphism is an isomorphism.
and similarly let g Notice that . We now want maps in the other direction but this will not work without increasing (φ 1 , φ 2 ). Suppose that we are given an element For each k this implies that {s ∈ S w k,s ≠ 0} is countable. Let S c be the subset of S defined by S c = S − {s ∈ S w k,s = 0 for all k}. Notice that S c is countable since it is a countable union of countable subsets: S c = ⋃ k∈K (S − {s ∈ S w k,s = 0}). Because of this countability, we can choose a collection of positive real numbers p s for s ∈ S c such that p = ∑ s∈Sc p s is finite.
Then for any s ∈ S c there exists a k s ∈ K such that we have sup . These three identities combined with Equation (5.5) imply that Now let U ∈ Ban R be arbitrary. Similarly to Lemma A.39 of [10] we can find a projective resolution K → P → U → 0 where all morphisms are strict epimorphisms, K = ∐ t∈T ≤1 R rt and P = ∐ s∈S ≤1 R rs . The fact that products are right exact immediately implies that U⊗ R ∏ k∈K W (k) ≅ ∏ k∈K (U⊗ R W (k) ).
Lemma 5.18. Let K be a countable set, For each k ∈ K suppose we are given an inductive system I k → Ban R given by the system of Banach modules W (k) is an isomorphism. Now let K be a category with a countable set of objects and morphisms. If V is metrizable and flat over R (or metrizable and nuclear) and K → Ind(Ban R ) is any functor then the natural morphism is an isomorphism.
because colimits over an ℵ 1 -filtered category commute with countable products by Lemma 5.4. Also by Lemma 5.17.
Lemma 5. 19. The converse to Lemma 5.18 holds in the sense that we can conclude that an object V ∈ Ind(Ban R ) is metrizable if and only the functor V⊗ R (−) commutes with countable products.
Proof. We will prove that if for some V ∈ Ind(Ban R ), that the natural morphism is an isomorphism, then V is metrizable. Consider the category J of all objects of Ban R mapping to V . Then of course V ≅ colim j∈J V j . The above isomorphism combined with Lemma

which tells us
there exists some j ∈ J such that all morphisms V j k → V factor through some V j → V . Therefore J is ℵ 1 -filtered and so V is metrizable.
Remark 5.20. This result is surprising since the analogous result in the purely algebraic case is not true. In fact, in the algebraic case the tensor product of a module will commute with all products of other modules if and only if the first module is finitely presented [23]. However, there is no contradiction here because if we take a ring, endow it with the discrete Banach structure, then we can consider the category of discrete modules over the ring but this category is not closed under the operation⊗ R .
Lemma 5.21. Say we fix A ∈ Comm(Ind(Ban R )). Let K be a countable set, For each k ∈ K suppose we are given an inductive system I k → Ban R given by the system of Banach modules W (k) i . Suppose we are given objects W (k) ∈ Mod(A) with underlying object "colim . Suppose the objects underlying A and V in Ind(Ban R ) are metrizable. Then the natural morphism is an isomorphism.
Proof. This follows from the case of A = R which was proven in Lemma 5.18 and the description of⊗ A as a coequalizer in Equation 3.7 together with the fact that⊗ R is right exact in each variable as discussed in subsection 3.5.
Corollary 5.22. As a corollary of Remark 5.14 (or of Lemma 5.17) and Lemma 5. 19, we see that if we have a countable collection V i of nuclear objects of Ind(Ban R ), their product is nuclear. A coproduct of any collection of nuclear objects is also nuclear.
Proof. Suppose that we have a countable collection of nuclear objects V i indexed by a countable set I. For any Banach module W the map ( breaks up as a product of maps V i⊗R W ∨ → Hom(W, V i ). Write the coproduct of nuclear objects V i over a countable set I as a filtered colimit of coproducts over finite subsets. The finite coproducts of V i are clearly nuclear. Then notice that both sides of the needed equation V⊗ R W ∨ = Hom(W, V ) are filtered colimits of true equations.

