On the sheafyness property of spectra of Banach rings

Abstract

Let $$R$$ be a non-Archimedean Banach ring, satisfying some mild technical hypothesis that we will specify later on. We prove that it is possible to associate to $$R$$ a homotopical Huber spectrum $${\rm Spa\,}^h(R)$$ via the introduction of the notion of derived rational localization. The spectrum so obtained is endowed with a derived structural sheaf $${\mathcal {O}}_{{\rm Spa\,}^h(R)}$$ of simplicial Banach algebras for which the derived C̆ech–Tate complex is strictly exact. Under some hypothesis, we can prove that there is a canonical morphism of underlying topological spaces $$|{\rm Spa\,}(R)| \rightarrow|{\rm Spa\,}^h(R)|$$ that is a homeomorphism in some well-known examples of non-sheafy Banach rings, where $${\rm Spa\,}(R)$$ is the usual Huber spectrum of $$R$$. This permits the use of the tools from derived geometry to understand the geometry of $${\rm Spa\,}(R)$$ in cases when the classical structure sheaf $$H^0({\mathcal {O}}_{{\rm Spa\,}(R)})$$ is not a sheaf.

1 INTRODUCTION

Geometry of abstract commutative Banach rings

A well-known limitation of analytic geometry with respect to algebraic geometry is that it lacks an abstract approach. By this we mean that given a general (commutative) Banach ring† $$R$$, it is not clear, not even in the case when $$R$$ is a Banach $$k$$-algebra over a complete valued field, how to interpret $$R$$ as a space of functions on some meaningful geometrical object. More precisely, it is not clear what the analytic version of Grothendieck's definition of the spectrum of a commutative ring is. Several notions of spectra for Banach rings have been proposed, like the Berkovich spectrum of $$R$$, here denoted by $${\mathcal {M}}(R)$$. But besides the extreme success of this notion in the classical setting of non-Archimedean geometry, Mihara showed in [22] that there exist Banach $$k$$-algebras for which the (natural) definition of the structural sheaf on $${\mathcal {M}}(R)$$ does not even give a well-defined pre-sheaf. Another possibility is to associate to $$R$$ an affinoid pair $$(R, R^\circ)$$ in the sense of Huber and to consider the adic spectrum $${\rm Spa\,}(R, R^\circ) = {\rm Spa\,}(R)$$. In this case, it is possible to show that the structural pre-sheaf is always well defined but it lacks the sheaf property in general (cf. [22] and [11] for counter-examples, or check Section 5 where an example is worked out in detail).

In [22] and [11], some conditions on $$R$$ that ensure the sheafyness of the structural pre-sheaf on $${\rm Spa\,}(R)$$ are given. One drawback of these conditions is that they are difficult to check in practice, and another one is that there is an increasing need for more general results for some applications of the theory (cf. [16] for example). In this work, we propose a solution to this problem based on the theory of quasi-abelian categories and on the methods developed in [4]. More precisely, we propose to use the methods of derived geometry in order to obtain a derived structural pre-sheaf that satisfies descent in the derived setting (the descent condition is the derived version of the sheafyness condition). This is done compatibly with the classical theory of Huber spaces and therefore provides an extension of it to a broader class of Banach rings that do not seem treatable with the classical methods.

Our main results

We now summarize the main ideas of this paper. Suppose that $$R$$ is a Banach $$A$$-algebra, where $$A$$ is a non-Archimedean strongly Notherian Tate ring (for simplicity the reader can consider the case when $$A$$ is a non-Archimedean complete valued field, that is already very important for applications). If $$R$$ is an affinoid $$A$$-algebra (cf. Definition 4.1), the classical theory of Huber spaces gives a structure sheaf of Banach algebras on $${\rm Spa\,}(R)$$. The topological space $${\rm Spa\,}(R)$$ is defined as a suitable subset of the set of Krull valuations on $$R$$ and its topology is defined by considering an analytic version of the principal Zariski localizations of algebraic geometry. These analytic localizations are called rational localizations and, roughly speaking, consist of subsets determined by inequalities of the form

$$$$\begin{equation*} U_{f_0, f_1, \ldots , f_n} = U {\left(\frac{f_1}{f_0}, \ldots , \frac{f_n}{f_0} \right)} = \lbrace x \in {\rm Spa\,}(R)||f_1(x)| \leqslant|f_0(x)|, \ldots ,|f_n(x)| \leqslant|f_0(x)| \rbrace \end{equation*}$$$$

by elements $$f_0, \ldots , f_n \in R$$ that generate the unit ideal in $$R$$. The structure pre-sheaf is defined to take values

$$$$\begin{equation*} {\mathcal {O}}_{{\rm Sp \,}(R)} (U_{f_0, f_1, \ldots , f_n}) = \frac{R \langle X_1, \ldots , X_n \rangle }{(f_0 X_1 - f_1, \ldots , f_0 X_n - f_n)}, \end{equation*}$$$$

where the quotient is computed in the category of Banach rings and therefore if the ideal is not closed, the quotient must be computed by taking the closure of the ideal.

One of the main ideas of this paper is to replace the notion of rational localization with its derived analog, which we call derived rational localization. Very basically, this means to replace the quotients described so far in the definition of rational localization by the homotopical quotients. Concretely, this means that in both situations we are dealing with a morphism of Banach $$R$$-modules

$$$$\begin{equation*} R \langle X_1, \ldots , X_n \rangle ^n \rightarrow R \langle X_1, \ldots , X_n \rangle \end{equation*}$$$$

whose image is the ideal $$(f_0 X_1 - f_1, \ldots , f_0 X_n - f_n)$$ and instead of computing the cokernel of this map as done classically, we compute its cone in the derived category of Banach $$R$$-modules. To achieve this, a mixture of functional analysis and homological algebra must be used because the Banach algebra structure is the relevant structure and the closedness properties of the image of morphisms and ideals play an important role. It is not possible to work out a properly working theory considering only the underlying algebraic categories. Also, the use of the completed tensor product is needed and hence the use of categories of Banach modules and similar functional analytic structures equipped with an algebraic structure compatible with an analytic structure is required. In our work, we found it convenient to use the class of bornological modules over $$R$$. This is a natural and simple extension, in our opinion, of the notion of Banach module over $$R$$. We will devote Section 3 to recall the main features of this theory. We also underline that this class does not form an abelian category but they form what is called a quasi-abelian category. Luckily, the theory of quasi-abelian categories has been well established in the last 20 years or so, mainly through the work of Schneiders [24]. We will recall the basic features of the theory in Section 2 of the paper and explain how we used it for doing derived geometry.

In [8], it has been proved in the specific case when $$A$$ is a non-Archimedean valued field, and $$R$$ an affinoid algebra over $$A$$ that affinoid localizations† are precisely characterized by a homological property that is called homotopical epimorphism. In particular, this implies that, under such hypotheses, the notions of derived rational localization and rational localization agree. Thus, our first main task in this work is to generalize these results to the case when $$R$$ is an affinoid algebra over a strongly Noetherian Tate ring. In order to prove this result, we need to add the hypothesis that $$A$$ has a topologically nilpotent uniform unit, that is, a unit $$u \in A^\times$$ such that $$|u^n| =|u|^n$$ for all $$n \in {\mathbb {Z}}$$. We conjecture that both the hypothesis that $$A$$ is a Tate ring and that it has a uniform unit can be removed with better strategies of proof. Under such hypotheses, we prove that all derived rational localizations of $$A$$-affinoid algebras are non-derived at the beginning of Section 4. This ensures the compatibility of the geometrical notions from the theory of Huber spaces and the notions from the derived geometry of bornological algebras we developed in our previous works (see [4]). So, we introduce the derived version of the notion of classical rational localization and we check that in the specific case of affinoid $$A$$-algebras the derived and classical notions coincide (see Proposition 4.15). Then, to analyze the general case, we write

$$$$\begin{equation*} R \cong {\mathop {\lim \limits _{\displaystyle \rightarrow }}_{i \in I}}^{\leqslant 1} R_i, \end{equation*}$$$$

where $$R_i$$ are affinoid algebras over $$A$$ and $${\mathop {\lim \limits _{ \rightarrow }}_{i \in I}}^{\leqslant 1}$$ means the colimit in the contracting category of Banach algebras (see Section 3 for the definition of the contracting category and the computation of limits and colimits in this category). We underlie that, contrary to the algebraic situation, the functor $${\mathop {\lim \limits _{ \rightarrow }}_{i \in I}}^{\leqslant 1}$$ is not exact, therefore it should be expected that computations that for $$R_i$$ give a result that is concentrated in degree 0 may have higher cohomology when the functor $${\mathop {\lim \limits _{ \rightarrow }}_{i \in I}}^{\leqslant 1}$$ is applied. This is a possible interpretation of the sheafyness issue that appears in the passage from (analytically) finite-dimensional algebras over $$A$$ to infinite-dimensional ones. So, for any Banach ring both the notion of rational localization and derived rational localization make sense and under the mentioned hypothesis, we are able to check that derived rational covers always give a cover in the sense of the machinery of derived geometry introduced in Section 2 (see Theorem 4.31). This permits us to associate to $$R$$ a homotopical version of the Huber spectrum that we denote $${\rm Spa\,}^h(R)$$ and endow it with the structure of derived scheme relative to derived geometry over bornological modules, we would call these objects derived analytic spaces (see Theorem 4.35). Of course, if $$R$$ is affinoid, we have a canonical homeomorphism of underlying topological spaces $$|{\rm Spa\,}(R)| \cong|{\rm Spa\,}^h(R)|$$, but we do not know how far this comparison of spectra extends as we do not know any counter-example.

After proving this main abstract result, we dedicate a section of the paper to working out in full detail a classical example of a non-sheafy Banach ring. So, this is a relatively simple example of Banach algebra over a valued field $$k$$ for which it is known that Huber's theory does not work and the classical definition of structure pre-sheaf is not a sheaf. We show that in this particular case, there are rational localizations that are not derived localizations, that their pathological behavior is precisely the source of non-sheafyness, and that their derived counterparts fix this issue. So, we recall the computations showing that the C̆ech–Tate complex of this cover is not exact. After that, we compute the derived version of the same constructions and we check explicitly that the derived C̆ech–Tate complex of the same cover is strictly exact, as implied by the abstract theory developed before. We also see how the appearance of higher cohomology groups on open subsets of the structural sheaf is precisely what is needed to compensate for the non-injectivity of the restriction map on the $$H^0$$-part.

Structure of the paper

The paper is structured as follows. In Section 2, we recall some basic language of the theory of quasi-abelian categories and some fundamental results from [4] that are used throughout the whole paper. We also recall how the homotopy Zariski topology is defined and how it can be used to define derived analytic spaces over any given Banach ring $$R$$. In Section 3, we recall the definitions of the quasi-abelian categories that are used in this paper, the categories of Banach modules, the contracting categories of Banach modules, and the categories of bornological modules over a fixed Banach (or bornological) ring. In Section 4, we prove our main results. We first prove that rational localizations of affinoid $$A$$-algebras, with $$A$$ any strongly Noetherian Tate ring, are open localizations for the homotopy Zariski topology giving a homological characterization of the Huber spectrum, generalizing the results of [8]. Then, we show how to associate to any Banach $$A$$-algebra $$R$$ (satisfying some mild hypothesis) a $$\infty$$-site $${\rm Spa\,}^h(R)$$ but differently from the case when $$R$$ is an affinoid $$A$$-algebra, the space $${\rm Spa\,}^h(R)$$ is equipped with a structure sheaf that makes it into a derived analytic space. We also describe how one can find a continuous map $${\rm Spa\,}(R) \rightarrow {\rm Spa\,}^h(R)$$ when $$R$$ is defined over a valued field. We conclude Section 4 by showing how these results can be extended to more general bornological rings.

Finally, in Section 5, we perform some explicit computations of the derived structural sheaf for spectra for which the standard definitions do not give a well-defined structural sheaf. We compute in detail the structural sheaf on a Laurent cover of a well-known non-sheafy Banach ring discussed by Buzzard–Verberkmoes [11] and Huber [14]. We show that in this case $${\rm Spa\,}^h(R) \cong {\rm Spa\,}(R)$$ so that our main results define a derived structural sheaf directly on $${\rm Spa\,}(R)$$ and we compare the derived structure with the non-derived one.

We conclude the paper discussing some possible and conjectural generalizations of the results of this paper, mainly via the use of the theory of reified spaces introduced by Kedlaya in [15].

Notation and conventions

In this work, we use standard conventions and terminology in addition to the following

• - the term ring always means commutative ring with unit, if not otherwise stated;
• - ring homomorphisms are always supposed to preserve identities;
• - if $${\mathbf {C}}$$ is a category, we adopt the common abuse of notation of identifying $${\mathbf {C}}$$ with its class of objects; therefore, $$X \in {\mathbf {C}}$$ means that $$X$$ is an object of $${\mathbf {C}}$$;
• - we suppose the existence of an uncountable strongly inaccessible cardinal and we fix one which will bound the class of morphisms of all our basic categories; this allows us to always consider categories of functors avoiding set-theoretic issues;
• - we will use cohomological indexing for chain complexes, that is, the differentials increase the degree.

