Abstract
We establish a connection between continuous Ktheory and integral cohomology of rigid spaces. Given a rigid analytic space over a complete discretely valued field, its continuous Kgroups vanish in degrees below the negative of the dimension. Likewise, the cohomology groups vanish in degrees above the dimension. The main result provides the existence of an isomorphism between the lowest possibly nonvanishing continuous Kgroup and the highest possibly nonvanishing cohomology group with integral coefficients. A key role in the proof is played by a comparison between cohomology groups of an admissible ZariskiRiemann space with respect to different topologies; namely, the rhtopology which is related to Ktheory as well as the Zariski topology whereon the cohomology groups in question rely.
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1 Introduction
The negative algebraic Ktheory of a scheme is related to its singularities. If X is a regular scheme, then \({{\,\textrm{K}\,}}_{i}(X)\) vanishes for \(i>0\). For an arbitrary noetherian scheme X of dimension d we know that

(i)
\({{\,\textrm{K}\,}}_{i}(X)=0\) for \(i>d\),

(ii)
\({{\,\textrm{K}\,}}_{i}(X) \cong {{\,\textrm{K}\,}}_{i}(X\times \textbf{A}^n)\) for \(i\ge d\), \(n\ge 1\), and

(iii)
\({{\,\textrm{K}\,}}_{d}(X) \cong {{\,\textrm{H}\,}}_{\textrm{cdh}}^d(X;\textbf{Z})\).
The affine case of (i) was a question of Weibel [52, 2.9] who proved (i) and (ii) for \(d\le 2\) [53, 2.3, 2.5, 4.4]. For varieties in characteristic zero (i)(iii) were proven by CortiñasHaesemeyerSchlichtingWeibel [13] and the general case is due to KerzStrunkTamme [31]. As an example for the lowest possibly nonvanishing group \({{\,\textrm{K}\,}}_{d}(X)\), the cusp \(C=\{y^2=x^3\}\) over a field has \({{\,\textrm{K}\,}}_{1}(C)=0\) whereas the node \(N = \{y^2=x^3+x^2\}\) over a field (of characteristic not 2) has \({{\,\textrm{K}\,}}_{1}(N) = \textbf{Z}\); more generally, for a nice curve the rank is the number of loops [53, 2.3]. The main result of this article are analogous statements of (i)(iii) for continuous Ktheory of rigid analytic spaces in the sense of Morrow [38].
There is a long history of versions of Ktheory for topological rings that take the topology into account. For instance, the higher algebraic Kgroups of a ring A can be defined via the classifying space \({{\,\textrm{BGL}\,}}(A)\) of the general linear group \({{\,\textrm{GL}\,}}(A)\). If A happens to be a Banach algebra over the complex numbers, it also makes sense to consider \({{\,\textrm{GL}\,}}(A)\) as a topological group and to define topological Ktheory \({{\,\textrm{K}\,}}^\textrm{top}(A)\) analogously in terms of the classifying space \({{\,\textrm{BGL}\,}}^\textrm{top}(A)\). This yields a better behaved Ktheory for complex Banach algebras which satisfies homotopy invariance and excision (which does not hold true in general for algebraic Ktheory). Unfortunately, a similar approach for nonarchimedean algebras does not behave well since the nonarchimedean topology is totally disconnected. KaroubiVillamayor [33] and Calvo [12] generalised topological Ktheory to arbitrary Banach algebras (either nonarchimedean or complex) in terms of the ring of power series converging on a unit disc. A different approach is to study continuous Ktheory which is the prospectrum
where R is an Iadic ring with respect to some ideal \(I\subset R\) (e.g. \(\textbf{Z}_p\) with the padic topology or \(\textbf{F}_p\llbracket t\rrbracket \) with the tadic topology). Such “continuous” objects have been studied amply in the literature – cf. Wagoner [50, 51], Dundas [14], GeisserHesselholt [17, 18], or Beilinson [3] – and they were related by BlochEsnaultKerz to the Hodge conjecture for abelian varieties [4] and the padic variational Hodge conjecture [5]. Morrow [38] suggested an extension of continuous Ktheory to rings A admitting an open subring \(A_0\) which is Iadic with respect to some ideal I of \(A_0\) (e.g. \(\textbf{Q}_p = \textbf{Z}_p[p^{1}]\) or \(\textbf{F}_p(\!(t)\!) = \textbf{F}_p\llbracket t\rrbracket [t^{1}]\)).^{Footnote 1} This notion was recently studied by KerzSaitoTamme [32] and they showed that it coincides in nonpositive degrees with the groups studied by KaroubiVillamayor and Calvo. For an affinoid algebra A over a discretely valued field, Kerz proved the corresponding analytical statements to (i) and (ii); that is replacing algebraic Ktheory by continuous Ktheory and the polynomial ring by the ring of power series converging on a unit disc [28]. Morrow showed that continuous Ktheory extends to a sheaf of prospectra on rigid kspaces for any discretely valued field k. The main result of this article provides analogous statements of (i)(iii) above for continuous Ktheory of rigid kspaces; the statements (i) and (ii) extend Kerz’ result to the global case and statement (iii) is entirely new.
Theorem A
(Theorem 8.8, Theorem 8.12) Let X be a quasicompact and quasiseparated rigid kspace of dimension d over a discretely valued field k. Then:

(i)
For \(i> d\) we have \({{\,\textrm{K}\,}}^\textrm{cont}_{i}(X)=0\).

(ii)
For \(i\ge d\) and \(n\ge 0\) the canonical map
$$\begin{aligned} {{\,\textrm{K}\,}}^\textrm{cont}_{i}(X) \overset{}{\rightarrow }{{\,\textrm{K}\,}}^\textrm{cont}_{i}(X\times {\textbf{B}}_k^n) \end{aligned}$$is an isomorphism where \({\textbf{B}}_k^n:={{\,\textrm{Spm}\,}}(k\langle t_1,\ldots ,t_n\rangle )\) is the rigid unit disc.