Spaces of Functions
In this section, we use the previous results to study rings of analytic functions and their modules on Stein spaces over Banach rings R. As usual, "affine" spaces are considered as the opposite category of commutative, associative, unital ring objects over Ind(Ban R ). As these form a huge category, one often wants to do geometry with a more manageable class of objects. Although we will not be precise here, one could define Stein algebras as limits of a sequence ⋯ → A 3 → A 2 → A 1 where the A i are quotients of Banach disk algebras by finitely generated ideals which are flat over R, and the morphisms are ring homomorphisms over R in Ind(Ban R ) which are injective, homotopy epimorphisms, nuclear, non-expanding, and dense. Similarly, dagger algebras over R are defined as colimits of systems made up of the same type of morphisms A 1 → A 2 → A 3 → ⋯. These limits and colimits take place in Ind(Ban R ). In this section we focus on limits of colimits of canonical maps of disk algebras, without quotienting by any ideals. It may be interesting to extend these results to more general contexts to develop a complete theory.
and if we apply Lemma 5.13 in the case These Stein rings are written here as formal ℵ 1 -filtered colimits of Banach modules. But the Banach modules themselves indexed by Υ and appearing within this colimit are not Banach rings because they are not closed under the multiplication of R [[x]]. In fact the same description can be done for any type of categorical limit lim V i of Banach modules as submodules of a given algebraic module where V n are the elements f in V 1 with f Vn < ∞: Proof. This map clearly decomposes into the completed tensor product of its factors R{ x i τ i } → R{ x i ρ i }. By the compatibility of nuclearity with the completed tensor product discussed in Lemma 4.2, it is enough to treat the one dimensional case and to show that for ρ < τ that the map R{ x τ } → R{ x ρ } is nuclear. For any τ and ρ the Banach module R{ x τ } ∨⊗ R R{ x ρ } can be described as The right hand side consists of elements (a ij ) i,j∈Z ≥0 such that first of all sup j∈Z ≥0 a ij ρ i τ −j < ∞ for any i ∈ Z ≥0 and that furthermore, In the case that a ij = δ ij which gives the restriction map, the first condition is vacuous because it just says that ( ρ τ ) i < ∞ for any i ∈ Z ≥0 . The second condition gives ∑ i∈Z ≥0 ( ρ τ ) i < ∞ which holds precisely when ρ < τ .
Let ψ be a non-decreasing sequence Z ≥0 → Z ≥1 . Define a Banach ring over R by Consider the morphisms and where the map σ is defined by summation. Here, s(f ) i = f i−1 for i ≥ 1 and s(f ) 0 = 0. The map s is contracting because it comes from the contracting maps R{ x r+(i−1) −1 } ψ(i) → R{ x r+i −1 } ψ(i) and similarly the map id is contracting because it comes from the contracting The map σ is non-expanding as it is induced by the obvious non-expanding morphisms The fact that δ is bounded follows from the estimate: Then we have σ ○ (id − s) = 0. Indeed the norm of ∑ N 1 (id − s)f is smaller than or equal to Because the terms of f are summable, f N → 0 as N goes to infinity and so the norm of σ ○ (id −s)f is zero for any f . Also notice that σ ○ δ = id R{ x r } ψ . We conclude that (6.1) splits on the right. By dualizing it and using Lemma 3.45 we get the identification where the morphism on the right is given by Lemma 6.4. For two sequences φ, ψ such that ψ(i) > 2 i φ(i) for all i the natural morphism Proof. As a module, we can identify R{ x An element of this space is just a collection (a j,l ) j,l such that sup l∈Z ≥0 a j,l (r + l −1 ) −l ψ(l) −1 (r + j −1 ) j φ(j) < ∞ for each j and The morphism we care about is nuclear if and only if a j,l = δ j,l satisfies these conditions. The first condition is obvious and the second reduces to checking that ∑ j∈Z ≥0 ψ(j) −1 φ(j) is finite, but this finiteness follows from our assumptions.