2 RELATIVE ALGEBRAIC GEOMETRY OVER QUASI-ABELIAN CATEGORIES

In this section, we recall some basic notions from the theory of quasi-abelian categories (cf. [24]) and we recall how they have been used in [4] (and also in [2, 3, 7-9]) for defining derived geometry over the category of (simplicial) commutative algebras over a symmetric monoidal quasi-abelian category. Currently, the most general version of this kind of theory goes beyond the theory of quasi-abelian categories and is developed in [18]. In this work, we will never leave the setting of the theory of quasi-abelian categories.

2.1 Quasi-abelian categories

Recall that a pre-abelian category is an additive category for which all morphisms have a kernel and a cokernel. Any morphism in such a category fits in the following canonical diagram:

(2.0.1)

where $${\rm Coim\,}(f) = {\rm Coker\,}({\rm Ker\,}(f) \rightarrow A)$$ and $${\rm Im\,}(f) = {\rm Ker\,}(B \rightarrow {\rm Coker\,}(f))$$. The morphism $$f: A \rightarrow B$$ is called strict if the canonical morphism $${\rm Coim\,}(f) \rightarrow {\rm Im\,}(f)$$ is an isomorphism. We recall that a morphism is strict if and only if it can be written as a composition of a strict epimorphism followed by a strict monomorphism (cf. [24, Remark 1.1.2(c)]). We say that a short exact sequence

$$$$\begin{equation*} 0 \rightarrow A \stackrel{f}{\rightarrow } B \stackrel{g}{\rightarrow } C \rightarrow 0 \end{equation*}$$$$

is strictly exact or that it is a strict short exact sequence if both $$f$$ and $$g$$ are strict morphisms. Such short exact sequences are also called kernel–cokernel sequences in literature.

A quasi-abelian category is a pre-abelian category such that the family of strict short exact sequences forms a Quillen exact structure. The definition of quasi-abelian category can be restated by saying that it is a pre-abelian category such that strict monomorphisms are stable by pushouts and strict epimorphisms are stable by pullbacks. Hence, the property of being quasi-abelian is an intrinsic property of a category. We remark that for all quasi-abelian categories the canonical morphism $${\rm Coim\,}(f) \rightarrow {\rm Im\,}(f)$$ is always a bimorphism, that is, it is both an epimorphism and a monomorphism (cf. [24, Corollary 1.1.5]). But this latter property does not characterize quasi-abelian categories.

In this section, $${\mathbf {C}}$$ denotes a quasi-abelian category. Later on we will also suppose that $${\mathbf {C}}$$ has a closed symmetric monoidal structure that we denote by $$({-}) {\overline{\otimes }}({-}): {\mathbf {C}}\times {\mathbf {C}}\rightarrow {\mathbf {C}}$$ and $$\underline{{\rm Hom \,}}({-}, {-}): {\mathbf {C}}^{\rm op\,}\times {\mathbf {C}}\rightarrow {\mathbf {C}}$$. We give some key examples of quasi-abelian categories focusing on those relevant to this paper.

Example 2.1.

• (1) The category of abelian groups is clearly quasi-abelian (as it is abelian), has a standard symmetric tensor product, and an internal hom. In the same fashion, the category of modules over a commutative ring is (quasi-)abelian with a symmetric tensor product and an internal hom.
• (2) Let $$R$$ be a non-Archimedean Banach ring. We denote by $$\mathbf {Ban}_R^{{\rm na}}$$ the category of non-Archimedean Banach $$R$$-modules with bounded morphisms, that is, homomorphism $$\phi: X \rightarrow Y$$ such that $$|\phi (x)| \leqslant C|x|$$ for a fixed $$C > 0$$. This category is quasi-abelian, closed symmetric monoidal (a structure that we describe in more detail later on), but it is not complete or cocomplete (cf. section 3.1 of [2]).
• (3) Let $$R$$ be any (commutative) Banach ring. The category $$\mathbf {Ban}_R$$ of Banach $$R$$-modules with bounded morphisms is quasi-abelian, closed symmetric monoidal, but not complete or cocomplete. Notice that if $$R$$ is non-Archimedean, the category $$\mathbf {Ban}_R$$ contains more objects than the category $$\mathbf {Ban}_R^{\rm na}$$ and their monoidal structures do not agree on $$\mathbf {Ban}_R^{\rm na}$$.
• (4) Let $$R$$ be a Banach ring. If $$R$$ is non-Archimedean, then the category $$\mathbf {Ban}_R^{\leqslant 1, {\rm na}}$$ of ultrametric Banach $$R$$-modules with contracting homomorphisms, that is, bounded homomorphism $$\phi: X \rightarrow Y$$ such that $$|\phi (x)| \leqslant|x|$$, is quasi-abelian. The category $$\mathbf {Ban}_R^{\leqslant 1, {\rm na}}$$ has a closed symmetric monoidal structure and is complete and cocomplete. Notice that it is possible to consider also the category $$\mathbf {Ban}_R^{\leqslant 1}$$, that is, the category of all Banach $$R$$-modules with contracting morphisms, but this category is not additive.
• (5) To remedy the fact that arbitrary limits and colimits do not exist in $$\mathbf {Ban}_R$$, we will consider the category $$\mathbf {Ind}(\mathbf {Ban}_R)$$ and its subcategory $$\mathbf {Born}_R$$ of (complete) bornological modules. These are complete and cocomplete closed symmetric monoidal quasi-abelian categories, whose definition will be recalled in detail in Section 3.3.

We now recall some notions and results about exactness of additive functors between quasi-abelian categories.

Definition 2.2.An additive functor $$F: {\mathbf {C}}\rightarrow {\mathbf {D}}$$ between two quasi-abelian categories is called left exact if for any exact sequence

$$$$\begin{equation*} 0 \rightarrow A \stackrel{f}{\rightarrow } B \stackrel{g}{\rightarrow } C, \end{equation*}$$$$

where $$f$$ and $$g$$ are strict morphisms, the sequence

$$$$\begin{equation*} 0 \rightarrow F(A) \stackrel{F(f)}{\rightarrow } F(B) \stackrel{F(g)}{\rightarrow } F(C) \end{equation*}$$$$

is exact with $$F(f)$$ strict. The functor $$F$$ is called strictly left exact if for any exact sequence

$$$$\begin{equation*} 0 \rightarrow A \stackrel{f}{\rightarrow } B \stackrel{g}{\rightarrow } C, \end{equation*}$$$$

where $$f$$ is a strict morphism, the sequence

$$$$\begin{equation*} 0 \rightarrow F(A) \stackrel{F(f)}{\rightarrow } F(B) \stackrel{F(g)}{\rightarrow } F(C) \end{equation*}$$$$

is exact with $$F(f)$$ strict. Dually one defines the notions of right exactness and strictly right exactness. We say that $$F$$ is exact (resp. strictly exact) if it is both left and right exact (resp. both strictly left and strictly right exact).

Definition 2.2 can be restated by saying that a functor is left exact if and only if it preserves kernels of strict morphisms and it is strictly left exact if and only if it preserves kernels of all morphisms.† The dual statement holds for right exact and strictly right exact functors.

The next proposition should be well known to experts but we cannot find a reference in literature where it is stated in this form.

Proposition 2.3.Let $$F: {\mathbf {C}}\rightarrow {\mathbf {D}}$$ be a functor between quasi-abelian categories.

• (1) If $$F$$ is right exact, then it is exact if and only if it preserves strict monomorphisms.
• (2) If $$F$$ is strictly right exact, then it is strictly exact if and only if it maps strict monomorphisms to strict monomorphisms and monomorphisms to monomorphisms.

Dually for left exact and strictly left exact functors.

Proof.

• (1) Left exact functors clearly preserve strict monomorphisms, so we only need to prove the converse. We first notice that $$F$$ maps strict morphisms to strict morphism because the class of strict morphism is precisely the class of morphism that can be written as a composition of a strict epimorphism followed by a strict monomorphism, and $$F$$ preserves both classes of morphisms. So, let us consider a strict morphism $$f$$ and write it as $$f = m \circ e$$ with $$e$$ a strict epimorphism and $$m$$ a strict monomorphism. Then, we have the exact sequence of strict morphisms
  $$$$\begin{equation*} 0 \rightarrow {\rm Ker\,}(f) \rightarrow B \stackrel{f}{\rightarrow } C \end{equation*}$$$$
  that is mapped to the sequence
  $$$$\begin{equation*} 0 \rightarrow F({\rm Ker\,}(f)) \rightarrow F(B) \stackrel{F(f)}{\rightarrow } F(C). \end{equation*}$$$$
  It is easy to check that $${\rm Ker\,}(f) \cong {\rm Ker\,}(e)$$ and since $$F(m)$$ is a strict monomorphism then we also have $${\rm Ker\,}(F(f)) \cong {\rm Ker\,}(F(e))$$. Thus, it is enough to consider the sequence
  $$$$\begin{equation*} 0 \rightarrow F({\rm Ker\,}(e)) \rightarrow F(B) \stackrel{F(e)}{\rightarrow } F({\rm Coim\,}(f)) \rightarrow 0 \end{equation*}$$$$
  in which $$F(e)$$ is a strict epimorphism because $$F$$ is right exact. Moreover, the sequence is exact in the middle because $$F$$ is right exact and the fact that the morphism $$F({\rm Ker\,}(e)) \rightarrow F(B)$$ is a strict monomorphism implies that the sequence is actually a strict short exact sequence. All together, this shows $$F({\rm Ker\,}(f)) \cong {\rm Ker\,}(F(f))$$ as claimed.
• (2) Strictly left exact functors clearly preserve strict monomorphisms and monomorphisms, so we only need to prove the converse. Conversely, consider now an exact sequence
  $$$$\begin{equation*} 0 \rightarrow A \rightarrow B \stackrel{f}{\rightarrow } C, \end{equation*}$$$$
  where $$A \cong {\rm Ker\,}(f)$$ but $$f$$ is not necessarily a strict morphism. Since $$F$$ preserves strict monomorphisms and is right exact, the sequence
  $$$$\begin{equation*} 0 \rightarrow F({\rm Im \,}(f)) \rightarrow F(C) \rightarrow F({\rm Coker\,}(f)) \rightarrow 0 \end{equation*}$$$$
  is strictly exact and thus the isomorphism $$F({\rm Coker\,}(f)) \cong {\rm Coker\,}(F(f))$$ implies $${\rm Im \,}(F(f)) \cong F({\rm Im\,}(f))$$. Since $$F$$ preserves monomorphisms we can deduce that the induced morphism $$F({\rm Coim\,}(f)) \rightarrow {\rm Coim\,}(F(f))$$ is a monomorphism. Indeed, in the canonical commutative diagram of canonical maps
  (2.3.1)
  the horizontal maps are bimorphisms and the right vertical map is an isomorphism, implying that the left vertical map is a monomorphism. But since both $${\rm Coim\,}(F(f))$$ and $$F({\rm Coim\,}(f))$$ are quotients of $$F(B)$$, this implies that they are actually isomorphic, and, as before, this implies $$F(A) \cong {\rm Ker\,}(F(f))$$ because we have proven that the sequence
  $$$$\begin{equation*} 0 \rightarrow F(A) \rightarrow F(B) \rightarrow {\rm Coim \,}(F(f)) \rightarrow 0 \end{equation*}$$$$
  is strictly exact.$$\Box$$

It is not enough for a strictly right exact functor to map strict monomorphisms to strict monomorphisms to deduce strict left exactness. Our main example of a functor that is strictly right exact and left exact without being strictly left exact is the completion functor $$\widehat{({-})}: \mathbf {Nr}_R \rightarrow \mathbf {Ban}_R$$ from the category of normed modules over a Banach ring $$R$$ to the category of Banach $$R$$-modules. This functor is strictly right exact because it is left adjoint to the embedding $$\mathbf {Ban}_R \rightarrow \mathbf {Nr}_R$$ and it can be checked that it preserves strict monomorphisms. But it is also easy to find examples of (non-strict) monomorphism that are not preserved by $$\widehat{({-})}$$.

It is possible to associate to $${\mathbf {C}}$$ its derived category $$D({\mathbf {C}})$$ and we now recall its definition as well as the definitions of the categories $$D^-({\mathbf {C}})$$, $$D^+({\mathbf {C}})$$ and $$D^{\leqslant 0}({\mathbf {C}})$$. Let $$K({\mathbf {C}})$$ be the homotopy category of $${\mathbf {C}}$$, that is, the localization of the category of unbounded complexes of objects of $${\mathbf {C}}$$ by homotopy equivalences. This is a triangulated category. Let us consider the full subcategory of $$N \subset K({\mathbf {C}})$$ of objects that have a representative that is a strictly exact complex. Recall, that a complex $$ \cdots \stackrel{d_{n - 1}}{\rightarrow } C_{n- 1} \stackrel{d_n}{\rightarrow } C_n \stackrel{d_{n + 1}}{\rightarrow } C_{n + 1} \rightarrow \cdots$$ is said to be strictly exact if $${\rm Ker\,}(d_n) \cong {\rm Im\,}(d_{n -1})$$ for all $$n$$ and all $$d_n$$s are strict morphisms. It is possible to prove that $$N$$ is a thick triangulated subcategory of $$K({\mathbf {C}})$$, see [24, Corollary 1.2.15]. We thus define

$$$$\begin{equation*} D({\mathbf {C}}) = \frac{K({\mathbf {C}})}{N}, \end{equation*}$$$$

where the quotient denotes the Verdier quotient of $$K({\mathbf {C}})$$ by $$N$$. This Verdier quotient is equivalent to inverting the morphisms in $$K({\mathbf {C}})$$ whose cone is a strictly exact complex. We remark that in the case when $${\mathbf {C}}$$ is abelian then $$D({\mathbf {C}})$$ is the derived category of $${\mathbf {C}}$$ in the usual sense, that is, the definition we gave of the derived category is compatible with the standard definition in the case when $${\mathbf {C}}$$ is abelian. The category $$D^-({\mathbf {C}})$$ (resp. $$D^+({\mathbf {C}})$$, resp. $$D^{\leqslant 0}({\mathbf {C}})$$) can be defined as the full subcategory of objects of $$D({\mathbf {C}})$$ whose objects have a representative that is eventually zero on the right (resp. on the left, resp. in degree $$\leqslant 0$$) or by a suitable modification the definition of $$D({\mathbf {C}})$$.