(iii)
If \(d\ge 2\) or if there exists a formal model of X which is algebraic (Definition 5.1, e.g. X is affinoid or projective), then there exists an isomorphism
$$\begin{aligned} {{\,\textrm{K}\,}}^\textrm{cont}_{d}(X) \cong {{\,\textrm{H}\,}}^d(X;\textbf{Z}) \end{aligned}$$where the righthand side is sheaf cohomology with respect to the admissible topology on the category of rigid kspaces.
There are several approaches to nonarchimedean analytic geometry. Our proof uses rigid analytic spaces in the sense of Tate [44] and adic spaces introduced by Huber [27]. Another approach is the one of Berkovich spaces [6] for which there is also a version of our main result as conjectured in the affinoid case by Kerz [28, Conj. 14].
Corollary B
(Corollary 8.10) Let X be a quasicompact and quasiseparated rigid analytic space of dimension d over a discretely valued field. Assume that \(d\ge 2\) or that there exists a formal model of X which is algebraic (e.g. X is affinoid or projective). Then there is an isomorphism
where \(X^\textrm{berk}\) is the Berkovich space associated with X.
If X is smooth over k or the completion of a kscheme of finite type, then there is an isomorphism
with singular cohomology by results of Berkovich [7] and HrushovskiLoeser [24]. The identification of Corollary B is very helpful since it is hard to actually compute Kgroups whereas the cohomology of Berkovich spaces is amenable for computations. For instance, the group \({{\,\textrm{H}\,}}^d(X^\textrm{berk};\textbf{Z})\) is finitely generated since \(X^\textrm{berk}\) has the homotopy type of a finite CWcomplex; such a finiteness statement is usually unknown for Ktheory.
An important tool within the proof of Theorem A is the admissible ZariskiRiemann space \(\langle X\rangle _U\) which we will associate, more generally, with every quasicompact and quasiseparated scheme X with open subscheme U. The admissible ZariskiRiemann space \(\langle X\rangle _U\) is given by the limit of all Umodifications of X in the category of locally ringed spaces (Definition 3.1). In our case of interest where A is a reduced affinoid algebra and \(A^\circ \) is its open subring of powerbounded elements, then we will set \(X={{\,\textrm{Spec}\,}}(A^\circ )\) and \(U={{\,\textrm{Spec}\,}}(A)\). We shall relate its Zariski cohomology to the cohomology with respect to the socalled rhtopology, i.e. the minimal topology generated by the Zariski topology and abstract blowup squares (Definition A.2). To every topology \(\tau \) on the category of schemes (e.g. Zar, Nis, rh, cdh), there is a corresponding appropriate site \(\textrm{Sch}_\tau (\langle X\rangle _U)\) for the admissible ZariskiRiemann space (Definition 4.10). We show the following statement which is later used in the proof of Theorem A and which is the main new contribution of this article.
Theorem C
(Theorem 4.16) For every constant abelian rhsheaf F on \(\textrm{Sch}(\langle X\rangle _U)\) the canonical map
is an isomorphism. In particular,
where \(\textrm{Mdf}(X,U)\) is the category of all Umodifications of X and \(X'\setminus U\) is equipped with the reduced scheme structure. The same statement also holds if one replaces ‘Zar’ by ‘Nis’ and ‘rh’ by ‘cdh’.
We also show an rhversion of a cdhresult of KerzStrunkTamme [31, 6.3]. This is not a new proof but the observation that the analogous proof goes through. The statement will enter in the proof of Theorem A.
Theorem D
(Theorem A.15) Let X be a finite dimensional noetherian scheme. Then the canonical maps of rhsheaves with values in spectra on \(\textrm{Sch}_X\)
are equivalences.
Proofsketch for the main result. We shall briefly sketch the proof of Theorem A(iii) in the affinoid case (Theorem 6.1). For every reduced affinoid algebra A and every model \(X'\overset{}{\rightarrow }{{\,\textrm{Spec}\,}}(A^\circ )\) over the subring \(A^\circ \) of powerbounded elements with pseudouniformiser \(\pi \) there exists a fibre sequence [32, 5.8]
Now let us for a moment assume that A is regular and that resolution of singularities is available so that we could choose a regular model \(X'\) whose special fibre \(X'/\pi \) is simple normal crossing. In this case, \({{\,\textrm{K}\,}}(X'\,\,\textrm{on}\,\,\pi )\) vanishes in negative degrees and hence we have
where (1) follows from \({{\,\textrm{K}\,}}^\textrm{cont}_{i}(X'\,\,\textrm{on}\,\,\pi )=0\) for \(i>0\), (2) from nilinvariance of Ktheory in degrees \(\ge d\) (Lemma 6.3), and (3) is (iii) above resp. [31, Cor. D]. Let \((D_i)_{i\in I}\) be the irreducible components of \(X'/\pi \). As \(X'/\pi \) is simple normal crossing, all intersections of the irreducible components are regular, hence their cdhcohomology equals their Zariski cohomology which is \(\textbf{Z}\) concentrated in degree zero; hence \({{\,\textrm{H}\,}}_{\textrm{cdh}}^d(X'/\pi ;\textbf{Z})\) can be computed by the Čech nerve of the cdhcover \(D:=\bigsqcup _{i\in I} D_i \overset{}{\rightarrow }X/\pi \). On the other hand, the Berkovich space \({{\,\textrm{Spb}\,}}(A)\) associated with A is homotopy equivalent to its skeleton which is homeomorphic to the intersection complex \(\Delta (D)\) [40, 2.4.6, 2.4.9]. Putting these together yiels \({{\,\textrm{K}\,}}^\textrm{cont}_{d}(A) \cong {{\,\textrm{H}\,}}^d({{\,\textrm{Spb}\,}}(A);\textbf{Z}) \cong {{\,\textrm{H}\,}}^d({{\,\textrm{Spa}\,}}(A,A^\circ );\textbf{Z})\).
In the general case where \(X'\) is an aribtrary model, we have do proceed differently. For \(n<0\) and \(\alpha \in {{\,\textrm{K}\,}}_n(X'\,\,\textrm{on}\,\,\pi )\) there exists by RaynaudGruson’s platification par éclatement an admissible blowup \(X''\overset{}{\rightarrow }X'\) such that the pullback of \(\alpha \) vanishes in \({{\,\textrm{K}\,}}_n(X''\,\,\textrm{on}\,\,\pi )\) [28, 7]. In the colimit over all models this yields that \({{\,\textrm{K}\,}}^\textrm{cont}_n(A) \cong {{\,\textrm{K}\,}}^\textrm{cont}_n(\langle A_0\rangle _A)\). For \(d=\dim (A)\) we have \({{\,\textrm{K}\,}}^\textrm{cont}_d(\langle A_0\rangle _A) \cong {{\,\textrm{K}\,}}_d(\langle A_0\rangle _A/\pi )\) and the latter is isomorphic to \({{\,\textrm{H}\,}}_{\textrm{rh}}^d(\langle A_0\rangle _A/\pi ;\textbf{Z})\) via a descent spectral sequence argument (Theorem A.20). Using Theorem C (Theorem 4.16) we can pass to Zariski cohomology. Now the result follows from identifying \(\langle A^\circ \rangle _A\) with the adic spectrum \({{\,\textrm{Spa}\,}}(A,A^\circ )\) (Theorem 5.8).
Leitfaden. In Sect. 2 we recall the definition of and some basic facts about continuous Ktheory. Then we introduce admissible ZariskiRiemann spaces in Sect. 3 and we establish a comparison between their rhcohomology and their Zariski cohomology (Theorem 4.16) in Sect. 4. Subsequently we recall the connection between formal ZariskiRiemann spaces and adic spaces in Sect. 5; this causes the adic spaces showing up in the main result. In Sect. 6 we prove the main result in the affine case (Theorem 6.1). In Sect. 7 we present, following Morrow, a global version of continuous Ktheory and in Sect. 8 we prove the main result in the global case (Theorem 8.8). Finally, there is an Appendix A about the rhtopology and rhversions of results for the cdhtopology.
Notation. Discrete categories are denoted by \(\textrm{upright}\) \(\textrm{letters}\) whereas genuine \(\infty \)categories are denoted by \(\textbf{bold}\) \(\textbf{letters}\). We denote by \(\textbf{Spc}\) the \(\infty \)category of spaces [34, 1.2.16.1] and by \(\textbf{Sp}\) the \(\infty \)category of spectra [35, 1.4.3.1]. Given a scheme X we denote by \(\textrm{Sch}_X\) the category of separated schemes of finite type over X. If X is noetherian, then every scheme in \(\textrm{Sch}_X\) is noetherian as well.