Definition 6.5. The dagger algebra [6] of overconvergent functions on the polydisk of polyradius (r 1 , . . . , r n ) ∈ R n ≥0 is the colimit of the monomorphic restrictions of the functions on closed polydisks in Comm(Ind(Ban R )): This also makes sense when r = (0, . . . , 0). This bornological ring can be realized as the subring of elements f = ∑ Lemma 6.6. There are canonical injective morphisms Proof. For the second, we first define a bounded pairing between O(D n <r −1 ,R ) and R{ x 1 r 1 , . . . , xn rn } † which is R-linear and non-degenerate in each variable. Consider the partially defined map It suffices to show that it restricts to a well defined, bounded non-degenerate morphism on the product The non-degeneracy is obvious. The fact that it is well defined follows from the estimate . . , xn ρn } and ρ > r. In order to show that it is bounded we need to take a bounded subset B 1 ⊂ R{ x 1 ρ 1 , . . . , xn ρn } for ρ > r and another bounded subset The first map is clearly bounded and so B 1 ×B 2 is still bounded in R{ x 1 ρ 1 , . . . , xn ρn }×O(D n <r −1 ,R ) ρ and so lands inside the product of disks D 1 × D 2 where D i consists of elements of norm less than d i . The estimate (6.3) again shows the image in R is bounded.
On the other hand, Together these inclusions give the desired isomorphisms.
On the other hand, Notice that for each ρ > r we have injective bounded maps This then shows that f ρ has norm less than or equal to one. For any η > ρ > r we have an injective bounded map given by sending any α to ∑ Notice that the composition g ρ,η ○ f ρ is simply restriction from the disk of radius ρ −1 to η −1 . The composition f η ○ g ρ,η is the identity. Therefore these maps give maps of systems which give the required isomorphisms: We can now finish showing that open disks and affine spaces over R are flat over R in any dimension.  (R{ . which is isomorphic to "colim ψ∈Ψ "(R{r 1 x 1 , . . . , r n x n } ψ ) ∨ by Equation 6.2. Lemma 6.4 implies that we can find a final indexing set so that all the morphisms in the system are nuclear. Therefore, by Lemma 4.13, O(D n <r −1 ,R ) is nuclear and hence flat by Lemma 4.19.
Lemma 6.10. Given a system ⋯ → V 2 → V 1 → V 0 in Ban R where all structure morphisms are dense, the canonical map Proof. Because the object Rlim i V i has bounded cohomological dimension (in fact with amplitude [0, 1] by [33]), it is enough to show that for any projective P that the natural morphism R Hom(P, lim Therefore, we need to show that the morphism is a quasi-isomorphism. In the system defined by the Hom(P, V i ) the structure maps are bounded and dense. By the Mittag-Leffler lemma for abelian groups equipped with compatible metrics given in the work of Palamadov [28], Vogt, and Retakh, we get a quasiisomorphism in the derived category of (the quasi-abelian category of) topological abelian groups. This implies that we have the needed quasi-isomorphism for the underlying abelian groups in equation (6.7). In Proof. First notice that both R{ρ 1 x 1 , . . . , ρ n x n } and O(D n <τ ) are metrizable by Corollary 5.13. Also, O(D n <τ ) is nuclear by Corollary 6.9 and hence flat by Lemma 4.19. Now the second statement in Lemma 5.17 shows that all the limits can be pulled outside. Explicitly ρ) ). (6.8)

Topologies and Descent
Different considerations of abstract topologies and descent that we know of have appeared for example in works of Orlov [27] and Kontsevich/Rosenberg [22]. We consider descent in the infinity-category version of the homotopy monomorphism, flat, and other topologies in our project on derived analytic geometry [11]. On the other hand, in this article we try to focus on non-derived categories of modules (i.e. homotopicaly discrete modules or complexes in degree 0) and the underived pullback functors of restriction. As this would not work for arbitrary modules concentrated in degree zero, we need