By [17, Corollary 5.18] the Dold–Kan correspondence† holds for all additive categories with kernels, in particular, it holds for quasi-abelian categories. This implies that we can describe the category $$D^{\leqslant 0}({\mathbf {C}})$$ as a localization of the category $$\mathbf {Simp}({\mathbf {C}})$$ of simplicial objects of $${\mathbf {C}}$$ endowed with a suitable model structure. We will say more about this model structure later on. We will often represent objects of $$D^{\leqslant 0}({\mathbf {C}})$$ by simplicial objects when studying derived geometry relative to $${\mathbf {C}}$$.

As for the case of the derived categories of abelian categories, objects of $$D({\mathbf {C}})$$ are usually studied via projective or injective resolutions. The quasi-abelian version of this process is very similar, compatible with the abelian one, and it is reviewed in what follows.

Definition 2.4.Let $$P \in {\mathbf {C}}$$. We say that $$P$$ is projective if the functor $${\rm Hom \,}(P, -)$$ is exact (in the sense of Definition 2.2).

More explicitly, using Proposition 2.3, $$P \in {\mathbf {C}}$$ is projective if $${\rm Hom \,}(P, -)$$ sends strict epimorphisms to surjections. We say that $${\mathbf {C}}$$ has enough projective objects if for any $$X \in {\mathbf {C}}$$ there exists a strict epimorphism $$P \rightarrow X$$ with $$P$$ projective. If $${\mathbf {C}}$$ has enough projective objects and is complete and cocomplete, then [4, Theorem 3.7] implies that on $$\mathbf {Simp}({\mathbf {C}})$$, the quasi-abelian category of simplicial objects on $${\mathbf {C}}$$, there is a symmetric monoidal combinatorial model structure whose homotopy category is equivalent to $$D^{\leqslant 0}({\mathbf {C}})$$, via the quasi-abelian Dold–Kan correspondence.

Let $$F: {\mathbf {C}}\rightarrow {\mathbf {D}}$$ be an additive functor between quasi-abelian categories. We now recall how to define its left and right derived functors. A subcategory $${\mathbf {P}}_F \subset {\mathbf {C}}$$ is called $$F$$-projective if

• (1) for any $$X \in {\mathbf {C}}$$, there exists a strict epimorphism $$P \rightarrow X$$ with $$P \in {\mathbf {P}}_F$$;
• (2) for any strictly exact sequence
  $$$$\begin{equation*} 0 \rightarrow P \rightarrow P^{\prime } \rightarrow P^{\prime \prime } \rightarrow 0 \end{equation*}$$$$
  with $$P^{\prime }, P^{\prime \prime } \in {\mathbf {P}}_F$$, one has $$P \in {\mathbf {P}}_F$$;
• (3) for any strictly exact sequence
  $$$$\begin{equation*} 0 \rightarrow P \rightarrow P^{\prime } \rightarrow P^{\prime \prime } \rightarrow 0 \end{equation*}$$$$
  in $${\mathbf {P}}_F$$, one has
  $$$$\begin{equation*} 0 \rightarrow F(P) \rightarrow F(P^{\prime }) \rightarrow F(P^{\prime \prime }) \rightarrow 0 \end{equation*}$$$$
  is strictly exact in $${\mathbf {D}}$$.

In a dual fashion, one can define the notion of $$F$$-injective category $${\mathbf {I}}_F$$. In the case when $$F$$ admits an $$F$$-projective category (resp. $$F$$-injective category), it has a left derived functor $${\mathbb {L}}F: D^-({\mathbf {C}}) \rightarrow D^-({\mathbf {D}})$$ (resp. right derived functor $${\mathbb {R}}F: D^+({\mathbf {C}}) \rightarrow D^+({\mathbf {D}})$$) defined using $$F$$-projective resolutions (resp. $$F$$-injective resolutions). An object $$P \in {\mathbf {C}}$$ is said $$F$$-acyclic if

$$$$\begin{equation*} {\mathbb {L}}F(P) \cong F(P) \ \ \ (\text{resp. } {\mathbb {R}}F(I) \cong F(I)) \end{equation*}$$$$

in $$D^-({\mathbf {D}})$$ (resp. $$D^+({\mathbf {D}})$$). In general, the class of $$F$$-acyclic objects may not form an $$F$$-projective (resp. $$F$$-injective) class.

Example 2.5.

• (1) The category of abelian groups has enough projective objects. Indeed, every free abelian group is projective and every abelian group is a quotient of a free abelian group.
• (2) Let $$R$$ be a Banach ring. The category $$\mathbf {Ban}_R$$ has enough projective objects. We will give a proof of this fact in Proposition 3.11.
• (3) Let $$R$$ be a non-Archimedean Banach ring. The category $$\mathbf {Ban}_R^{\leqslant 1, {\rm na}}$$ does not have enough projective objects, in contrast with the category $$\mathbf {Ban}_R^{{\rm na}}$$ which, similarly to $$\mathbf {Ban}_R$$, has enough projective objects.
• (4) Both categories $$\mathbf {Born}$$ and $$\mathbf {Ind}(\mathbf {Ban})$$ have enough projective objects. Moreover, they are derived equivalent, that is, $$D(\mathbf {Ind}(\mathbf {Ban}_R)) \cong D(\mathbf {Born}_R)$$, and we will show in Proposition 3.19 that the equivalence preserves the monoidal structure (a fact for which we cannot find a reference in literature). We will prefer to work with $$\mathbf {Born}_R$$ but, from the perspective of the derived geometry that will be introduced in the following, these two categories are essentially equivalent.

2.2 The left heart of a quasi-abelian category

The derived category $$D({\mathbf {C}})$$ has a t-structure called the left t-structure (of course there exists also a right t-structure, but we only use the left t-structure in this work) whose associated truncation functors are denoted by $$\tau _L^{\leqslant n}$$ and $$\tau _L^{\geqslant n}$$. The explicit definition of this t-structure is not important for our discussion as we will need only the properties that we discuss in this section.† The heart of the left t-structure is denoted by $${\rm LH\,}({\mathbf {C}})$$ and it is an abelian category. Moreover, one has that $$D({\mathbf {C}}) \cong D({\rm LH\,}({\mathbf {C}}))$$, that is, $${\mathbf {C}}$$ and $${\rm LH\,}({\mathbf {C}})$$ are derived equivalent. We also notice that the left t-structure gives the correct notion of the cohomology of an object $$X \in D({\mathbf {C}})$$, given by

$$$$\begin{equation*} {\rm LH\,}^n(X) = \tau _L^{\leqslant n}{\left(\tau _L^{\geqslant n}(X)\right)} \in {\rm LH\,}({\mathbf {C}}) \end{equation*}$$$$

because $$X \cong 0$$ in $$D({\mathbf {C}})$$ if and only if $${\rm LH\,}^n(X) \cong 0$$ for all $$n$$.

Proposition 2.6.The objects of $${\rm LH\,}({\mathbf {C}})$$ can be described as complexes of objects of $${\mathbf {C}}$$ of the form $$[ 0 \rightarrow E \rightarrow F \rightarrow 0]$$ (with $$F$$ in degree 0), where $$E \rightarrow F$$ is a monomorphism. The morphisms of such complexes are commutative squares localized by the multiplicative system generated by the ones that are simultaneously cartesian and cocartesian.

Proof.This is in [24, Corollary 1.2.20]. Explicitly, given two objects $$[ 0 \rightarrow E_0 \rightarrow F_0 \rightarrow 0]$$ and $$[ 0 \rightarrow E_1 \rightarrow F_1 \rightarrow 0]$$ and a morphism $$f = (f_E, f_F)$$ defined by

(2.6.1)

then $${\rm cone\,}(f)$$ is the complex

$$$$\begin{equation*} 0 \rightarrow E_0 \stackrel{\binom{-d_0}{f_E}}{\rightarrow } F_0 \oplus E_1 \stackrel{(f_F, d_1)}{\rightarrow } F_1 \rightarrow 0 \end{equation*}$$$$

from which it follows immediately that $${\rm cone\,}(f)$$ is strictly exact if and only if the commutative square of (2.6.1) is both cartesian and cocartesian.$$\Box$$

One important property of $${\rm LH\,}({\mathbf {C}})$$, for our scopes, is the following.

Proposition 2.7.The category $${\mathbf {C}}$$ is a reflective subcategory of $${\rm LH\,}({\mathbf {C}})$$. The embedding functor $$i: {\mathbf {C}}\rightarrow {\rm LH\,}({\mathbf {C}})$$ sends an object of $${\mathbf {C}}$$ to a complex concentrated in degree 0, and its adjoint sends an object $$[ 0 \rightarrow E \rightarrow F \rightarrow 0]$$ in $${\rm LH\,}({\mathbf {C}})$$ to the cokernel $${\rm Coker\,}(E \rightarrow F)$$ computed in $${\mathbf {C}}$$ (it is easy to check that the cokernel does not depend on the representative using Proposition 2.6, cf. [24, Lemma 1.2.25]).

Proof. ([[24], Corollary 1.2.20])$$\Box$$

The left adjoint of the embedding functor $$i: {\mathbf {C}}\rightarrow {\rm LH\,}({\mathbf {C}})$$ is denoted by $$c: {\rm LH\,}({\mathbf {C}}) \rightarrow {\mathbf {C}}$$ and is called the classical part functor.

Corollary 2.8.The embedding functor $$i: {\mathbf {C}}\rightarrow {\rm LH\,}({\mathbf {C}})$$ preserves monomorphisms.

Corollary 2.9.With the same notation of Proposition 2.6, the morphism $$f$$ is a monomorphism if and only if the sequence

$$$$\begin{equation*} 0 \rightarrow E_0 \stackrel{\binom{-d_0}{f_E}}{\rightarrow } F_0 \oplus E_1 \stackrel{(f_F, d_1)}{\rightarrow } F_1 \end{equation*}$$$$

is exact and $$\binom{-d_0}{f_E}$$ is a strict morphism, whereas $$f$$ is an epimorphism if and only if $$(f_F, d_1)$$ is a strict epimorphism.

Proof.To simplify notation, let us write $${\mathcal {E}}_0 = [0 \rightarrow E_0 \rightarrow F_0 \rightarrow 0]$$ and $${\mathcal {E}}_1 = [0 \rightarrow E_1 \rightarrow F_1 \rightarrow 0]$$. By [24, Proposition 1.2.32], the embedding functor $$i: {\mathbf {C}}\rightarrow {\rm LH\,}({\mathbf {C}})$$ induces a derived equivalence that identifies the left t-structure on $$D({\mathbf {C}})$$ with the canonical t-structure on $$D({\rm LH\,}({\mathbf {C}}))$$. This means that $$f$$ is a monomorphism if and only if $${\rm LH\,}^{-1}({\rm cone\,}(f)) = 0$$ and $$f$$ is an epimorphism if and only if $${\rm LH\,}^0({\rm cone\,}(f)) = 0$$, as the left-heart cohomology coincides with the usual cohomology of complexes of $${\rm LH\,}({\mathbf {C}})$$ as an abelian category. In the proof of Proposition 2.6, we described $${\rm cone\,}(f)$$ as the complex

$$$$\begin{equation*} 0 \rightarrow E_0 \stackrel{\binom{-d_0}{f_E}}{\rightarrow } F_0 \oplus E_1 \stackrel{(f_F, d_1)}{\rightarrow } F_1 \rightarrow 0, \end{equation*}$$$$

where $$F_1$$ is placed in degree 0. By [24, Corollary 1.2.20] the left-heart cohomology of $${\rm cone\,}(f)$$ at degree $$n$$ is zero if and only if $${\rm cone\,}(f)$$ is strictly exact in degree $$n$$, that is precisely the condition in the statement of the corollary.$$\Box$$

Proposition 2.6 suggests an interpretation of the objects of $${\rm LH\,}({\mathbf {C}})$$ as quotients of objects of $${\mathbf {C}}$$ as follows. In $${\mathbf {C}}$$ the canonical maps from kernels of morphisms are strict monomorphisms, thus the computation of a cokernel always yields a strict short exact sequence. But if we consider a generic monomorphism $$E \rightarrow F$$ in $${\mathbf {C}}$$, then this monomorphism fits into the short exact sequence

$$$$\begin{equation*} 0 \rightarrow i(E) \rightarrow i(F) \rightarrow [0 \rightarrow E \rightarrow F \rightarrow 0] \rightarrow 0 \end{equation*}$$$$

in $${\rm LH\,}({\mathbf {C}})$$. In this sense, we think of $$[0 \rightarrow E \rightarrow F \rightarrow 0]$$ as the quotient of $$F$$ by $$E$$, that is in general different from its quotient in $${\mathbf {C}}$$ which is given by its classical part. We further emphasize this point with an example from functional analysis.