2 Continuous Ktheory for Tate rings
In this section we recall the definition of continuous Ktheory as defined by Morrow [38] and further studied by KerzSaitoTamme [32].
Definition 2.1
Let X be a scheme. We denote by \({{\,\textrm{K}\,}}(X)\) the nonconnective Ktheory spectrum \({{\,\textrm{K}\,}}(\textbf{Perf}(X))\) à la BlumbergGepnerTabuada associated with the \(\infty \)category \(\textbf{Perf}(X)\) of perfect complexes on X [9, §7.1, §9.1]. For a ring A, we write \({{\,\textrm{K}\,}}(A)\) denoting \({{\,\textrm{K}\,}}({{\,\textrm{Spec}\,}}(A))\). For \(i\in \textbf{Z}\) we denote by \({{\,\textrm{K}\,}}_i(X)\) and \({{\,\textrm{K}\,}}_i(A)\) the ith homotopy group of \({{\,\textrm{K}\,}}(X)\) and \({{\,\textrm{K}\,}}(A)\), respectively.
Remark 2.2
For a scheme X the homotopy category \(\textrm{Ho}(\textbf{Perf}(X))\) is equivalent to the derived category of perfect complexes \(\textrm{Perf}(X)\) and the Ktheory spectrum \({{\,\textrm{K}\,}}(X)\) is equivalent to the one constructed by ThomasonTrobaugh [46, §3]. Every relevant scheme in this article is quasiprojective over an affine scheme, hence admits an ample family of line bundles. Thus Ktheory can be computed in terms of the category \(\textrm{Vec}(X)\) of vector bundles (i.e. locally free \({\mathcal {O}}_X\)modules). In view of Bass’ Fundamental Theorem, for \(n\ge 1\) the group \({{\,\textrm{K}\,}}_{n}(X)\) is a quotient of \({{\,\textrm{K}\,}}_0(X\times {\textbf{G}}_m^n)\) wherein elements coming from \({{\,\textrm{K}\,}}_0(X\times {\textbf{A}}^n)\) vanish.
In order to define continous Ktheory for adic rings, we give some reminders about adic rings and proobjects.
Reminder 2.3
Let \(A_0\) be be a ring and let I be an ideal of \(A_0\). Then the ideals \((I^n)_{n\ge 0}\) form a basis of neighbourhoods of zero in the socalled Iadic topology. An adic ring is a topological ring \(A_0\) such that its topology coincides with the Iadic topology for some ideal I of \(A_0\). We say that I is an ideal of definition. Note that adic rings have usually more than one ideal of definition. If the ideal I is finitely generated, the completion \({\hat{A}}_0\) is naturally isomorphic to the limit \(\lim _{n\ge 1}A_0/I^n\).
Reminder 2.4
We briefly recall the notion of proobjects and, in particular, of prospectra. For proofs or references we refer to KerzSaitoTamme [32, §2].
Given an \(\infty \)category \({\mathcal {C}}\) which is assumed to be accessible and to admit finite limits, one can built its procategory
where \({{\,\textrm{Fun}\,}}^{\textrm{lex,acc}}({\mathcal {C}},\textbf{Spc})\) is the full subcategory of \({{\,\textrm{Fun}\,}}({\mathcal {C}},\textbf{Spc})\) consisting of functors which are accessible (i.e. preserve \(\kappa \)small colimits for some regular cardinal number \(\kappa \)) and leftexact (i.e. commute with finite limits). The category \(\textbf{Pro}({\mathcal {C}})\) has finite limits and, if \({\mathcal {C}}\) has, also finite colimits which both can be computed levelwise. If \({\mathcal {C}}\) is stable, then also \(\textbf{Pro}({\mathcal {C}})\) is.
As a matter of fact, a proobject in \({\mathcal {C}}\) can be represented by a functor \(X:I\overset{}{\rightarrow }{\mathcal {C}}\) where I is a small cofiltered \(\infty \)category. In this case, we write \(\mathop {\mathrm {``lim''}}\limits _{i\in I}X_i\) for the corresponding object in \(\textbf{Pro}({\mathcal {C}})\). In our situations, the index category I will always be the poset of natural numbers \(\textbf{N}\).
Our main example of interest is the category \(\textbf{Pro}(\textbf{Sp})\) of prospectra whereas we are interested in another notion of equivalence. For this purpose, let \(\iota :\textbf{Sp}^+ \overset{}{\hookrightarrow }\textbf{Sp}\) be the inclusion of the full stable subcategory spanned by bounded above spectra (i.e. whose higher homotopy groups eventually vanish). The induced inclusion \(\textbf{Pro}(\iota ) :\textbf{Pro}(\textbf{Sp}^+) \overset{}{\hookrightarrow }\textbf{Pro}(\textbf{Sp})\) is rightadjoint to the restriction functor \(\iota ^* :\textbf{Pro}(\textbf{Sp}) \overset{}{\rightarrow }\textbf{Pro}(\textbf{Sp}^+)\).
A map \( X \overset{}{\rightarrow }Y\) of prospectra is said to be a weak equivalence iff the induced map \(\iota ^*X\overset{}{\rightarrow }\iota ^*Y\) is an equivalence in \(\textbf{Pro}(\textbf{Sp}^+)\). This nomenclature is justified by the fact that the map \(X\overset{}{\rightarrow }Y\) is a weak equivalence if and only if some truncation is an equivalence and the induced map on prohomotopy groups are proisomorphisms. Similarly, one defines the notions of weak fibre sequence and weak pullback.
Definition 2.5
Let \(A_0\) be a complete Iadic ring for some ideal I of \(A_0\). The continuous Ktheory of \(A_0\) is defined as the prospectrum
where \({{\,\textrm{K}\,}}\) is nonconnective algebraic Ktheory (Definition 2.1). This is independent of the choice of the ideal of definition.
Definition 2.6
A topological ring A is called a Tate ring if there exists an open subring \({A_0}\subset A\) which is a complete \(\pi \)adic ring (i.e. it is complete with respect to the \((\pi )\)adic topology) for some \(\pi \in {A_0}\) such that \(A = {A_0}[\pi ^{1}]\). We call such a subring \({A_0}\) a ring of definition of A and such an element \(\pi \) a pseudouniformiser. A Tate pair \((A,A_0)\) is a Tate ring together with the choice of a ring of definition and a Tate triple \((A,{A_0},\pi )\) is a Tate pair together with the choice of a pseudouniformiser.^{Footnote 2}
Example 2.7
Every affinoid algebra A over a complete nonarchimedean field k is a Tate ring. One can take \({A_0}\) to be those elements x which have residue norm \(x_\alpha \le 1\) with respect to some presentation \(k\langle t_1,\ldots ,t_n\rangle \overset{\alpha }{\twoheadrightarrow } A\) and any \(\pi \in k\) with \(\pi <1\) is a pseudouniformiser.
Definition 2.8
Let \((A,{A_0},\pi ) \) be a Tate triple. We define the continuous Ktheory \({{\,\textrm{K}\,}}^\textrm{cont}(A)\) of A as the pushout
in the \(\infty \)category \(\textbf{Pro}(\textbf{Sp})\) of prospectra.
Remark 2.9
In the situation of Definition 2.8 we obtain a fibre sequence
of prospectra. If \(A = A^\prime _0[\lambda ^{1}]\) for another complete \(\lambda \)adic ring \(A^\prime _0\), one obtains a weakly equivalent prospectrum, i.e. there is a zigzag of maps inducing proisomorphisms on prohomotopy groups [32, Prop. 5.4].
For regular rings algebraic Ktheory vanishes in negative degrees. For continuous Ktheory this may not be the case since it sees the reduction type of a ring of definition.
Example 2.10
Let \((A,A_0)\) be a Tate pair. By definition there is an exact sequence
If both A and \(A_0\) are regular, it follows that \({{\,\textrm{K}\,}}^\textrm{cont}_{1}(A) \cong {{\,\textrm{K}\,}}_{1}(A_0)\). If \(A_0\) is a \(\pi \)adic ring, then \({{\,\textrm{K}\,}}^\textrm{cont}_{1}(A_0) = {{\,\textrm{K}\,}}_{1}(A_0/\pi )\) due to nilinvariance of negative algebraic Ktheory (which follows from nilinvariance of \({{\,\textrm{K}\,}}_0\) [54, II. Lem. 2.2] and the definition of negative Ktheory in terms of \({{\,\textrm{K}\,}}_0\) [54, III. Def. 4.1]). Now let k be a discretely valued field and let \(\pi \in k^\circ \) be a uniformiser.