to specialize to quasi-coherent modules. The Grothendieck pre-topology that we work with is not quasi-compact, it has covers consisting of a countable collection of homotopy monomorphisms spec(A i ) → spec(A) such that given a morphism f ∶ M → N in Mod(A) with M⊗ A A i → N⊗ A A i an isomorphism for all i, then f is an isomorphism. This property is called being conservative. It is expected to correspond to surjectivity of the cover for the topos-theoretic notion of points. For instance, it is known that the Huber points correspond with the topos-theoretic notion of points in the rigid-analytic context with the G-topology (see [10] where more explanation and citations are given). In most geometric settings [10] [11] [8] [9], conservativity on the level of quasicoherent modules agrees with surjectivity of the map ∪ i max(A i ) → max(A) on the maximal ideal spectrum, which then agrees with the pullback map from quasi-coherent modules to the descent category for quasi-coherent modules being fully faithful. Comm(Ind(Ban R )) where B is metrizable that M is transverse to B over A. The full subcategory of quasi-coherent modules is denoted by Mod RR F (A). Our notation "RR" is in credit to Ramis and Ruget who introduced a similar notion in the context of complex analysis in [35].
Example 7.4. Let A ∈ Comm(Ind(Ban R )) and V ∈ Ind(Ban R ). Assume that both A and V are flat over R and metrizable. Then A⊗ R V ∈ Mod RR F (A). Lemma 7.5. Say that C is a category with countably many objects and morphisms. Then given any functor F ∶ C → Mod RR F (A) the limit computed in Mod(A) lives in Mod RR F (A). 7.2. General Results on Descent.
Lemma 7.6. Let A ∈ Comm(Ind(Ban R )), and let {E i } i∈I be a projective system in Mod(A) indexed by the countable poset I. Let F be an object in Mod(A). Suppose that the underlying objects of A and F in Ind(Ban R ) are metrizable and flat over R. Suppose in addition F is transverse to E i over A for each i and the system {E i } i∈I is lim i∈I -acyclic. Then {F⊗ A E i } i∈I is a lim i∈I -acyclic projective system, F is transverse to lim i∈I E i over A and the natural morphism is an isomorphism. If instead of the condition that F is flat over R we have that both the E i and lim i∈I E i are flat over R then the same conclusion holds.
Proof. Recall that the Bar complex L • A (F ) is strictly quasi-isomorphic to F and that F⊗ Using the explicit form of this complex together with the fact that F⊗ R (−) and A⊗ R (−) commute with products by Lemma 5.18, we can see that it interacts well with the Roos complex of {E i } i∈I in the sense that there is a strict quasiisomorphism . The left hand side computes F⊗ L A (Rlim i∈I E i ) and the right hand side computes Rlim Using that F is transverse to E i over A for each i and the system {E i } i∈I is lim i∈I -acyclic the above equation simplifies to a quasi-isomorphism From which the rest of the claims follow immediately as one side is in non-negative degrees and the other is in non-positive degrees and so both are concentrated in degree zero.
Corollary 7.7. Let A ∈ Comm(Ind(Ban R )) presented by a system A i indexed by i in a countable poset I in Comm(Ind(Ban R )) A . Suppose that the system {A i } i∈I is lim Proof. Given an object M ∈ Mod RR F (A) and A i → B is a homotopy epimorphism we have we can rewrite f using Lemma 7.6 as lim Therefore, the collection is conservative. Conversely, if the collection is conservative let π ∶ A → lim i∈I A i be the canonical morphism. To show it is an isomorphism it is enough to know that it becomes so after applying the A j⊗A (−). But after doing this we get using Lemma 7.6 which is an isomorphism. The essential surjectivity holds because again using Lemma 7.6 we have for any M ∈ Mod RR F (A). Lemma 7.8. Let A ∈ Comm(Ind(Ban R )) and say we are given a countable poset A → A i of epimorphisms of Comm(Ind(Ban R )) A . The functor is fully faithful.