Example 2.10.Consider the quasi-abelian category $$\mathbf {Ban}_R$$ of Banach modules over a Banach ring $$R$$. Strict monomorphisms in this case are given by inclusions of closed subspaces and monomorphisms just by injective morphisms. Thus, if $$E \rightarrow F$$ is a monomorphism, then in $$\mathbf {Ban}_R$$

$$$$\begin{equation*} {\rm Coker\,}(E \rightarrow F) \cong \frac{F}{\; \overline{E} \;}, \end{equation*}$$$$

where $$\overline{E}$$ is the closure of the image of $$E$$ in $$F$$. Whereas the object $$[0 \rightarrow E \rightarrow F \rightarrow 0]$$ of $${\rm LH\,}(\mathbf {Ban}_R)$$ can be thought of as the quotient of $$F$$ by $$E$$, without the closure operation. These quotients will be important later on.

We will use the following property several times.

Lemma 2.11.Let $$F: {\mathbf {C}}\rightarrow {\mathbf {D}}$$ be a left derivable functor between quasi-abelian categories. Then, $${\mathbb {L}}F$$ is isomorphic to $${\mathbb {L}}({\rm LH\,}^0({\mathbb {L}}F))$$ and the restriction of $${\rm LH\,}^0({\mathbb {L}}F): {\rm LH\,}({\mathbf {C}}) \rightarrow {\rm LH\,}({\mathbf {D}})$$ is isomorphic to $$F$$ if and only if $$F$$ is regular† and right exact. The dual statement holds to right derivable functors.

Proof.This is proved in [24, Proposition 1.3.15].$$\Box$$

It is natural to ask what is the relation between the left t-structure and the notions of exactness introduced so far, that is, to find conditions that permit to check when a functor $$F: {\mathbf {C}}\rightarrow {\mathbf {D}}$$ ‘derives trivially’, that is, it has $${\rm LH\,}^n(D(F)) = 0$$ for all $$n \ne 0$$, where $$D(F)$$ denotes the derived functor of $$F$$, in terms of the exactness properties of $$F$$.

Proposition 2.12.Let $$F: {\mathbf {C}}\rightarrow {\mathbf {D}}$$ be a right exact functor between quasi-abelian categories and assume that $$F$$ is left derivable to a functor $${\mathbb {L}}F: D^-({\mathbf {C}}) \rightarrow D^-({\mathbf {D}})$$. Then, $${\rm LH\,}^n({\mathbb {L}}F) = 0$$ for all $$n \ne 0$$ if and only if $$F$$ is strictly left exact.

Proof.Suppose that $$F$$ is strictly left exact. By [24, Proposition 1.3.15] the functor $${\rm LH\,}^0({\mathbb {L}}F)$$ is right exact, therefore to check that it is left exact it is enough to check that it sends monomorphisms (of $${\rm LH\,}({\mathbf {C}})$$) to monomorphisms (of $${\rm LH\,}({\mathbf {D}})$$). The left and right exactness of $$F$$ imply that it preserve strict monomorphisms and strict epimorphisms and hence $$F$$ is a regular functor. This permits to apply Lemma 2.11 to deduce that the restriction of $${\rm LH\,}^0({\mathbb {L}}F)$$ to $${\mathbf {C}}$$ is isomorphic to $$F$$ and since $$F$$ preserves monomorphisms, because it is strictly left exact, then we have that

$$$$\begin{equation*} ({\rm LH\,}^0({\mathbb {L}}F))([E_0 \rightarrow F_0]) \cong [F(E_0) \rightarrow F(F_0)]. \end{equation*}$$$$

By Corollary 2.9, a morphism $$f: E = [E_0 \rightarrow F_0] \rightarrow F = [E_1 \rightarrow F_1]$$ in $${\rm LH\,}({\mathbf {C}})$$ is a monomorphism if and only if the associated sequence

$$$$\begin{equation*} 0 \rightarrow E_0 \stackrel{\alpha }{\rightarrow } F_0 \oplus E_1 \stackrel{\beta }{\rightarrow } F_1 \end{equation*}$$$$

is strictly exact at $$F_0 \oplus E_1$$, that is, the morphism $$\alpha$$ is the kernel of $$\beta$$ where $$\beta$$ is a strict morphism (that is, not necessarily strict). Here $$\alpha$$ and $$\beta$$ denote the morphisms described explicitly in Corollary 2.9. We thus have that $$({\rm LH\,}^0({\mathbb {L}}F))(f)$$ is given, up to canonical isomorphism, by the morphism

that is a monomorphism if and only if the sequence

$$$$\begin{equation*} 0 \rightarrow F(E_0) \stackrel{F(\alpha)}{\rightarrow } F(F_0 \oplus E_1) \stackrel{F(\beta)}{\rightarrow } F(F_1) \end{equation*}$$$$

is strictly exact, again by applying Corollary 2.9. But this sequence is strictly exact because $$F$$ is strictly left exact, proving that $${\rm LH\,}^0({\mathbb {L}}F)$$ is an exact functor.

On the other hand, if we suppose that $$F$$ is not left exact then $${\rm LH\,}^0({\mathbb {L}}F)$$ is not exact because it does not preserve strict exactness of complexes of $$D({\mathbf {C}})$$, otherwise $$F$$ would preserve strict exactness too. If we suppose that $$F$$ is left exact but not strictly left exact then there exists a monomorphism in $${\mathbf {C}}$$ whose kernel is not preserved by $$F$$. If $$A \rightarrow B$$ is such a monomorphism, then

$$$$\begin{equation*} 0 \rightarrow A \rightarrow B \rightarrow [A \rightarrow B] \rightarrow 0 \end{equation*}$$$$

is an exact sequence in $${\rm LH\,}({\mathbf {C}})$$ that is mapped to

$$$$\begin{equation*} 0 \rightarrow F(A) \rightarrow F(B) \rightarrow {\rm LH\,}^0({\mathbb {L}}F)([A \rightarrow B]) \rightarrow 0 \end{equation*}$$$$

because the restriction of $${\rm LH\,}^0({\mathbb {L}}F)$$ to $${\mathbf {C}}$$ is isomorphic to $$F$$ in this case by [24, Proposition 1.3.15]. But by hypothesis $$F(A) \rightarrow F(B)$$ is not a monomorphism and thus $${\rm LH\,}^0({\mathbb {L}}F)$$ not exact.$$\Box$$

Corollary 2.13.Let $$F: {\mathbf {C}}\rightarrow {\mathbf {D}}$$ be left strictly exact functor and right exact functor. Then, $${\rm LH\,}^n({\mathbb {L}}F) = 0$$ for all $$n \ne 0$$ and $${\rm LH\,}^0(F)$$ restricts to $$F$$ on $${\mathbf {C}}$$.

Proof.Proposition 2.12 and Lemma 2.11 prove the claim.$$\Box$$

We stress once more the fact that it is not enough that $$F$$ is exact for having $${\rm LH\,}^n({\mathbb {L}}F) = 0$$ for all $$n \ne 0$$, in contrast with the case of additive functors between abelian categories. So, the terminologies used in the theory of quasi-abelian categories and that of abelian categories are not in perfect agreement. Having vanishing higher order derived functors for the left-heart cohomology for a functor between quasi-abelian categories is equivalent to being right exact and strictly left exact and dually having vanishing higher order derived functors for the right-heart cohomology is equivalent to being strictly right exact and left exact. The latter claim can be proved but dualizing the argument provided so far for the left-heart cohomology. We deduce that for testing the strict exactness of a functor one must compute its higher derived functors both for the left-heart and right-heart cohomology. The main example that we will meet in applications of an exact functor for which the higher derived functors do not vanish is the completion functor $$\widehat{({-})}: \mathbf {Nr}_R \rightarrow \mathbf {Ban}_R$$. This functor is strictly right exact and left exact but not strictly left exact. Thus, although $$\widehat{({-})}$$ is an exact functor for the right-heart cohomology, it is not for the left-heart cohomology and thus it is a non-trivial functor homologically, with non-vanishing higher derived functors for the left-heart cohomology. We will also use the following lemma.

Lemma 2.14.Let $${\mathbf {C}}$$ be a quasi-abelian category with enough projective objects and $$F: {\mathbf {C}}\rightarrow {\mathbf {D}}$$ a right exact functor to another quasi-abelian category. Then, the following diagram of functors is commutative up to natural isomorphism:

(2.14.1)

Proof.It is enough to check that the functors $$F$$ and $$c \circ {\rm LH\,}(F) \circ i$$ have the same values on objects. Let $$X \in {\mathbf {C}}$$. We can write

$$$$\begin{equation*} X \cong {\rm Coker\,}(P^1 \rightarrow P^0), \end{equation*}$$$$

where the $$P^i$$ are the first two terms of a projective resolution of $$X$$. Notice that the morphism $$P^1 \rightarrow P^0$$ is strict because it is the composition of the strict epimorphism $$P^1 \rightarrow {\rm Ker\,}(P^0 \rightarrow X)$$ with the strict monomorphism $${\rm Ker\,}(P^0 \rightarrow X) \rightarrow X$$, and the post-composition of a strict epimorphism with a strict monomorphism is a strict morphism. Thus, since $$F$$ is right exact, we have

$$$$\begin{equation*} F(X) \cong {\rm Coker\,}(F(P^1) \rightarrow F(P^0)). \end{equation*}$$$$

By [24, Proposition 1.3.24], the category $${\rm LH\,}^0({\mathbf {C}})$$ has enough projective objects and the projective objects of $${\rm LH\,}^0({\mathbf {C}})$$ are isomorphic to objects of the form $$i(P)$$ for $$P$$ projective in $${\mathbf {C}}$$. Therefore, we have

$$$$\begin{equation*} i(X) \cong {\rm Coker\,}(i(P^1) \rightarrow i(P^0)) \end{equation*}$$$$

because $$i(P^1)$$ and $$i(P^0)$$ are the first two terms of a projective resolution of $$i(X)$$ (notice that the functor $$i$$ sends strictly exact sequences to exact sequences, cf. [24, Corollary 1.2.27]). Thus,

$$$$\begin{equation*} {\rm LH\,}^0(F)(X) \cong {\rm Coker\,}({\rm LH\,}^0(F)(i(P^1)) \rightarrow {\rm LH\,}^0(F)(i(P^0))) \end{equation*}$$$$

because $${\rm LH\,}^0(F)$$ is right exact and since $$c$$ is a left adjoint functor, we have

$$$$\begin{equation*} c({\rm LH\,}^0(F)(X)) \cong c({\rm Coker\,}({\rm LH\,}^0(F)(i(P^1)) \rightarrow {\rm LH\,}^0(F)(i(P^0)))) \cong \end{equation*}$$$$

$$$$\begin{equation*} {\rm Coker\,}(c({\rm LH\,}^0(F))(i(P^1)) \rightarrow c({\rm LH\,}^0(F))(i(P^0))) \cong {\rm Coker\,}(F(P^1) \rightarrow F(P^0)) \end{equation*}$$$$

because

$$$$\begin{equation*} {\rm LH\,}^0(F)(i(P)) \cong {\mathbb {L}}F(P) \cong F(X) \end{equation*}$$$$

for all projective objects of $${\mathbf {C}}$$.$$\Box$$

We now consider $${\mathbf {C}}$$ to be closed symmetric monoidal.

Proposition 2.15.Let $$({\mathbf {C}}, {\overline{\otimes }})$$ be a closed symmetric monoidal quasi-abelian category with enough projective objects. Suppose also that for all projective objects $$P, Q \in {\mathbf {C}}$$, one has that $$P {\overline{\otimes }}Q$$ is projective. Then, $${\rm LH\,}({\mathbf {C}})$$ is a closed symmetric monoidal abelian category equipped with the functors $${\rm LH\,}^0(({-}) {\overline{\otimes }}^{\mathbb {L}}({-}))$$ and $${\rm LH\,}^0({\mathbb {R}}\underline{{\rm Hom \,}}({-}, {-}))$$.

Proof.This is Proposition 1.5.3 and Corollary 1.5.4 [24].$$\Box$$

Proposition 2.15 has the following important consequence (we refer to the beginning of the next sub-section for the definition of the $$\infty$$-categories associated to $${\mathbf {C}}$$, $${\rm LH\,}({\mathbf {C}})$$ and their categories of monoids).

Corollary 2.16.In the hypothesis of Proposition 2.15, the symmetric monoidal $$\infty$$-categories $$\infty {-}{\rm LH\,}({\mathbf {C}})$$ and $$\infty {-}{\mathbf {C}}$$ are monoidally Quillen equivalent. Hence, the $$\infty$$-categories $$\infty {-}\mathbf {Comm}({\rm LH\,}({\mathbf {C}}))$$ and $$\infty {-}\mathbf {Comm}({\mathbf {C}})$$ are Quillen equivalent.