(i)
If \(A \cong k\langle x,y\rangle /(x^3y^2+\pi )\), we can choose \(A_0 := k^\circ \langle x,y\rangle /(x^3y^2+\pi )\) so that both A and \(A_0\) are regular. The reduction \(A_0/\pi \cong {\tilde{k}}\langle x,y\rangle /(x^3y^2)\) is the “cusp” over \({\tilde{k}}\). Thus \({{\,\textrm{K}\,}}^\textrm{cont}_{1}(A) = {{\,\textrm{K}\,}}(A_0/\pi ) = 0\) [53, 2.4].

(ii)
If \(A \cong k\langle x,y\rangle /(x^3+x^2y^2+\pi )\), we can choose \(A_0 := k^\circ \langle x,y\rangle /(x^3+x^2y^2+\pi )\) so that both A and \(A_0\) are regular. The reduction \(A_0/\pi \cong {\tilde{k}}\langle x,y\rangle /(x^3+x^2y^2)\) is the “node” over \({\tilde{k}}\). If \(\textrm{char}(k)\ne 2\), then \({{\,\textrm{K}\,}}^\textrm{cont}_{1}(A) = {{\,\textrm{K}\,}}(A_0/\pi ) = \textbf{Z}\) does not vanish [53, 2.4].
For the reader’s intuition we state some properties of continuous Ktheory.
Proposition 2.11
(KerzSaitoTamme) Let \((A,A_0,\pi )\) be a Tate triple.

(i)
The canonical map \({{\,\textrm{K}\,}}_0(A) \overset{}{\rightarrow }{{\,\textrm{K}\,}}^\textrm{cont}_0(A)\) is an isomorphism.

(ii)
\({{\,\textrm{K}\,}}^\textrm{cont}_1(A) \cong \mathop {\mathrm {``lim''}}\limits \limits _n {{\,\textrm{K}\,}}_1(A)/(1+\pi ^nA_0)\).