Proof. The natural "pushforward" functors Mod(A i ) → Mod(A) are fully faithful. The limit of these functors is therefore fully faithful.
By combining Corollary 7.7 and Lemma 7.8 we have Theorem 7.9. Let A ∈ Comm(Ind(Ban R )) and A → A i are homotopy epimorphisms indexed by i in a countable poset I in Comm(Ind(Ban R )) A whose underlying modules are in Mod RR F (A). Suppose that the system {A i } i∈I is lim i∈I -acyclic and that the objects of Ind(Ban R ) underlying A i are metrizable and flat over R. Assume the collection of functors (−)⊗ A A i is conservative. When this holds the natural functor This theorem can be used in the case of hypercovers. We now give a more explicit proof in the case of covers which can be easily adapted to general posets. Theorem 7.10. Let A ∈ Comm(Ind(Ban R )) and say we are given a countable collection A → A i of objects of Comm(Ind(Ban R )) A indexed by i ∈ S. Suppose that A and A i are metrizable objects which are flat over R, in Mod RR F (A). Suppose that each morphism A → A i is a homotopy epimorphism and the collection of functors Mod RR F (A) → Mod RR F (A i ) is conservative. Suppose that the corresponding system A w = A i 1⊗ A ⋯⊗ A A im for words w in S is a lim w∈P -acyclic projective system as above. Then the canonical functor Proof. Given N w ∈ Mod RR F (A w ) and suppose that A → B is a homotopy epimorphism.
showing that D ○ R is naturally equivalent to the identity. Using Lemma 7.6 and Corrollary 7.7 we have lim showing that R ○ D is naturally equivalent to the identity.  Remark 7.12. The non-archimedean version of this (the proof is the same) can give new settings for Tate's acyclicty theorem. We expect that the hypothesis of Lemma 7.11 will be satisfied whenever A → A i are homotopy epimorphisms and the topological spaces associated to the A i form a cover of the topological space associated to A.
This article was originally motivated by a desire to make "more categorical" the results on descent for Stein algebras from [8] (see also [5]). We believe that we have succeeded in a large aspect in terms of the issues surrounding infinite products and completed tensor products and their interaction. Unfortunately, we have not been able to make categorical the Mittag-Leffler aspects which involve dense maps of algebras in the projective system and lim-acyclicity. In standard complex analysis one often exhausts a Stein open subset by an increasing union of compact, convex subsets with the Noether property. Then one would like to understand how certain quasi-abelian categories of quasi-coherent modules on the Stein open are constructed as categorical limits of the similar categories on the compact subsets. More precisely, can a nice enough module over the algebra of holomorphic functions on the Stein open be determined in terms of gluing data for modules on the compact subsets. These questions also have a rich history in the non-archimedean literature, for example see work of Ardakov and Wadsley [3], where one uses affinoids in place of compact convex subsets. In our desire for a unified approach to the archimedean and non-archimedean case, we can restate Theorem 7.9 in this case. Let A i be Banach rings, flat over R together with a sequence of dense, nuclear, homotopy epimorphisms ⋯ → A 2 → A 1 . Any quasi-coherent, metrizable, ind-Banach module M flat over R over A = lim A i can be expressed as a limit in Mod(A) of a sequence ⋯ → M 2 → M 1 where each M i is a nuclear, metrizable ind-Banach object of Mod(A i ) flat over R and the morphisms are consistent with this action in the sense that there are isomorphisms A i+1⊗ compatible with one another and with the maps in the sequence. Any element of Hom(M, N) in the category of ind-Banach A-modules is a consistent limit of elements of Hom(M i , N i ) in the category of ind-Banach A i -modules. This should have applications in non-archimedean geometry for instance in the case of analytic differential operators as appear in work of Ardakov and Wadsley (see also [4]) which are Fréchet (and as we have shown therefore metrizable) modules which are not coherent over the functions, but which can be shown to be quasi-coherent in our definition. A version of the results in this subsection was given in [8] over a complete valuation field but there we needed to separately prove the theorem in the archimedean and non-archimedean cases whereas in this article we provide a single proof over a Banach ring that works in the archimedean or non-archimedean case. Lemma 7.13. Suppose A ∈ Comm(Ind(Ban R )), M is a metrizable A-module in Ind(Ban R ) and both are flat over R. Say that we have by  (a 1 , a 2 , . . . ) ↦ (a 2 − a 1 , a 3 − a 2 , . . . ) is a strict epimorphism with kernel (with the induced subspace structure) isomorphic to A and M is transverse to each A i over A. Then we can conclude that ∏ is a strict epimorphism with kernel (with the induced structure) isomorphic to M.