Proof.The fact that the $$\infty$$-categories $$\infty {-}{\rm LH\,}({\mathbf {C}})$$ and $$\infty {-}{\mathbf {C}}$$ are Quillen equivalent is just a restatement of the fact that the homotopy categories of both categories are equivalent to $$D^{\leqslant 0}({\mathbf {C}})$$ and the equivalences are induced by the adjunction $$c: {\rm LH\,}({\mathbf {C}}) \leftrightarrows {\mathbf {C}}: i$$, discussed in Proposition 2.7. Another way of thinking about this is to notice that the objects of both $$D^{\leqslant 0}({\mathbf {C}})$$ and $$D^{\leqslant 0}({\rm LH\,}({\mathbf {C}}))$$ can be described in terms of the projective objects of $${\mathbf {C}}$$ and $${\rm LH\,}({\mathbf {C}})$$, respectively. But since $${\mathbf {C}}$$ has enough projective the class of projective objects of $${\mathbf {C}}$$ and $${\rm LH\,}({\mathbf {C}})$$ agree, cf. [24, Proposition 1.3.24]. It is thus clear that, under the hypothesis of the corollary, this equivalence preserves the tensor products and closed structures. The assertion about the categories of commutative algebras is then a formal consequence of the monoidal Quillen equivalence.$$\Box$$

In general, the categories $$\mathbf {Comm}({\mathbf {C}})$$ and $$\mathbf {Comm}({\rm LH\,}({\mathbf {C}}))$$ are not equivalent. Moreover, there seems to be no reason for which in general the adjunction $$i: {\rm LH\,}({\mathbf {C}}) \rightleftarrows {\mathbf {C}}:c$$ must be a monoidal adjunction without any assumption on $${\mathbf {C}}$$.

Proposition 2.17.Suppose that $$({\mathbf {C}}, {\overline{\otimes }})$$ has enough projective objects, then in the adjunction of Proposition 2.7

$$$$\begin{equation} c: {\rm LH\,}({\mathbf {C}}) \rightleftarrows {\mathbf {C}}: i, \end{equation}$$$$(2.17.1)

$$i$$ is a lax monoidal functor and $$c$$ is a strong monoidal functor.

Proof.Since $$c$$ is left adjoint it is enough to show that $$c$$ is strongly monoidal as then $$i$$ is automatically lax monoidal. Since $${\mathbf {C}}$$ has enough projective objects, the class of projective objects is $${\overline{\otimes }}$$-acyclic (cf. [24, Remark 1.3.21]) and the class of projective objects of $${\mathbf {C}}$$ and $${\rm LH\,}({\mathbf {C}})$$ agree (cf. [24, Proposition 1.3.24]), then every object of $${\rm LH\,}({\mathbf {C}})$$ can be written as a complex of the form

$$$$\begin{equation*} [E \rightarrow P] \cong {\rm Coker\,}(i(E) \rightarrow i(P)), \end{equation*}$$$$

where $$E, P \in {\mathbf {C}}$$ and $$P$$ is a projective object. We denote by $${\overline{\otimes }}^{\mathbb {L}}$$ and $${\overline{\otimes }}^{\rm LH\,}$$ the monoidal structures of $$D({\mathbf {C}})$$ and of $${\rm LH\,}({\mathbf {C}})$$, respectively. Consider two such objects $$X = [E \rightarrow P]$$ and $$X^{\prime } = [E^{\prime } \rightarrow P^{\prime }]$$ then

$$$$\begin{equation*} X {\overline{\otimes }}^{\rm LH\,}X^{\prime } \cong {\rm LH\,}^0{\left(X {\overline{\otimes }}^{\mathbb {L}}X^{\prime }\right)}. \end{equation*}$$$$

We thus have

$$$$\begin{align*} X {\overline{\otimes }}^{\rm LH\,}X^{\prime } &\cong {\rm Coker \,}({\rm Coker\,}(i(E) {\overline{\otimes }}^{\rm LH\,}i(E^{\prime }) \rightarrow i(E) {\overline{\otimes }}^{\rm LH\,}i(P^{\prime })) \\ &\quad\rightarrow {\rm Coker\,}(i(P) {\overline{\otimes }}^{\rm LH\,}i(E^{\prime }) \rightarrow i(P) {\overline{\otimes }}^{\rm LH\,}i(P^{\prime }))), \end{align*}$$$$

where we have just commuted the cokernels with the functor $${\overline{\otimes }}^{\rm LH\,}$$. In order to prove the claim we have to show

$$$$\begin{equation*} c{\left(X {\overline{\otimes }}^{\rm LH\,}X^{\prime }\right)} \cong c(X) {\overline{\otimes }}c(X^{\prime }). \end{equation*}$$$$

Since $$c$$ is a left adjoint functor it commutes with cokernels, so

$$$$\begin{align*} c{\left(X {\overline{\otimes }}^{\rm LH\,}X^{\prime }\right)} &\cong {\rm Coker \,}({\rm Coker\,}(c(i(E) {\overline{\otimes }}^{\rm LH\,}i(E^{\prime })) \rightarrow c(i(E) {\overline{\otimes }}^{\rm LH\,}i(P^{\prime }))) \rightarrow {\rm Coker\,}(c(i(P) {\overline{\otimes }}^{\rm LH\,}i(E^{\prime }))\\ &\quad \rightarrow c(i(P) {\overline{\otimes }}^{\rm LH\,}i(P^{\prime })))). \end{align*}$$$$

We can now repeatedly apply Lemma 2.14 to deduce the isomorphisms

$$$$\begin{equation*} c{\left(i(E) {\overline{\otimes }}^{\rm LH\,}i(E^{\prime })\right)} \cong E {\overline{\otimes }}E^{\prime }, \ \ c{\left(i(E) {\overline{\otimes }}^{\rm LH\,}i(P^{\prime })\right)} \cong E {\overline{\otimes }}P^{\prime }, \end{equation*}$$$$

$$$$\begin{equation*} c{\left(i(P) {\overline{\otimes }}^{\rm LH\,}i(E^{\prime })\right)} \cong P {\overline{\otimes }}E^{\prime }, \ \ c{\left(i(P) {\overline{\otimes }}^{\rm LH\,}i(P^{\prime })\right)} \cong P {\overline{\otimes }}P^{\prime }. \end{equation*}$$$$

We thus get

$$$$\begin{equation*} c{\left(X {\overline{\otimes }}^{\rm LH\,}X^{\prime }\right)} \cong {\rm Coker\,}(E \rightarrow P) {\overline{\otimes }}{\rm Coker\,}(E^{\prime } \rightarrow P^{\prime }) \cong c(X) {\overline{\otimes }}c(X^{\prime }) \end{equation*}$$$$

because the functor $${\overline{\otimes }}$$ commutes with cokernels.$$\Box$$

We notice that it is easy to prove that the functor $$i: {\mathbf {C}}\rightarrow {\rm LH\,}({\mathbf {C}})$$ is strongly monoidal in the case when $${\overline{\otimes }}$$ is a regular functor (that is, preserves the strictness of morphisms). Indeed, in this case, we can directly apply [24, Proposition 1.3.15] to deduce that $$i(X) {\overline{\otimes }}^{\rm LH\,}i(X^{\prime }) \cong i(X {\overline{\otimes }}X^{\prime })$$ for all $$X, X^{\prime } \in {\mathbf {C}}$$. Unfortunately, in cases of interest, the monoidal structure is not regular. We show this fact in the case of Banach spaces over $${\mathbb {C}}$$, for the sake of simplicity. Consider the Banach space $$C([0, 1])$$ of continuous functions on the real interval $$[0, 1]$$. The Banach–Mazur Theorem gives a strict monomorphism $$\ell ^2 \rightarrow C([0, 1])$$, where $$\ell ^2$$ is the space of square summable sequences. Notice that $$\ell ^2$$ is reflexive and it is not a direct factor of $$C([0, 1])$$. Now let us consider the morphism

$$$$\begin{equation*} \phi: \ell ^2 \widehat{\otimes }\ell ^2 \rightarrow C([0, 1]) \widehat{\otimes }\ell ^2. \end{equation*}$$$$

Since Hilbert spaces have the approximation property, the morphism $$\phi$$ is injective. Let us show that it is not strict. Suppose that it is strict. This implies that it is a strict monomorphism and hence

$$$$\begin{equation*} \phi ^*: {\rm Hom \,}(C([0, 1]) \widehat{\otimes }\ell ^2, {\mathbb {C}}) \rightarrow {\rm Hom \,}(\ell ^2 \widehat{\otimes }\ell ^2, {\mathbb {C}}) \end{equation*}$$$$

is surjective. By the hom-tensor adjunction, we have the isomorphisms

$$$$\begin{equation*} {\rm Hom \,}(C([0, 1]) \widehat{\otimes }\ell ^2, {\mathbb {C}}) \cong {\rm Hom \,}(C([0, 1]), \ell ^2), \ \ {\rm Hom \,}(\ell ^2 \widehat{\otimes }\ell ^2, {\mathbb {C}}) \cong {\rm Hom \,}(\ell ^2, \ell ^2), \end{equation*}$$$$

where we used the fact that $$\ell ^2$$ is isomorphic to its dual. This implies that the strict monomorphism $$\ell ^2 \rightarrow C([0, 1])$$ has a left inverse $$C([0, 1]) \rightarrow \ell ^2$$, but this is impossible because $$\ell ^2$$ is not a direct factor of $$C([0, 1])$$. Nevertheless, it is easy to show that the monoidal structure on $$\mathbf {Ban}_k^{{\rm na}}$$, for $$k$$ a non-Archimedean field, is regular because it is a strictly exact functor. We conjecture that the monoidal structure on $$\mathbf {Ban}_R^{{\rm na}}$$ is a regular functor for all non-Archimedean Banach rings.

2.3 The homotopy Zariski topology

Henceforth, we will use the language of $$\infty$$-categories. The theory of $$\infty$$-categories can be embodied by several different concrete models (for example, using weak Kan simplicial complexes) and the resulting theory is independent of the particular chosen model. In this work, we adopt a model-free approach, and thus we just speak of $$\infty$$-categories and we use the formal properties of the resulting theory. One of the main use of the concept of $$\infty$$-category is that it permits better handling of the operation of localization of a category and gives tools for computing objects, and morphism in the localized category. Moreover, the theory permits the characterization of objects by universal properties that are not accessible by looking at the localized category as a classical category. The main example the reader should keep in mind is the localization of the category of chain complexes of an abelian category that gives rise to the derived category. The resulting category is triangulated and all monomorphisms are split, a fact that implies that the cone construction is a limit or a colimit in the classical sense. When the category of chain complexes of an abelian category is enriched with a suitable structure of $$\infty$$-category, then the cone construction becomes a cokernel in the $$\infty$$-categorical sense.

One way of constructing an $$\infty$$-category is via a model category. A model category $${\mathbf {C}}$$ is a category equipped with an extra structure that we do not specify in detail here, see [13, Definition 1.1.3] for a precise definition of model category. Among the data needed to specify a model structure, there is a class of morphisms $$W$$ whose elements are called weak equivalences. The formal inversion of the morphisms in $$W$$ gives the homotopy category $$\mathbf {Ho}({\mathbf {C}}) = {\mathbf {C}}[W^{-1}]$$. For example, in the case of chain complexes discussed so far, the class of weak equivalences is the class of quasi-isomorphisms of chain complexes. In general, there is a precise construction, called simplicial localization, that associates to any model category an $$\infty$$-category and this construction is functorial in an appropriate sense. Although not all $$\infty$$-categories can be obtained via this construction, in this paper only $$\infty$$-categories coming from model categories will be considered and for all the purposes of this paper, the terms model category and $$\infty$$-category can be considered synonymous. It is important to keep in mind the difference between the use of model categories and $$\infty$$-categories. Model categories usually permit to do explicit computations that can be hardily accessible via the abstract machinery of $$\infty$$-categories. Whereas, $$\infty$$-categories give the correct meaning and theoretical interpretation to the computations to these computations that the theory of model categories cannot provide. Indeed, in a precise sense, the theory of $$\infty$$-categories is a faithful extension of the theory of 1-categories whereas the theory of model categories is not. It is therefore fruitful to use both theories to exploit their respective advantages.

We now describe the model categories that appear in this work. Once we fix a quasi-abelian category $${\mathbf {C}}$$ that has all limits and colimits, with enough projective objects we can associate to it the model category obtained by equipping the category of simplicial objects on $${\mathbf {C}}$$, denoted $$\mathbf {Simp}({\mathbf {C}})$$, with the projective model structure (cf. [18, Theorem 4.6] for an explicit description of the projective model structure in the more general context of exact categories). The $$\infty$$-category obtained as the simplicial localization of this model category is denoted $$\infty {-}{\mathbf {C}}$$. Via the quasi-abelian version of the Dold–Kan correspondence, it is possible to identify simplicial objects of $${\mathbf {C}}$$ with chain complexes concentrated in negative degrees and the weak equivalences of the projective model structure on simplicial objects with the strict quasi-isomorphisms of chain complexes introduced so far. We will therefore switch between the simplicial objects and chain complexes as needed. We find it more convenient to do computations with chain complexes but, when $${\mathbf {C}}$$ is equipped with a symmetric monoidal structure the Dold–Kan correspondence is not symmetric monoidal. In this situation, the correct model category for $$\infty {-}{\mathbf {C}}$$ is the category of simplicial objects when considering algebras over $$\infty {-}{\mathbf {C}}$$. Therefore, it turns out that the homotopy category of $$\infty {-}{\mathbf {C}}$$ is equivalent to a full subcategory of $$D({\mathbf {C}})$$ that we denote $$D^{\leqslant 0}({\mathbf {C}})$$.