(iii)
Continuous Ktheory satifies an analytic version of Bass Fundamental Theorem; more precisely, for \(i\in \textbf{Z}\) there is an exact sequence
$$\begin{aligned} 0&\overset{}{\rightarrow }&{{\,\textrm{K}\,}}^\textrm{cont}_i(A) \overset{}{\rightarrow }{{\,\textrm{K}\,}}^\textrm{cont}_i(A\langle t\rangle )\oplus {{\,\textrm{K}\,}}^\textrm{cont}_i(A\langle t^{1}\rangle ) \overset{}{\rightarrow }{{\,\textrm{K}\,}}^\textrm{cont}_i(A\langle t,t^{1}\rangle )\\ {}&\overset{}{\rightarrow }&{{\,\textrm{K}\,}}^\textrm{cont}_{i1}(A) \overset{}{\rightarrow }0. \end{aligned}$$ 
(iv)
Continuous Ktheory coincides in negative degrees with the groups defined by KaroubiVillamayor [33, 7.7]^{Footnote 3} and Calvo [12, 3.2].
Proof
The statements (i), (iii), and (iv) are [32, 5.10] and (ii) is [32, 5.5]. \(\square \)
There are not always rings of definition which behave nice enough so that we will have to deal with other models which may not be affine. Hence we define similarly to Definition 2.8 the following.
Definition 2.12
Let X be a scheme over a \(\pi \)adic ring \(A_0\). Its continuous Ktheory is
where \(X/\pi ^n := X \times _{{{\,\textrm{Spec}\,}}(A_0)} {{\,\textrm{Spec}\,}}(A_0/\pi ^n)\).
Proposition 2.13
(KerzSaitoTamme [32, 5.8]) Let \((A,A_0,\pi )\) be a Tate triple such that \(A_0\) is noetherian and let \(X\overset{}{\rightarrow }{{\,\textrm{Spec}\,}}(A_0)\) be an admissible blowup, i.e. a proper morphism which is an isomorphism over \({{\,\textrm{Spec}\,}}(A)\). Then there exists a weak fibre sequence
of prospectra.
For a more detailed account of continuous Ktheory we refer the reader to [32, §6].
3 Admissible ZariskiRiemann spaces
Using a regular model \(X'\) of a regular affinoid algebra A makes the fibre sequence (Proposition 2.13)
much easier as the lefthand term vanishes in negative degrees, cf. the proofsketch for the main result (p. 3). Unfortunately, resolution of singularities is not available at the moment in positive characteristic. A good workaround for this inconvenience is to work with a ZariskiRiemann type space which is defined as the inverse limit of all models, taken in the category of locally ringed spaces. This is not a scheme anymore, but behaves in the world of Ktheory almost as good as a regular model does. For instance \({{\,\textrm{K}\,}}^\textrm{cont}_n(A) \cong {{\,\textrm{K}\,}}^\textrm{cont}_n(\langle A_0\rangle _A)\) for negative n where \(\langle A_0\rangle _A\) is the admissible ZariskiRiemann space associated with A (Definition 3.6).
The key part of this article is a comparison of rhcohomology and Zariski cohomology for admissible ZariskiRiemann spaces (Theorem 4.16). Furthermore, we will see later that ZariskiRiemann spaces for formal schemes are closely related to adic spaces (Theorem 5.8).
Notation
In this section let X be a reduced quasicompact and quasiseparated scheme and let U be a quasicompact open subscheme of X.
Definition 3.1
A Umodification of X is a projective morphism \(X'\overset{}{\rightarrow }X\) of schemes which is an isomorphism over U. Denote by \(\textrm{Mdf}(X,U)\) the category of Umodifications of X with morphisms over X. We define the Uadmissible ZariskiRiemann space of X to be the limit
in the category of locally ringed spaces; it exists due to [15, ch. 0, 4.1.10].
Lemma 3.2
The underlying topological space of \(\langle X\rangle _U\) is coherent and sober and for any \(X'\in \textrm{Mdf}(X,U)\) the projection \(\langle X\rangle _U\overset{}{\rightarrow }X'\) is quasicompact.
Proof
This is a special case of [15, ch. 0, 2.2.10]. \(\square \)
The notion of a Uadmissible modification is quite general. However, one can restrict to a more concrete notion, namely Uadmissible blowups.
Definition 3.3
A Uadmissible blowup is a blowup \({{\,\textrm{Bl}\,}}_Z(X) \overset{}{\rightarrow }X\) whose centre Z is finitely presented and contained in \(X\setminus U\). Denote by \(\textrm{Bl}(X,U)\) the category of Uadmissible blowups with morphisms over X.
Proposition 3.4
The inclusion \(\textrm{Bl}(X,U)\overset{}{\hookrightarrow }\textrm{Mdf}(X,U)\) is cofinal. In particular, the canonical morphism
is an isomorphism of locally ringed spaces.
Proof
Since a blowup in a finitely presented centre is projective and an isomorphism outside its centre, \(\textrm{Bl}(X,U)\) lies in \(\textrm{Mdf}(X,U)\). On the other hand, every Umodification is dominated by a Uadmissible blowup [45, Lem. 2.1.5]. Hence the inclusion is cofinal and the limits agree.^{Footnote 4}\(\square \)
Lemma 3.5
The full subcategory \({\text {Mdf}}^\mathrm {\,\,red}(X,U)\) spanned by reduced schemes is cofinal in \(\textrm{Mdf}(X,U)\).
Proof
As U is reduced by assumption, the map \(X'_\textrm{red}\overset{}{\hookrightarrow }X'\) is a Uadmissible blowup for every \(X'\in \textrm{Mdf}(X,U)\). \(\square \)
The remainder of this section is merely fixing notation for the application of admissible ZariskiRiemann spaces to the context of Tate rings.
Definition 3.6
Let \((A,{A_0},\pi )\) be a Tate triple (Definition 2.6). Setting \(X={{\,\textrm{Spec}\,}}({A_0})\) and \(U={{\,\textrm{Spec}\,}}(A)\), we are in the situation of Definition 3.1. For simplicity we denote
and call its objects admissible blowups. Furthermore, we call the locally ringed space
the admissible ZariskiRiemann space associated to the pair \((A,{A_0})\).
Remark 3.7
The admissible ZariskiRiemann space \(\langle A_0\rangle _A\) depends on the choice of the ring of definition \({A_0}\). However, if \(B_0\) is another ring of definition, then also the intersection \(C_0 := A_0 \cap B_0\) is. Hence we get a cospan
which is compatible with the inclusions of \({{\,\textrm{Spec}\,}}(A)\) into these. Hence every admissible blowup \(X \overset{}{\rightarrow }{{\,\textrm{Spec}\,}}(C_0)\) induces by pulling back an admissible blowup \(X_{A_0} \overset{}{\rightarrow }{{\,\textrm{Spec}\,}}(A_0)\) and a morphism \(X \overset{}{\rightarrow }X_{A_0}\). Precomposed with the canonical projections we obtain a map \(\langle A_0\rangle _A \overset{}{\rightarrow }X\). Hence the universal property yields a morphism \(\langle A_0\rangle _A \overset{}{\rightarrow }\langle C_0\rangle _A\). The same way, we get a morphism \(\langle B_0\rangle _A \overset{}{\rightarrow }\langle C_0\rangle _A\). One checks that the category of all admissible ZariskiRiemann spaces associated with A is filtered.
4 Cohomology of admissible ZariskiRiemann spaces
This section is the heart of this article providing the key ingredient for the proof of our main result; namely, a comparison of Zariski cohomology and rhcohomology for admissible ZariskiRiemann spaces (Theorem 4.16). This will be done in two steps passing through the biZariski topology.
Definition 4.1
Let S be a noetherian scheme. The biZariski topology is the topology generated by Zariski covers as well as by closed covers, i.e. covers of the form \(\{Z_i\overset{}{\rightarrow }X\}_i\) where \(X\in \textrm{Sch}_S\) and the \(Z_i\) are finitely many jointly surjective closed subschemes of X. This yields a site \(\textrm{Sch}_S^\textrm{biZar}\).
Lemma 4.2
The points on the biZariski site (in the sense of GoodwillieLichtenbaum [20, §2]) are precisely the spectra of integral local rings.
Proof
This follows from the fact that local rings are points for the Zariski topology and integral rings are points for the closed topology [19]. \(\square \)
Lemma 4.3
Let X be a noetherian scheme. The cover of X by its irreducible components refines every closed cover.
Proof
Let \((X_i)_i\) be the irreducible components of X with generic points \(\eta _i\in X_i\). Let \(X = \bigcup _\alpha Z_\alpha \) be a closed cover. For every i there exists an \(\alpha \) such that \(\eta _i\in Z_\alpha \), hence \(X_i = \overline{\{\eta _i\}} \subseteq {{\overline{Z}}}_\alpha = Z_\alpha \). By maximality of the irreducible components we have equality. \(\square \)
Lemma 4.4
Let S be a noetherian scheme. Every constant Zariski sheaf on \(\textrm{Sch}_S\) is already a biZariski sheaf.
Proof
Let A be an abelian group. For an open subset U of \(X\in \textrm{Sch}_S\), the sections over U are precisely the locally constant functions \(f:U \overset{}{\rightarrow }A\). By Lemma 4.3, it suffices to check the sheaf condition for the cover of U by its irreducible components \((U_i)_i\). We only have to show the glueing property. If \(f_i:U_i \overset{}{\rightarrow }A\) are locally constant functions which agree on all intersections, then they glue to a function \(f:U\overset{}{\rightarrow }A\). We have to show that f is locally constant. If \(x\in U\), for every i such that \(x\in U_i\) there exists an open neighbourhood \(V_i\) of x in U such that f becomes constant when restricted to \(U_i\cap V_i\). Hence f becomes also constant when restricted to the intersection of all these \(V_i\). Thus f is locally constant. \(\square \)
Lemma 4.5
Let S be a noetherian scheme and let \(X\in \textrm{Sch}_S\). For any constant sheaf A on \(\textrm{Sch}_S^\textrm{Zar}\) we have \({{\,\textrm{H}\,}}_{\textrm{Zar}}^*(X;A) \cong {{\,\textrm{H}\,}}_{\textrm{biZar}}^*(X;A)\).
Proof
Let \(u:\textrm{Sch}_S^\textrm{biZar}\overset{}{\rightarrow }\textrm{Sch}_S^\textrm{Zar}\) be the change of topology morphism of sites. Using the Leray spectral sequence
it is enough to show that the higher images \(R^qu_*A\) vanish for \(q>0\). We know that \(R^qu_*A\) is the Zariski sheaf associated with the presheaf
and that its stalks are given by \({{\,\textrm{H}\,}}_{\textrm{biZar}}^q(X;A)\) for X a local scheme (i.e. the spectrum of a local ring). As the biZariski sheafification of \(R^qu_*A\) is zero and using Lemma 4.3, we see that \({{\,\textrm{H}\,}}_{\textrm{biZar}}^q(X;A)=0\) for every irreducible local scheme X. For a general local scheme X we can reduce to the case where X is covered by two irreducible components \(Z_1\) and \(Z_2\).