As we have proven that the last complex (representing Rlim in the derived category, we are done.
Remark 7.14. A situation where Lemma 7.13 can be used is the definition of a Stein by its defining affinoid cover. In fact, the category of modules over a Stein with an exhaustive affinoid cover which we define includes fully faithfully the category of co-admissible modules of Schneider and Teitelbaum.

The Fargues-Fontaine Curve
A thorough treatment of the Fargues-Fontaine curve from the point of view of Banach algebraic geometry appears in [9]. Therefore, we only focus on the aspects here which are relevant to the current article. Let For r 2 < r 1 < 1, the non-expanding morphism Z{( x r 1 ) 1 n } → Z{( x r 2 ) 1 n } is nuclear, being a sum of nuclear morphisms. Using the non-expanding morphisms α n,m ∶ Z{( are nuclear for all r 2 < r 1 < 1. Let E be the field E = F p ((Q)) = { γ∈Q a γ x γ a γ ∈ F p , support(a γ ) well ordered}.
It is equipped with the valuation given by v γ∈Q a γ x γ = min{γ ∶ a γ ≠ 0}.
The associated valuation ring is a γ x γ a γ ∈ F p , support(a γ ) well ordered}.
In Fargues-Fontaine theory one encounters a scheme Y E whose set of closed points Y E parametrize un-tilts of E. An un-tilt of E is an isomorphism class of pairs (F, ι) where F is a perfectoid field of characteristic 0, ι ∶ E → F ♭ is a embedding of topological fields and the quotient is a finite extension. Here F ♭ = Frac( lim x↦x p O F p) where O F is the ring of integers of F . Let W denote the Witt vectors construction. Let Z r be the Banach Z-module which is Z with norm r ⋅ where ⋅ is the usual absolute value. For any M ∈ Ban Z , let S ≤1 (M) be the symmetric ring construction in the category Ban ≤1 Z consisting of Banach modules with non-expanding morphisms, i.e. S ≤1 (M) = ∐ n=0,...,∞ ≤1 (M⊗ n Z Σ n ) where the coproduct is taken in Ban ≤1 Z . Consider the colimit in Ban ≤1 Z of the l-th power morphisms x ↦ x l in the ring of functions on the "closed 1-dimensional disk of radius r" given by the contracting coproduct S ≤1 (Z r ). One then has that for each prime p, The importance of this completion is that the closed maximal ideals of the localization at p are in bijection with Y E Z where n ∈ Z acts by by (F, ι) ↦ (F, ι ○ φ n ) where φ the p-th power Frobenius automorphism of E. In [12], Cuntz and Deninger found a nice description of the additive group structure on the ring of p-typical Witt vectors of a perfect F p -algebra with basis b. They found it to be simply the p-adic completion of the free Z-algebra with basis b.
Lemma 8.2. The natural functor is fully-faithful for any poset K with cardinality less than ℵ 1 .