We now suppose that $${\mathbf {C}}$$ is equipped with a symmetric monoidal structure $${\overline{\otimes }}: {\mathbf {C}}\times {\mathbf {C}}\rightarrow {\mathbf {C}}$$. We suppose that $${\overline{\otimes }}$$ that is a left Quillen bifunctor for the projective model structure introduced so far on $${\mathbf {C}}$$, that is, the pair $$({\mathbf {C}}, {\overline{\otimes }})$$ is a symmetric monoidal model category. This implies that $$\infty {-}{\mathbf {C}}$$ becomes a closed symmetric monoidal $$\infty$$-category and the symmetric monoidal structure on $$\infty {-}{\mathbf {C}}$$ induces a closed monoidal structure on $$D^{\leqslant 0}({\mathbf {C}})$$. Imposing some further conditions on $${\mathbf {C}}$$, we obtain what is called a HAG context (that is, a Homotopical Algebraic Geometry Context). We refer to [26] for the precise list of technical assumptions needed on $${\mathbf {C}}$$, all of which are satisfied by the category of bornological modules we will use later on, we only recall one of the most important: We suppose that the monoidal structure $${\overline{\otimes }}$$ satisfies the monoid and the symmetric monoid axioms when extended to simplicial objects. This condition implies that the $$\infty$$-category $$\mathbf {Comm}(\infty {-}{\mathbf {C}})$$ of commutative algebras can be described as the $$\infty$$-category associated with the model category of commutative simplicial monoids over $$({\mathbf {C}}, {\overline{\otimes }})$$ where the weak equivalences are the same as the ones of the projective model structure, that is, a morphism of commutative simplicial monoids is a weak equivalence if and only if it is a weak equivalence of the underlying simplicial objects of $$\mathbf {Simp}({\mathbf {C}})$$. More precisely, the model structure on $$\mathbf {Simp}({\mathbf {C}})$$ is transferred to $$\mathbf {Comm}(\mathbf {Simp}({\mathbf {C}}))$$ via the adjunction $$\mathbf {Simp}({\mathbf {C}}) \leftrightarrows \mathbf {Comm}(\mathbf {Simp}({\mathbf {C}}))$$.

In the theory of derived geometry with respect to $${\mathbf {C}}$$, the category $$\mathbf {Comm}(\infty {-}{\mathbf {C}})$$ has the same role as the category of commutative rings in algebraic geometry or the category of commutative simplicial rings in derived algebraic geometry. It is therefore natural to give the following definition for the category of affine-derived schemes over $${\mathbf {C}}$$.

Definition 2.18.The $$\infty$$-category $$\mathbf {Comm}(\infty {-}{\mathbf {C}})^{\rm op\,}$$ is called the category of affine $$\infty$$-schemes over $${\mathbf {C}}$$ and denoted $$\mathbf {Aff}(\infty {-}{\mathbf {C}})$$. Its homotopy category is called category of affine-derived schemes and denoted $$\mathbf {dAff}({\mathbf {C}})$$.

Having defined the category $$\mathbf {Aff}(\infty {-}{\mathbf {C}})$$, we can proceed to define schemes and stacks over $${\mathbf {C}}$$ using the functor of points point of view. These will be a specific kind of $$\infty$$-functors $$\mathbf {Aff}(\infty {-}{\mathbf {C}})^{\rm op\,}\rightarrow \infty {-}\mathbf {Sets}$$, where $$\infty {-}\mathbf {Sets}$$ is the $$\infty$$-category of spaces, that is, the $$\infty$$-category obtained from the category of simplicial sets equipped with the standard model structure. To do this we need to equip $$\mathbf {Aff}(\infty {-}{\mathbf {C}})$$ with a Grothendieck topology. There are many possible choices of Grothendieck topologies, some of which are direct generalizations of the most common topologies used in algebraic geometry, like the Zariski topology and the étale topology. We limit the discussion to the homotopical analog of the Zariski topology because it is the only one relevant to this work.

By definition, a Grothendieck topology on $$\mathbf {Aff}(\infty {-}{\mathbf {C}})$$ is the datum of a Grothendieck topology on the homotopy category $$\mathbf {dAff}({\mathbf {C}})$$. Thus, we have just to deal with the common concept of Grothendieck topology. For doing that, by the assumption that $${\mathbf {C}}$$ is a HAG context, we notice that for any object $$A \in \mathbf {Comm}(\infty {-}{\mathbf {C}})$$, it makes sense to consider the symmetric monoidal $$\infty$$-category of $$A$$-modules, denoted by $$(\infty {-}\mathbf {Mod}_A, {\overline{\otimes }}_A)$$ whose homotopy category is denoted by $$(D^{\leqslant 0}(\mathbf {Mod}_A), {\overline{\otimes }}_A^{\mathbb {L}})$$. If $$A \in \mathbf {Comm}(\infty {-}{\mathbf {C}})$$, we denote the corresponding object of $$\mathbf {Aff}(\infty {-}{\mathbf {C}})$$ by $${\rm Spec \,}(A)$$. The same notation will be used for derived affine schemes, that is, for objects of the homotopy category of $$\mathbf {Aff}(\infty {-}{\mathbf {C}})$$ because the two categories have the same class of objects. We are ready to define the homotopy Zariski topology.

Definition 2.19.A morphism $${\rm Spec \,}(A) \rightarrow {\rm Spec \,}(B)$$ in $$\mathbf {Aff}(\infty {-}{\mathbf {C}})$$ is called (formal) homotopy Zariski open immersion† if the induced morphism

$$$$\begin{equation} A {\overline{\otimes }}_B A \rightarrow A \end{equation}$$$$(2.19.1)

is an equivalence in $$\infty {-}\mathbf {Mod}_A$$ (that is, an isomorphism in the homotopy category). A family $$\lbrace {\rm Spec \,}(A_i) \rightarrow {\rm Spec \,}(B) \rbrace _{i \in I}$$ of homotopy Zariski open immersions is a cover if there exists a finite subfamily $$J \subset I$$ such that the pullback functors

$$$$\begin{equation*} \lbrace ({-}) {\overline{\otimes }}_B A_i: \infty {-}\mathbf {Mod}_B \rightarrow \infty {-}\mathbf {Mod}_{A_i} \rbrace _{i \in J} \end{equation*}$$$$

form a conservative family of functors.

A morphism $$B \rightarrow A$$ whose dual $${\rm Spec \,}(A) \rightarrow {\rm Spec \,}(B)$$ is a homotopy Zariski open immersion is called a homotopy Zariski open localization.

Proposition 2.20.The homotopy Zariski open immersions and covers define a Grothendieck topology on $$\mathbf {Aff}(\infty {-}{\mathbf {C}})$$ that we call the homotopy Zariski topology.

Proof.By definition, a topology on $$\mathbf {Aff}(\infty {-}{\mathbf {C}})$$ is a usual Grothendick topology on its homotopy category. The axioms of Grothendick topology are straightforward (we skip the details) to check because equivalences are clearly homotopy Zariski open immersion, homotopy pullbacks of affine schemes (corresponding to tensor products of algebras) clearly preserve homotopy Zariski open immersion (see Proposition 2.27 below) and covers, and finally, it is also easy to check that the composition of two homotopy Zariski open immersions is a homotopy Zariski open immersion and the composition of covers give a cover.$$\Box$$

The notion of (formal) homotopy Zariski open immersion of Definition 2.19 is an analog of the notion of formal Zariski open immersion of algebraic geometry. So, it does not require any finite presentation for the morphisms. This creates the issue that the family of homotopy Zariski open immersions is huge and difficult to describe. The focus of this work will be to find for any $$X \in \mathbf {Comm}(\infty {-}{\mathbf {C}})$$, in the case when $${\mathbf {C}}$$ is the quasi-abelian category of bornological modules, a subfamily of homotopy Zariski open immersion that will provide the association to $$X$$ of a site of reasonable size, similar to the definition of the small Zariski or small étale site of scheme theory.

Given a morphism $$f: B \rightarrow A$$ in $$\mathbf {Comm}(\infty {-}{\mathbf {C}})$$, the condition of Equation (2.19.1) can usually be checked effectively through explicit computations. We will perform these computations in the specific context of analytic geometry in later sections. Instead, the cover condition of Definition 2.19 is not stated in directly computable terms. So, we would like to find a formulation of the conservativity condition that is more amenable to computations. For doing so, we need to recall the following constructions and lemmas. Consider a linear diagram of objects

$$$$\begin{equation*} \cdots \rightarrow M_n \rightarrow M_{n+1} \rightarrow \cdots \end{equation*}$$$$

in $$D({\mathbf {C}})$$ (or more generally $$D(\mathbf {Mod}_A)$$ for $$A \in \mathbf {Comm}(\infty {-}{\mathbf {C}})$$). The colimit of this diagram does not exist in $$D({\mathbf {C}})$$, in general, and is substituted by the homotopy-colimit. This latter object can be described at the level of $$\infty$$-categories, but we limit our discussion to $$D({\mathbf {C}})$$. The abstract definition of the homotopy-colimit is given as the cofiber (that is, the cone) of the canonical morphism

$$$$\begin{equation*} \bigoplus _{n \in {\mathbb {N}}} M_n \rightarrow \bigoplus _{n \in {\mathbb {N}}} M_n \end{equation*}$$$$

induced by the inductive system. But in our specific situation, the objects $$M_n$$ can be represented by complexes of objects of $${\mathbf {C}}$$, and if all the maps of the system can be represented by morphisms of the complexes $$M_n$$ we obtain a double complex whose total complex computes the homotopy-colimit. In formulas, we have

$$$$\begin{equation*} \mathop {\mathop {\rm hocolim}\limits _{\displaystyle \longrightarrow }}_{n \in {\mathbb {N}}} M_n \cong {\rm Tot\,}{\left(\cdots \rightarrow M_n \rightarrow M_{n+1} \rightarrow \cdots \right)} \end{equation*}$$$$

in $$D({\mathbf {C}})$$. We should mention that the double complex construction can be given in two different fashions, one using direct sums and another using direct products. In this case, the one with the direct sums must be taken, although in this work (almost) all the computations will be done with finite complexes and the systems will always be eventually zero. Thus, only finitely many direct sums will be computed. We need the following lemma.

Lemma 2.21.Let $$\lbrace {\rm Spec \,}(A_i) \rightarrow {\rm Spec \,}(B) \rbrace _{i \in I}$$ be a cover for the homotopy Zariski topology, then for any $$M \in \infty {-}\mathbf {Mod}_B$$ there is a quasi-isomorphism

$$$$\begin{equation*} {\rm Tot\,}{\left(0 \rightarrow \prod _{i \in I} A_i {\overline{\otimes }}_B^{\mathbb {L}}M \rightarrow \prod _{(i,j) \in I^2} A_i {\overline{\otimes }}_B^{\mathbb {L}}A_j {\overline{\otimes }}_B^{\mathbb {L}}M \rightarrow \cdots \right)} \cong \end{equation*}$$$$

$$$$\begin{equation*} {\rm Tot\,}{\left(0 \rightarrow \prod _{i \in I} A_i {\overline{\otimes }}_B^{\mathbb {L}}M \rightarrow \prod _{i < j \in I} A_i {\overline{\otimes }}_B^{\mathbb {L}}A_j {\overline{\otimes }}_B^{\mathbb {L}}M \rightarrow \cdots \right)} \end{equation*}$$$$

between the derived C̆ech–Tate complex of the cover and the alternating derived C̆ech–Tate complex.

Proof.An easy way of proving this lemma is by using Mitchell's Embedding Theorem. We notice that once we fix a cofibrant replacement of $$M$$ and all $$A_j$$ the $${\rm Tot\,}({-})$$ functor is applied to an explicit bicomplex. This bicomplex can be seen as an object in the category of complexes over $$D^-({\mathbf {C}})$$, we denote it $$\mathbf {Ch}^+(D^-({\mathbf {C}}))$$. The category $$D^-({\mathbf {C}})$$ is additive and idempotent complete, so by Quillen's Embedding Theorem it can be embedded into an abelian category once $$D^-({\mathbf {C}})$$ is equipped with the split exact structure. Then, Mitchell's Theorem implies that we can embed the resulting abelian category into a category of modules over a ring. We notice that since our set $$I$$ is finite and the embedding functors are additive the embeddings preserve finite products. Therefore, the condition that $$B \rightarrow A_i$$ is a homotopy Zariski open embedding (and hence $$A_i \widehat{\otimes }_B^{\mathbb {L}}A_i \rightarrow A_i$$ is an isomorphism in $$D^-({\mathbf {C}})$$) implies that the resulting complexes are isomorphic, respectively, to the C̆ech complex and the alternating C̆ech complex of a sheaf valued in the category of modules over a ring, and hence they are homotopic by classical computations. Since the $${\rm Tot\,}({-})$$ of homotopic complexes are quasi-isomorphic complexes of $$D^-({\mathbf {C}})$$, we obtain the claimed quasi-isomorphism.$$\Box$$

Remark 2.22.The use of Quillen's Embedding Theorem and Mitchell's Embedding Theorem in Lemma 2.21 is a bit disappointing. It is possible to write a more explicit proof of the lemma but it requires entering some detailed computations that do not fit well into this paper. We plan to write them down in future work.

We can finally reformulate the condition of being a cover for the homotopy Zariski topology as follows.