First, let \(q=1\). We have an exact MayerVietoris sequence
Since local schemes are connected, the map \(\alpha \) is surjective, hence \(\partial =0\) and the second line remains exact with a zero added on the left. Thus \({{\,\textrm{H}\,}}_{\textrm{biZar}}^1(X;A)=0\) for any local scheme, hence \(R^1u_*A\) vanishes. For \(q>1\) we proceed by induction. Let
be an exact sequence of biZariski sheaves such that I is injective. This yields a commutative diagram with exact rows and columns
Being a closed subscheme of a local scheme, \(Z_1\cap Z_2\) is also a local scheme. By the case \(q=1\), the group \({{\,\textrm{H}\,}}_{\textrm{biZar}}^1(Z_1\cap Z_2;A)\) vanishes. Hence the map \(\beta \) is surjective. Using the analogous MayerVietoris sequence \((\Delta )\) above for G instead of A, we can conclude that \(R^2u_*A \cong R^1u_*G = 0\). Going on, we get the desired vanishing of \(R^qu_*A\) for every \(q>0\). \(\square \)
The remainder of the sections deals with the rhtopology defined by GoodwillieLichtenbaum [20]. We freely use results which are treated in a more detailed way in Appendix A.
Definition 4.6
An abstract blowup square is a cartesian diagram of schemes
where \(Z\overset{}{\rightarrow }X\) is a closed immersion, \({{\tilde{X}}}\overset{}{\rightarrow }X\) is proper, and the induced morphism \({{\tilde{X}}} \setminus E \overset{}{\rightarrow }X \setminus Z\) is an isomorphism. For any noetherian scheme S, the rhtopology on \(\textrm{Sch}_S\) is the topology generated by Zariski squares and covers \(\{Z\overset{}{\rightarrow }X, {\tilde{X}}\overset{}{\rightarrow }X\}\) for every abstract blowup square (abs) as well as the empty cover of the empty scheme.
Notation
For the rest of this section, let X be a reduced quasicompact and quasiseparated scheme and let U be a quasicompact dense open subscheme of X. We denote by Z the closed complement equipped with the reduced scheme structure.
Definition 4.7
For any morphism \(p:X'\overset{}{\rightarrow }X\) we get an analogous decomposition
where \(X'_Z := X' \times _XZ\) and \(X'_U := X'\times _XU\). By abuse of nomenclature, we call \(X_Z\) the special fibre of \(X'\) and \(X_U\) the generic fibre of \(X'\). An (abstract) admissible blowup of \(X'\) is a proper map \(X''\overset{}{\rightarrow }X'\) inducing an isomorphism \(X''_U\underset{}{\overset{\cong }{\longrightarrow }}X'_U\) over X. In particular, one obtains an abstract blowup square
At the end of this section, we will see that the Zariski cohomology and the rhcohomology on the ZariskiRiemann space coincide for constant sheaves (Theorem 4.16). The following proposition will be used in the proof to reduce from the rhtopology to the biZariski topology.
Proposition 4.8
Assume X to be noetherian and let \(X'\in \textrm{Sch}_X\). Then for every proper rhcover of the special fibre \(X'_Z\) there exists an admissible blowup \(X'' \overset{}{\rightarrow }X'\) such that the induced rhcover of \( X''_Z\) can be refined by a closed cover.
Proof
We may assume that \(X'\) is reduced. Every proper rhcover can be refined by a birational proper rhcover (Lemma A.6). Thus a cover yields a blowup square which can be refined by an honest blowup square
i.e. an abstract blowup square where \(Y'={{\,\textrm{Bl}\,}}_{V'}(X'_Z)\) (Lemma A.13). We consider the honest blowup square
which is an admissible blowup as \(V'\subseteq X'_Z\) and decomposes into two cartesian squares
where all the horizontal maps are closed immersions. By functoriality of blowups, we obtain a commutative diagram
wherein both horizontal maps are closed immersions and the right vertical map is an isomorphism by the universal property of the blowup. Thus \({{\,\textrm{Bl}\,}}_{V''}(X''_Z) \overset{}{\rightarrow }X''_Z\) is a closed immersion [21, Rem. 9.11]. Functoriality of blowups yields a commutative square
By the universal property of the pullback, there exists a unique map \({{\,\textrm{Bl}\,}}_{V''}(X''_Z) \overset{}{\rightarrow }Y'' := Y' \times _{X'_Z} X''_Z\) such that following diagram commutes.
To sum up, we have shown that the pullback of the proper rhcover \(V'\sqcup Y'\overset{}{\rightarrow }X'_Z\) along \(X''_Z \overset{}{\rightarrow }X'_Z\) can be refined by the closed cover \(V'' \sqcup {{\,\textrm{Bl}\,}}_{V''}(X''_Z) \overset{}{\rightarrow }X''_Z\) which was to be shown. \(\square \)
Given a topology on (some appropriate subcategory of) the category of schemes, we want to have a corresponding topology on admissible ZariskiRiemann spaces. For this purpose, we will work with an appropriate site.
Remark 4.9
Let \(\tau \) be a topology on the category \(\textrm{Sch}_X\). It restricts to a topology on the category \(\textrm{Sch}^\textrm{qc}_X\) of quasicompact Xschemes. One obtains compatible topologies on the slice categories \(\textrm{Sch}^\textrm{qc}_{X'} = (\textrm{Sch}^\textrm{qc}_X)_{/X'}\) for all Umodifications \(X'\in \textrm{Mdf}(X,U)\).
Definition 4.10
Consider the category
More precisely, the set of objects is the set of morphisms of schemes \(Y'\overset{}{\rightarrow }X'\) for some \(X'\in \textrm{Mdf}(X,U)\). The set of morphisms between two objects \(Y'\overset{}{\rightarrow }X'\) and \(Y''\overset{}{\rightarrow }X''\) is given by
where \({\tilde{X}}\) runs over all modifications \({\tilde{X}}\in \textrm{Mdf}(X,U)\) dominating both \(X'\) and \(X''\). Analogously, define the category
where the \(X'\setminus U\) are equipped with the reduced scheme structure.
Definition 4.11
Let \(Y'\overset{}{\rightarrow }X'\) be an object of \(\textrm{Sch}^\textrm{qc}(\langle X\rangle _U)\). We declare a sieve R on \(Y'\) to be a \(\tau \) covering sieve of \(Y'\overset{}{\rightarrow }X'\) iff there exists a Umodification \(p:X''\overset{}{\rightarrow }X'\) such that the pullback sieve \(p^*R\) lies in \(\tau (Y'\times _{X'}X'')\). Analogously we define \(\tau \)covering sieves in \(\textrm{Sch}^\textrm{qc}(\langle X\rangle _U\setminus U)\).
Lemma 4.12
The collection of \(\tau \)covering sieves in Definition 4.11 defines topologies on the categories \(\textrm{Sch}^\textrm{qc}(\langle X\rangle _U)\) and \(\textrm{Sch}^\textrm{qc}(\langle X\rangle _U\setminus U)\) which we will refer to with the same symbol \(\tau \).
Proof
This follows immedeately from the construction. \(\square \)
Remark 4.13
In practice, for working with the site \((\textrm{Sch}^\textrm{qc}(\langle X\rangle _U),\tau )\) it is enough to consider \(\tau \)covers in the category \(\textrm{Sch}^\textrm{qc}_X\) and identifying them with their pullbacks along Umodifications.
Caveat 4.14
The category \(\textrm{Sch}^\textrm{qc}(\langle X\rangle _U)\) is not a slice category, i.e. a scheme Y together with a morphism of locally ringed spaces \(Y \overset{}{\rightarrow }\langle X\rangle _U\) does not necessarily yield an object of \(\textrm{Sch}^\textrm{qc}(\langle X\rangle _U)\). Such objects were studied e.g. by Hakim [23]. In contrast, an object of \(\textrm{Sch}^\textrm{qc}(\langle X\rangle _U)\) is given by a scheme morphism \(Y \overset{}{\rightarrow }X'\) for some \(X'\in \textrm{Mdf}(X,U)\) and it is isomorphic to its pullbacks along admissible blowups.
In the proof of the main theorem we will need the following statement which follows from the construction of our site.
Proposition 4.15
Let F be a constant sheaf of abelian groups on \(\textrm{Sch}^\textrm{qc}(\langle X\rangle _U)\). Then the canonical morphism
is an isomorphism. Analogously, if F is a constant sheaf of abelian groups on \(\textrm{Sch}^\textrm{qc}(\langle X\rangle _U\setminus U)\), then the canonical morphism
is an isomorphism.
Proof
This is a special case of [42, Tag 09YP] where the statement is given for any compatible system of abelian sheaves. \(\square \)
Theorem 4.16
For any constant sheaf F on \(\textrm{Sch}^\textrm{qc}_\textrm{rh}(\langle X\rangle _U)\), the canonical map
is an isomorphism.
Proof
By construction, any rhcover of \(\langle X\rangle _U\setminus U\) is represented by an rhcover of \(X'_Z\) for some \(X'\in \textrm{Adm}({A_0})\). We find a refinement \({\tilde{V}}\overset{q}{\rightarrow }{\tilde{Y}}\overset{p}{\rightarrow }X'_Z\) where p is a proper rhcover and q is a Zariski cover (Proposition A.14). The rhcover \({\tilde{Y}}\overset{}{\rightarrow }X'_Z\) is given by \(Y'\sqcup V'\overset{}{\rightarrow }X\) for an abstract blowup square
This is the situation of Proposition 4.8. Thus there exists an admissible blowup \(X''\overset{}{\rightarrow }X'\) and a refinement \(V'' \sqcup {{\,\textrm{Bl}\,}}_{V''}(X''_Z) \overset{}{\rightarrow }X''_Z\) of the pulled back cover which consists of two closed immersions. Hence we have refined our given cover of \(\langle X\rangle _U\setminus U\) by a composition of a Zariski cover and a closed cover which yields a biZariski cover. This implies that \({{\,\textrm{H}\,}}_{\textrm{rh}}^*(\langle X\rangle _U\setminus U;F)\) equals \({{\,\textrm{H}\,}}_{\textrm{biZar}}(\langle X\rangle _U\setminus U;F)\). Now the assertion follows from Lemma 4.5. \(\square \)
Corollary 4.17
For any constant sheaf F, we have
Proof
This is a formal consequence of the construction of the topology on \(\textrm{Sch}^\textrm{qc}(\langle X\rangle _U)\) since the cohomology of a limit site is the colimit of the cohomologies [42, Tag 09YP]. \(\square \)
5 Formal ZariskiRiemann spaces and adic spaces
In this section we deal with ZariskiRiemann spaces which arise from formal schemes. According to a result of Scholze they are isomorphic to certain adic spaces (Theorem 5.8). This identification is used in the proof of the main theorem (Theorem 6.1) to obtain the adic spectrum \({{\,\textrm{Spa}\,}}(A,A^\circ )\) in the statement. We start with some preliminaries on formal schemes; for a detailled account of the subject we refer to Bosch’s lecture notes [10, pt. II].
Notation
In this section, let R be a ring of one of the following types (cf. [10, §7.3]):