Proof. Given objects X ∶ k ↦ "colim t∈T "X k t and Y ∶ k ↦ "colim s∈S "Y k s of Ind Hom(X k t , Y k s ) where ∫ k∈K is a limit over the usual diagram used to define morphisms in diagram categories. This is a limit over a diagram with cardinality less than ℵ 1 since K itself is such a diagram. It is a limit in the category of sets of a diagram of sets whose vertices are of the form Hom(X k t , Y l s ). On the other hand, Hom(X k t , Y k s ).
The term ∫ k∈K colim s∈S Hom(X k t , Y k s ) is a limit in the category of sets over the same diagram whose vertices are of the form colim   This is well defined because the full subcategory of Ind   Therefore, under this equivalence, we simply have This functor colim N ≤1 F is an exact functor from a full subcategory of Ind(Ban R ) N to Ind(Ban R ) (takes kernels to kernels) because both the ordinary non-exapanding colimit and the formal filtered colimit are exact functors. The functor we have described commutes with V⊗ R (−) for any V ∈ Ban R . Lemma 8.4. Consider a functor K × N → Ban ≤1 R where K is a countable category. There exists a chain of isomorphisms: Proof. The first and last morphisms are determined by the description of limits found in Corollary 5.13 in which they are shown to be isomorphisms. The second morphisms is an isomorphism as a consequence of Definition 8.3. The natural third morphism is an isomorphism because the non-expanding colimit functor is exact by Lemma 3.32 (see also Lemma 3.38). is an isomorphism and this object of Comm(Ind(Ban Z )) is flat over Z.
Proof. The first statement follows immediately from Lemma 8.4 because taking a cofinal system with r within a countable set, lim r<1 Z{( x r ) 1 l } is ℵ 1 -filtered by Corollary 5.13. Given any V ∈ Ban Z and any F as in Definition 8.3 admitting a suitable liftF , then V⊗ Z F admits V⊗ ZF as a suitable lift and therefore, the exact functor colim is an isomorphism where F p carries the residue norm from Z. This question is addressed in [9].

Appendix
As remarked above, most of this article has a non-archiedean version in the case that R is non-archimedean, so in this appendix, let R be a non-archimedean Banach ring. For k a nonarchimedean field, the standard Tate algebra representing an affinoid disk is k{ x 1 r 1 , . . . , xn rn }. In order to compare this with the "archimedean" disk algebra we used in this article which we denote the non-archimedean version by R{ x 1 r 1 , . . . , xn rn } na . Interestingly, Stein and Dagger algebras as defined in the introduction to Section 6 constructed from these two versions of disk algebras actually agree as we show in this informal Appendix. For any r > 0 there is an injective map Comm(CBorn R )

R{
x 1 r 1 , . . . , x n r n } → R{ x 1 r 1 , . . . , x n r n } na which by density is an epimorphism. Let A i = R{ x 1 r 1 −i −1 , . . . , xn rn−i −1 } and C i = R{ x 1 r 1 −i −1 , . . . , xn rn−i −1 } na . We have not only morphisms A i ⊂ C i but also C i ⊂ A i−1 because for any s < t we have Therefore, we get isomorphisms A = lim A i ≅ lim C i = C. Similarly, this should hold for general Stein or dagger algebras as defined in the introduction to Section 6 described in the two different ways (using archimedian or non-archimedean disk algebras or their quotients) for a non-archimedean Banach ring R.
Consider the category D whose objects are pairs consisting of a sequence of objects M i ∈ Mod(A i ) and a collection of compatible isomorphisms M i⊗A i A i−1 → M i−1 where morphisms are the obvious thing. Similarly there is the category D na whose objects are pairs consisting of a sequence of objects N i ∈ Mod na (C i ) and a collection of compatible isomorphisms N i⊗ na C i C i−1 → N i−1 where morphisms are the obvious thing. These categories are isomorphic and if we specialize to the nuclear metrizable modules and algebras flat over Z we get by descent (Theorem 7.9) an equivalence of categories for these modules on A and C.

Conflict of interest statement
On behalf of all authors, the corresponding author states that there is no conflict of interest.