Theorem 2.23. (Derived Tate's acyclicity)Let $$\lbrace {\rm Spec \,}(A_i) \rightarrow {\rm Spec \,}(B) \rbrace _{i \in I}$$ be a finite family of homotopy Zariski open embeddings. Then, the family is a cover if and only if the associated augmented derived C̆ech–Tate complex

$$$$\begin{equation} {\rm Tot\,}{\left(0 \rightarrow B \rightarrow \prod _{i \in I} A_i \rightarrow \prod _{i,j \in I} A_i {\overline{\otimes }}_B^{\mathbb {L}}A_j \rightarrow \cdots \right)} \end{equation}$$$$(2.23.1)

is strictly acyclic.

Proof.If the complex of (2.23.1) is strictly exact and $$M \rightarrow M^{\prime }$$ is a morphism in $$\infty {-}\mathbf {Mod}_B$$ such that $$M {\overline{\otimes }}_B^{\mathbb {L}}A_i \rightarrow M^{\prime } {\overline{\otimes }}_B^{\mathbb {L}}A_i$$ is an equivalence for all $$i$$, then

$$$$\begin{equation*} M \cong M {\overline{\otimes }}_B^{\mathbb {L}}{\rm Tot\,}{\left(\prod _{i \in I} A_i \rightarrow \prod _{i,j \in I} A_i {\overline{\otimes }}_B^{\mathbb {L}}A_j \rightarrow \cdots) \cong {\rm Tot\,}(\prod _{i \in I} M {\overline{\otimes }}_B^{\mathbb {L}}A_i \rightarrow \prod _{i,j \in I} M {\overline{\otimes }}_B^{\mathbb {L}}A_i {\overline{\otimes }}_B^{\mathbb {L}}A_j \rightarrow \cdots \right)} \end{equation*}$$$$

$$$$\begin{equation*} \cong {\rm Tot\,}{\left(\prod _{i \in I} M^{\prime } {\overline{\otimes }}_B^{\mathbb {L}}A_i \rightarrow \prod _{i,j \in I} M^{\prime } {\overline{\otimes }}_B^{\mathbb {L}}A_i {\overline{\otimes }}_B^{\mathbb {L}}A_j \rightarrow \cdots \right)} \cong M^{\prime }. \end{equation*}$$$$

The converse implication is an application of the Barr–Beck Theorem. An easy way to see this is to notice that the condition of strict exactness of the augmented derived C̆ech–Tate complex† is equivalent to the condition of $$\prod _{i \in I} A_i$$ admitting descent of [21, Proposition 3.20].$$\Box$$

Theorem 2.23 gives a computational way of checking the cover condition of Definition 2.19 but still the complex of Equation (2.23.1) is not easy to compute in practice. Much easier is to use the alternating version of the complex.

Corollary 2.24.With the same notation and hypothesis of Theorem 2.23, the augmented alternating derived C̆ech–Tate complex

$$$$\begin{equation} {\rm Tot\,}{\left(0 \rightarrow B \rightarrow \prod _{i \in I} A_i \rightarrow \prod _{i < j \in I} A_i {\overline{\otimes }}_B^{\mathbb {L}}A_j \rightarrow \cdots \right)} \end{equation}$$$$(2.24.1)

is strictly acyclic if and only if the family of morphisms is a cover.

Proof.The corollary is just a combination of Theorem 2.23 and Lemma 2.21.$$\Box$$

2.4 The homotopy Zariski site

In the previous section, we have proved that the notions of homotopy Zariski open immersion and homotopy Zariski cover define a Grothendieck topology on $$\infty {-}\mathbf {Aff}({\mathbf {C}})$$. This means that by fixing any object $$X \in \infty {-}\mathbf {Aff}({\mathbf {C}})$$, we can restrict this topology to the slice category of objects over $$X$$. Using the duality $$\infty {-}\mathbf {Aff}({\mathbf {C}}) = \infty {-}\mathbf {Comm}({\mathbf {C}})^{\rm op\,}$$ this is the same as contravariantly associating an $$\infty$$-site to any object of $$\infty {-}\mathbf {Comm}({\mathbf {C}})$$. We give a name to this site.

Definition 2.25.Let $$A \in \infty {-}\mathbf {Comm}({\mathbf {C}})$$ we define the homotopy Zariski $$\infty$$-site associated to $$A$$ as the $$\infty$$-site obtained by restricting the homotopy Zariski topology of $$\infty {-}\mathbf {Aff}({\mathbf {C}})$$ to the slice category over $${\rm Spec \,}(A)$$. We denote it by $${\rm Zar\,}_A$$.

In our setting the category $${\mathbf {C}}$$ is small with respect to a bigger strongly inaccessible cardinal than the one we fixed from the beginning. Therefore, the next theorem holds true.

Theorem 2.26.If $$A \in \infty {-}\mathbf {Comm}({\mathbf {C}})$$, then the $$\infty$$-topos $${\rm Zar\,}_A$$ defined by the (formal) homotopy Zariski topology has enough points.

Proof.This is a particular case of Theorem 4.1 of [20] because the (formal) homotopy Zariski topology is finitary (cf. Definition 3.17 of [20]).$$\Box$$

One issue of Theorem 2.26 is in the word ‘formal’ which we usually ignore in our discussions. The morphisms of $$\infty {-}\mathbf {Aff}_A$$ do not have any size restriction, therefore this class of morphisms seems difficult to describe in full generality. One of the main goals of this work is to fix this issue in the case when $$A$$ is a non-Archimedean Banach ring, or a bornological ring, by finding an explicitly describable canonical sub-$$\infty$$-site of $$\infty {-}\mathbf {Aff}_A$$ and relate the $$\infty$$-site we obtain with the adic spectrum of $$A$$. The next, easy, proposition is a step in the proof that the homotopy Zariski topology is well defined, and thus already used so far. We find it convenient to single this property out for referring to it.

Proposition 2.27.Let $${\rm Spec \,}(A) \rightarrow {\rm Spec \,}(B)$$ in $$\infty {-}\mathbf {Aff}({\mathbf {C}})$$ be a homotopy Zariski open immersion, then for any $${\rm Spec \,}(C) \rightarrow {\rm Spec \,}(B)$$ the homotopy base change $${\rm Spec \,}(A) \times _{{\rm Spec \,}(B)}^{\mathbb {R}}{\rm Spec \,}(C) \rightarrow {\rm Spec \,}(B)$$ is a homotopy Zariski open immersion.

Proof.By the duality $$\infty {-}\mathbf {Aff}({\mathbf {C}}) \cong \infty {-}\mathbf {Comm}({\mathbf {C}})^{\rm op\,}$$, homotopy pullbacks in $$\infty {-}\mathbf {Aff}({\mathbf {C}})$$ correspond to homotopy pushouts in $$\infty {-}\mathbf {Comm}({\mathbf {C}})$$. The latter are computed via the derived tensor product. Thus, if we have $$B \rightarrow A$$ is a homotopy Zariski open localization and we are given a $$B \rightarrow C$$, then

$$$$\begin{equation*} {\left(C {\overline{\otimes }}_B^{\mathbb {L}}A\right)} {\overline{\otimes }}_C^{\mathbb {L}}{\left(C {\overline{\otimes }}_B^{\mathbb {L}}A\right)} \cong {\left(A {\overline{\otimes }}_B^{\mathbb {L}}C\right)} {\overline{\otimes }}_C^{\mathbb {L}}{\left(C {\overline{\otimes }}_B^{\mathbb {L}}A\right)} \cong A {\overline{\otimes }}_B^{\mathbb {L}}C {\overline{\otimes }}_B^{\mathbb {L}}A \cong C {\overline{\otimes }}_B^{\mathbb {L}}A {\overline{\otimes }}_B^{\mathbb {L}}A \cong C {\overline{\otimes }}_B^{\mathbb {L}}A \end{equation*}$$$$

proving that $$C \rightarrow C {\overline{\otimes }}_B^{\mathbb {L}}A$$ is a homotopy Zariski open localization.$$\Box$$

Since the condition

$$$$\begin{equation*} A {\overline{\otimes }}_B^{\mathbb {L}}A \stackrel{\cong }{\rightarrow } A \end{equation*}$$$$

of Definition 2.19 reminds of the condition for the map $$B \rightarrow A$$ being an epimorphism, we will often use the name homotopy epimorphism to refer to homotopy Zariski open localizations. We also are interested in understanding homotopy filtered colimits in $$\infty {-}\mathbf {Comm}({\mathbf {C}})$$.

Proposition 2.28.Let $$({\mathbf {C}}, {\overline{\otimes }})$$ be a closed symmetric monoidal quasi-abelian category as above. Let $$\lbrace f_i: A_i \rightarrow B_i \rbrace _{i \in I}$$ be a filtered family of homotopy epimorphisms in $$\infty {-}\mathbf {Comm}({\mathbf {C}})$$, then

$$$$\begin{equation*} {\mathbb {L}}\mathop {\lim \limits _{\displaystyle \rightarrow }}_{i \in I} f_i: {\mathbb {L}}\mathop {\lim \limits _{\displaystyle \rightarrow }}_{i \in I} A_i \rightarrow {\mathbb {L}}\mathop {\lim \limits _{\displaystyle \rightarrow }}_{i \in I} B_i \end{equation*}$$$$

is a homotopy epimorphism.

Proof.For any $$i$$ we have that the morphism

$$$$\begin{equation*} A_i {\overline{\otimes }}_{B_i}^{\mathbb {L}}A_i \rightarrow A_i \end{equation*}$$$$

induced by $$f_i$$ is an isomorphism in the homotopy category. Therefore, we get an equivalence

$$$$\begin{equation*} {\mathbb {L}}\mathop {\lim \limits _{\displaystyle \rightarrow }}_{i \in I} A_i {\overline{\otimes }}_{B_i}^{\mathbb {L}}A_i \rightarrow {\mathbb {L}}\mathop {\lim \limits _{\displaystyle \rightarrow }}_{i \in I} A_i \end{equation*}$$$$

and since $${\overline{\otimes }}_{B_i}^{\mathbb {L}}$$ commutes with $${\mathbb {L}}\mathop {\lim \limits _{ \rightarrow }}$$ and the colimit is filtered, we get an isomorphism

$$$$\begin{equation*} ({\mathbb {L}}\mathop {\lim \limits _{\displaystyle \rightarrow }}_{i \in I} A_i) {\overline{\otimes }}_{{\mathbb {L}}\underset{i \in I}{\mathop {\lim \limits _{\displaystyle \rightarrow }}}B_i}^{\mathbb {L}}({\mathbb {L}}\mathop {\lim \limits _{\displaystyle \rightarrow }}_{i \in I} A_i) \rightarrow {\mathbb {L}}\mathop {\lim \limits _{\displaystyle \rightarrow }}_{i \in I} A_i \end{equation*}$$$$

because of the functorial isomorphism

$$$$\begin{equation*} {\mathbb {L}}\mathop {\lim \limits _{\displaystyle \rightarrow }}_{i \in I} ({-}) {\overline{\otimes }}_{B_i}^{\mathbb {L}}({-}) \cong ({-}) {\overline{\otimes }}_{{\mathbb {L}}\underset{i \in I}{\mathop {\lim \limits _{\displaystyle \rightarrow }}}B_i}^{\mathbb {L}}({-}). \end{equation*}$$$$

$$\Box$$

3 QUASI-ABELIAN CATEGORIES FOR ANALYTIC GEOMETRY

In this section, we consider particular cases of the symmetric monoidal quasi-abelian categories discussed in Section 2 that are relevant in analytic geometry. These categories are the category of Banach modules, the contracting category of Banach modules, and the category of (complete) bornological modules. We now recall their definitions and basic properties.

3.1 The category of Banach modules

Let $$R$$ be a Banach ring. By this we mean that $$R$$ is a ring equipped with a Banach norm such that the multiplication and addition maps are bounded morphisms of Banach abelian groups (more precisely the addition is supposed to be a contracting map, that is, the triangle inequality holds). In this work, we also suppose $$R$$ to be non-Archimedean and that Banach modules over $$R$$ are equipped with a non-Archimedean norm although none of these restrictions are necessary for the theory to work. We will comment more on the differences between the general case and the non-Archimedean case when these occur later on. Therefore, from now on Banach rings or modules are always supposed to be equipped with a non-Archimedean norm if not stated otherwise. Thus, from here on, to simplify the notation, $$\mathbf {Ban}_R$$ will denote what in the previous section has been denoted by $$\mathbf {Ban}_R^{\rm na}$$.

The category of (non-Archimedean) Banach rings has an initial object given by $${\mathbb {Z}}_{\rm triv}= ({\mathbb {Z}},|{-}|_0)$$ where $$|{-}|_0$$ is the trivial norm that assumes the value 1 on all $$n \ne 0$$.† The category $$\mathbf {Ban}_R$$ of Banach $$R$$-modules is defined as the category of Banach abelian groups‡ equipped with a bounded $$R$$-action and bounded morphisms between them. The completed projective tensor product of two objects $$M, N \in \mathbf {Ban}_R$$ is defined as

$$$$\begin{equation*} M \widehat{\otimes }_R N = \widehat{(M \otimes _R N,|{-}|_{M \otimes _R N}),} \end{equation*}$$$$

where $$\widehat{({-})}$$ denotes the separated completion and

$$$$\begin{equation*} |x|_{M \otimes _R N} = \inf {\left\lbrace \max|a_i||b_i| \ \big\vert \ x = \sum a_i \otimes b_i \right\rbrace} \end{equation*}$$$$

for any $$x \in M \otimes _R N$$. We recall the following result.