(V)
R is an adic valuation ring with finitely generated ideal of definition I.

(N)
R is a noetherian adic ring with ideal of definition I such that R does not have Itorsion.
An Ralgebra is called admissible iff it is of topologically finite presentation and without Itorsion [10, §7.3, Def. 3]. A formal Rscheme is called admissible iff it has a cover by affine formal Rschemes of the form \({{\,\textrm{Spf}\,}}(A_0)\) for admissible Ralgebras \(A_0\), cf. [10, §7.4, Def. 1].
Definition 5.1
For a scheme X over \({{\,\textrm{Spec}\,}}(R)\) we denote by \({\hat{X}}\) its associated formal scheme \( \mathop {\textrm{colim}}\limits _n X/I^n\) over \({{\,\textrm{Spf}\,}}(R)\). A formal scheme which is isomorphic to some \({\hat{X}}\) is called algebraic. Setting \(U:=X\setminus (X/I)\), for every Uadmissible blowup \(X' \overset{}{\rightarrow }X\) the induced morphism of \(\hat{X'} \overset{}{\rightarrow }{\hat{X}}\) is an admissible formal blowup [1, 3.1.3]. An admissible formal blowup \({\mathcal {X}}' \overset{}{\rightarrow }{\hat{X}}\) of an algebraic formal scheme is called algebraic whenever it is induced from a Uadmissible blow up of X.
Example 5.2

(i)
Any quasiaffine formal scheme is algebraic. Indeed, an affine formal scheme \({{\,\textrm{Spf}\,}}({A_0})\) is isomorphic to the formal completion of \({{\,\textrm{Spec}\,}}({A_0})\). The quasiaffine case is Lemma 5.5 below.