Proposition 3.1.The category $$\mathbf {Ban}_R$$ is quasi-abelian. Moreover, the monoidal structure given by the completed projective tensor product is closed, and its right adjoint is given by the hom-sets equipped with the Banach $$R$$-module structure given by the operator norm.

Proof.Cf. [2, Proposition 3.15 and Proposition 3.17].$$\Box$$

The main drawback of the category $$\mathbf {Ban}_R$$ is that it does not have any infinite products or any infinite coproducts. To remedy this issue, we will introduce the category of bornological modules. Other choices are possible, but we hope to convince the reader that this is the best choice (known to the authors) for our goals. We now introduce flatness in the context of Banach modules.

Definition 3.2.We say that a Banach $$R$$-module $$M$$ is flat if the functor $$({-}) \widehat{\otimes }_R M$$ is strictly exact (in the sense of Definition 2.2).

More explicitly, $$M \in \mathbf {Ban}_R$$ is flat if the functor $$({-}) \widehat{\otimes }_R M$$ preserves the kernel of any morphism. In the next section, when we will study the contracting category of Banach modules, we will prove that all projective objects of $$\mathbf {Ban}_R$$ are flat (cf. Proposition 3.11) and in particular $$\widehat{\otimes }_R$$-acyclic (as a consequence of Corollary 2.13). Therefore, $$\mathbf {Ban}_R$$ has enough $$\widehat{\otimes }_R$$-acyclic objects and it follows from Proposition 2.17 that the inclusion functor $$\mathbf {Ban}_R \rightarrow {\rm LH\,}(\mathbf {Ban}_R)$$ is lax monoidal and that its adjoint $${\rm LH\,}(\mathbf {Ban}_R) \rightarrow \mathbf {Ban}_R$$ is strongly monoidal.

The next category that we describe is the contracting category of Banach modules.

3.2 The contracting category of Banach modules

Let $$R$$ be a (non-Archimedean) Banach ring.

Definition 3.3.The contracting category of Banach $$R$$-modules is the subcategory $$\mathbf {Ban}_R^{\leqslant 1} \subset \mathbf {Ban}_R$$ where the hom-sets are given by considering only contracting morphisms.

Notice that the categories $$\mathbf {Ban}_R^{\leqslant 1}$$ and $$\mathbf {Ban}_R$$ have the same class of objects and they only differ for the hom-sets. Moreover, isomorphism classes of objects in $$\mathbf {Ban}_R^{\leqslant 1}$$ and $$\mathbf {Ban}_R$$ differ because in the former category, modules are isomorphic if and only if they are isometrically isomorphic whereas in the latter isomorphic modules are equipped with equivalent norms.

Proposition 3.4.The category $$\mathbf {Ban}_R^{\leqslant 1}$$ is quasi-abelian and is complete and cocomplete. Moreover, the closed monoidal structure of $$\mathbf {Ban}_R$$ restricts to a well-defined closed monoidal structure on $$\mathbf {Ban}_R^{\leqslant 1}$$.

Proof.This can be checked in the same way one checks that $$\mathbf {Ban}_R$$ is quasi-abelian noticing that the property that all norms involved must be ultrametric is necessary to ensure that the hom-sets are abelian groups. We omit the details. A proof of the fact that $$\mathbf {Ban}_R^{\leqslant 1}$$ has all limits and colimits can be found in [2, Proposition 3.21]. The assertion about the closed monoidal structure immediately follows from the explicit definitions given by the formulas for the norms of both the tensor product and the operator norm.$$\Box$$

Remark 3.5.The property of $$\mathbf {Ban}_R^{\leqslant 1}$$ being quasi-abelian is a very distinctive feature of ultrametric Banach rings. Indeed, if $$R$$ is not equipped with an ultrametric norm, then $$\mathbf {Ban}_R^{\leqslant 1}$$ is not an additive category, but besides the lack of additivity, it has all the other properties discussed so far. We do not discuss this version of the theory as the non-additivity of $$\mathbf {Ban}_R^{\leqslant 1}$$ would force us to introduce more abstract constructions that will lead us too far astray from the main results of this work.

We give an explicit description of products and coproducts in $$\mathbf {Ban}_R^{\leqslant 1}$$.

Proposition 3.6.Let $$\lbrace M_i \rbrace _{i \in I}$$ be a small family of objects of $$\mathbf {Ban}_R^{\leqslant 1}$$. Then, their coproduct is given by

$$$$\begin{equation*} {\coprod _{i \in I}}^{\leqslant 1} M_i = {\left\lbrace (m_i) \in \prod _{i \in I} M_i| \lim _{i \in I}|m_i| = 0 \right\rbrace}, \end{equation*}$$$$

equipped with the $$\sup$$-norm and the product is given by

$$$$\begin{equation*} {\prod _{i \in I}}^{\leqslant 1} M_i = {\left\lbrace (m_i) \in \prod _{i \in I} M_i| \sup _{i \in I}|m_i| < \infty \right\rbrace}, \end{equation*}$$$$

equipped with the $$\sup$$-norm.

Proof.See [2, Proposition 3.21].$$\Box$$

Definition 3.7.A Banach ring $$R$$ is said uniform if its norm is equivalent to the spectral semi-norm.†

For uniform Banach rings, we usually suppose that the norm is equal to the spectral norm as this holds up to isomorphism in $$\mathbf {Ban}_{{\mathbb {Z}}_{\rm triv}}$$.

Definition 3.8.We define the (1-dimensional) Tate algebra‡ over $${\mathbb {Z}}_{\rm triv}$$ of radius $$\rho$$ as

$$$$\begin{equation*} T_{{\mathbb {Z}}_{\rm triv}}(\rho) = {\mathbb {Z}}_{\rm triv}\langle \rho ^{-1} T \rangle = {\left\lbrace \sum _{i = 0}^\infty a_i T^i \in {\mathbb {Z}}[\![ T]\!] \Bigm| \lim _{i \rightarrow \infty }|a_i|_0 \rho ^i = 0 \right\rbrace}. \end{equation*}$$$$

For any Banach ring $$R$$, the (1-dimensional) Tate algebra over $$R$$ of radius $$\rho$$ is defined as

$$$$\begin{equation*} T_{{\mathbb {Z}}_{\rm triv}}(\rho) \widehat{\otimes }_{{\mathbb {Z}}_{\rm triv}} R = R \langle \rho ^{-1} T \rangle = {\left\lbrace \sum _{i = 0}^\infty a_i T^i \in R [\![ T]\!] \Bigm| \lim _{i \rightarrow \infty }|a_i|_0 \rho ^i = 0 \right\rbrace}. \end{equation*}$$$$

The definition of Tate algebra can be easily generalized to any finite set of variables just by inductively defining

$$$$\begin{equation*} T_{{\mathbb {Z}}_{\rm triv}}(\rho _1, \ldots , \rho _n) = {\mathbb {Z}}_{\rm triv}\langle \rho _1^{-1} T_1, \ldots , \rho _n^{-1} T_n \rangle \doteq {\mathbb {Z}}_{\rm triv}\langle \rho _1^{-1} T_1, \ldots , \rho _{n - 1}^{-1} T_{n-1} \rangle \langle \rho _n^{-1} T_n \rangle. \end{equation*}$$$$

Then, we define

$$$$\begin{equation*} T_R(\rho _1, \ldots , \rho _n) = T_{{\mathbb {Z}}_{\rm triv}}(\rho _1, \ldots , \rho _n) \widehat{\otimes }_{{\mathbb {Z}}_{\rm triv}} R. \end{equation*}$$$$

We now study flatness properties of projective objects of $$\mathbf {Ban}_R$$. We recall from section 1 of [4] that a normed set is a pointed set $$(X, \star)$$ equipped with a function $$|{-}|_X: X \rightarrow {\mathbb {R}}_{\geqslant 0}$$ such that $$|x|_X = 0 \iff x = \star$$ for $$x \in X$$. For any normed set $$(X,|{-}|_X)$$, we define the topologically free Banach $$R$$-module

$$$$\begin{equation} c^0(X) = {\coprod _{\star \ne x \in X}}^{\leqslant 1} R_{|x|_X}, \end{equation}$$$$(3.8.1)

where $$R_{|x|_X}$$ denotes $$R$$ considered as a Banach module over itself where the norm has been rescaled by the real number $$|x|_X$$. For example, if $$(X,|{-}|_X) = ({\mathbb {N}},|{-}|_0)$$, where $$|n|_0 = 1$$ for all $$n \in {\mathbb {N}}$$, then

$$$$\begin{equation*} c^0({\mathbb {N}}) = {\left\lbrace (a_n) \in R^{\mathbb {N}}\Bigm| \lim _{n \rightarrow \infty }|a_n| = 0 \right\rbrace} \end{equation*}$$$$

equipped with the $$\max$$ norm.

Proposition 3.9.For any normed set $$(X,|{-}|_X)$$ the Banach $$R$$-module $$c^0(X)$$ is projective in $$\mathbf {Ban}_R$$.

Proof.It is easy to see that the objects $$R_{|x|_X}$$ are projective in $$R$$ and coproducts of projective objects are projective objects. For more details, see [2, Lemma 3.26].$$\Box$$

Remark 3.10.We warn the reader that the objects $$c^0(X)$$ are not projective in $$\mathbf {Ban}_R^{\leqslant 1}$$. Therefore, $$\mathbf {Ban}_R$$ and $$\mathbf {Ban}_R^{\leqslant 1}$$ are very different from the point of view of homological algebra.

In particular, for any object $$M \in \mathbf {Ban}_R$$, we can consider

$$$$\begin{equation*} c^0(M) = {\coprod _{0 \ne m \in M}}^{\leqslant 1} R_{|m|}, \end{equation*}$$$$

where $$M$$ is considered as a normed set by forgetting his $$R$$-module structure.

Proposition 3.11.The canonical morphism $$c^0(M) \rightarrow M$$ is a strict epimorphism in $$\mathbf {Ban}_R^{\leqslant 1}$$. In particular, $$\mathbf {Ban}_R$$ has enough projective objects.

Proof.See [2, Lemma 3.27].$$\Box$$

Proposition 3.11 immediately implies the following corollary.

Corollary 3.12.All projective objects of $$\mathbf {Ban}_R$$ are direct summands of some $$c^0(X)$$.

Proof.Projective objects splits strict epimorphism. Therefore, if $$P$$ is projective, then the strict epimorphism $$c^0(P) \rightarrow P$$ splits.$$\Box$$

Now that we know how projective objects of $$\mathbf {Ban}_R$$ look like we are ready to prove that they are flat.

Proposition 3.13.In $$\mathbf {Ban}_R$$, projective objects are flat.†

Proof.By Corollary 3.12, we need to check only that projective objects of the form $$c^0(X)$$, for some normed set $$(X,|{-}|_X)$$, are flat. Since $$({-}) \widehat{\otimes }_R ({-})$$ is a left adjoint functor, it is a strictly right exact functor. Hence, by Proposition 2.3 to check that $$c^0(X)$$ is flat, we only need to check that the functor $$({-}) \widehat{\otimes }_R c^0(X)$$ preserves monomorphisms and strict monomorphisms.

Let $$f: M \rightarrow N$$ be a monomorphism in $$\mathbf {Ban}_R$$, that is, this means that $$f$$ is an injective map. Then, since the functor $$\widehat{\otimes }_R$$ commutes with contracting colimits (as it is a left adjoint functor),

$$$$\begin{equation*} c^0(X) \widehat{\otimes }_R M = {\coprod _{\star \ne x \in X}}^{\leqslant 1} R_{|x|_X} \widehat{\otimes }_R M = {\coprod _{\star \ne x \in X}}^{\leqslant 1} M_{|x|_X}, \end{equation*}$$$$

where the notation is self-explanatory enough, and the same for $$N$$. Therefore, from this explicit description, it is clear that the induced map

$$$$\begin{equation*} {\rm Id\,}\widehat{\otimes }f: c^0(X) \widehat{\otimes }_R M \rightarrow c^0(X) \widehat{\otimes }_R N \end{equation*}$$$$

is injective. Now, let us suppose that $$f$$ is a strict monomorphism in $$\mathbf {Ban}_R$$. We need to show that the induced injective morphism

$$$$\begin{equation*} {\coprod _{\star \ne x \in X}}^{\leqslant 1} M_{|x|_X} \rightarrow {\coprod _{\star \ne x \in X}}^{\leqslant 1} N_{|x|_X} \end{equation*}$$$$

has closed image. This follows immediately from the fact that the inclusion $$i_x: N_{|x|_X} \rightarrow {\coprod _{\star \ne x \in X}}^{\leqslant 1} N_{|x|_X}$$ has a bounded inverse $$\pi _x: {\coprod _{\star \ne x \in X}}^{\leqslant 1} N_{|x|_X} \rightarrow N_{|x|_X}$$ and therefore every Cauchy sequence of elements of $${\coprod _{\star \ne x \in X}}^{\leqslant 1} M_{|x|_X}$$ with respect to the norm on $${\coprod _{\star \ne x \in X}}^{\leqslant 1} N_{|x|_X}$$ is sent to a Cauchy sequence in $N_{{|x|}_{}}$