(ii)
For a nonarchimedean field k, every projective rigid kspace has an algebraic model. In fact, any closed subspace of the rigid analytic space \(\textbf{P}_k^{n,\textrm{an}}\) is the analytification of a closed subspace of \(\textbf{P}_k^n\) by a GAGAtype theorem [16, 4.10.5]. Since \(\textbf{P}_k^{n,\textrm{an}}\) can be obtained by glueing \(n+1\) closed unit discs \(\textbf{B}^n_k={{\,\textrm{Spm}\,}}(k\langle t_1,\ldots ,t_n\rangle )\) along algebraic maps [16, 4.3.4], the rigid space \(\textbf{P}_k^{n,\textrm{an}}\) is (isomorphic to) the generic fibre of the formal completion \((\textbf{P}_{k^\circ }^n)^\wedge \) of the \(k^\circ \)scheme \(\textbf{P}_{k^\circ }^n\); this argument also holds for closed subspaces. Hence every projective rigid kspace has an algebraic model.
Lemma 5.3
Let X be an Rscheme locally of finite type. Assume that R is of type (N) or that X is without Itorsion (e.g. flat over R). Then every admissible formal blowup of \({\hat{X}}\) is algebraic.
Proof
If R is of type (N), then \(R\langle t_1,\ldots ,t_n\rangle \) is noetherian [10, §7.3 Rem. 1] so that \({\hat{X}}\) is locally of topologically finite presentation. If X is without Itorsion, then \({\hat{X}}\) is locally of topologically finite presentation [10, §7.3, Cor. 5]. Hence in both cases the notion of an admissible formal blowup [10, §8.2, Def. 3] is defined. Set \(X/I:=X\times _{{{\,\textrm{Spec}\,}}(R)}{{\,\textrm{Spec}\,}}(R/I)\) and let \({\mathcal {I}}\) be the ideal sheaf of \({\mathcal {O}}_X\) defining X/I. Let \({\mathcal {X}}'\overset{}{\rightarrow }{\hat{X}}\) be an admissible formal blowup defined by an open ideal \({\mathcal {A}}\) of \({\mathcal {O}}_{{\hat{X}}}\). In particular, there exists an \(n\in \textbf{N}\) such that \({\mathcal {I}}^n{\mathcal {O}}_{{\hat{X}}}\subset {\mathcal {A}}\). Let \(Z_n := X/I^n\) be the closed subscheme of X defined by \({\mathcal {I}}^n\). This yields a surjective map \(\varphi =i^\#:{\mathcal {O}}_X\overset{}{\rightarrow }i_*{\mathcal {O}}_{Z_n}\) of sheaves on X where \(i :Z_n\overset{}{\rightarrow }X\) denotes the inclusion. Let \({{\tilde{{\mathcal {A}}}}} := \varphi ^{1}\bigl ( {\mathcal {A}}/({\mathcal {I}}^n{\mathcal {O}}_{{\hat{X}}})\bigr )\). By construction, \(i^{1}{{\tilde{{\mathcal {A}}}}}={\mathcal {A}}\) since both have the same pullback to \(Z_k = (Z,{\mathcal {O}}_X/{\mathcal {I}}^k) = (Z,{\mathcal {O}}_{{\hat{X}}}/{\mathcal {I}}^k)\). Thus \({\mathcal {X}}= {\hat{X}}_{{{\tilde{{\mathcal {A}}}}}}\). \(\square \)
Lemma 5.4
For every Ralgebra \(A_0\), the family \(\bigl ({{\,\textrm{Spf}\,}}(A_0\langle f^{1}\rangle \bigr )_{f\in {A_0}}\) is a basis of the topology of \({{\,\textrm{Spf}\,}}(A_0)\).
Proof
The family \(\bigl (({{\,\textrm{Spec}\,}}(A_0[f^{1}])\bigr )_{f\in A_0}\) forms a basis of the topology of \({{\,\textrm{Spec}\,}}(A_0)\). Topologically, \({{\,\textrm{Spf}\,}}(A_0)\) is a closed subspace of \({{\,\textrm{Spec}\,}}(A_0)\). Thus the induced family \(\bigl ({{\,\text {Spec}\,}}(A_0[f^{1}])\cap {{\,\text {Spf}\,}}(A_0)\bigr )_{f\in A_0}\) is a basis of the topology of \({{\,\textrm{Spf}\,}}(A_0)\). As topological spaces, \({{\,\textrm{Spf}\,}}(A_0\langle f^{1}\rangle ) = {{\,\textrm{Spec}\,}}(A_0[f^{1}]) \cap {{\,\textrm{Spf}\,}}(A_0)\). Hence we are done. \(\square \)
Lemma 5.5
Every admissible formal blowup of a quasiaffine admissible formal scheme is algebraic.
Proof
Let \(j :{\mathcal {U}}\overset{}{\hookrightarrow }{\mathcal {X}}={{\,\textrm{Spf}\,}}({A_0})\) be the inclusion of an open formal subscheme. and let \({\mathcal {U}}' \overset{}{\rightarrow }{\mathcal {U}}\) be an admissible formal blowup defined by a coherent open ideal \({\mathcal {A}}_U\subseteq {\mathcal {O}}_{\mathcal {U}}\). Then there exists a coherent open ideal \({\mathcal {A}}\subseteq {\mathcal {O}}_{\mathcal {X}}\) such that \({\mathcal {A}}_U\cong {\mathcal {A}}_U\) and \({\mathcal {A}}_V\cong {\mathcal {O}}_V\) whenever \(V\cap U = \emptyset \) [10, §8.2, Prop. 13]. In particular, \({\mathcal {U}}' \overset{}{\rightarrow }{\mathcal {U}}\) extends to an admissible formal blowup \({\mathcal {X}}' \overset{}{\rightarrow }{\mathcal {X}}\). By Lemma 5.3, this blowup comes from an admissible blowup \(p :X' \overset{}{\rightarrow }X={{\,\textrm{Spec}\,}}(A_0)\). By Lemma 5.4, we can write \({\mathcal {U}}= \bigcup _{i=1}^n {\mathcal {U}}_i\) with \({\mathcal {U}}_i = {{\,\textrm{Spf}\,}}(A_0\langle f_i^{1}\rangle )\) for suitable \(f_1,\ldots ,f_n\in A_0\). Setting \(U_i := {{\,\textrm{Spec}\,}}(A_0[f_i^{1}])\) and \(U'_i := p^{1}(U_i)\) and \(U':= \bigcup _{i=1}^n U'_i\) the union in \(X'\), then we obtain that
which finishes the proof. \(\square \)
Definition 5.6
For a formal scheme \({\mathcal {X}}\) locally of topologically finite presentation over R its associated formal ZariskiRiemann space is defined to be the limit
in the category of locally topologically ringed spaces where \(\textrm{Adm}({\mathcal {X}})\) denotes the category of all admissible formal blowups of \({\mathcal {X}}\).
Lemma 5.7
Assume that the ideal I is principal, say generated by \(\pi \). Let X be an Rscheme locally of finite type. Assume that R is of type (N) or that X is without \(\pi \)torsion (e.g. flat over R). Then its formal completion \({\hat{X}}\) is homeomorphic to the special fibre \(X/\pi = X\times _{{{\,\textrm{Spec}\,}}(R)}{{\,\textrm{Spec}\,}}(R/\pi )\). Consequently, the formal ZariskiRiemann space \(\langle {\hat{X}}\rangle \) is homeomorphic to \(\langle X\rangle _U/\pi = \langle X\rangle _U\setminus U\) where \(U={{\,\textrm{Spec}\,}}(R[\pi ^{1}])\).
Proof
This is a direct consequence of the definition of a formal scheme [10, §7.2] and Lemma 5.3. \(\square \)
Theorem 5.8
( [41, 2.22]) Let k be a complete nonarchimedean field, i.e. a topological field whose topology is induced by a nonarchimedean norm, and let \(k^\circ \) be its valuation ring. Let \(X^\textrm{ad}\) be a quasicompact and quasiseparated adic space locally of finite type over k. Then there exists an admissible formal model \({\mathcal {X}}\) of \(X^\textrm{ad}\) and there is a homeomorphism \(X^\textrm{ad}\overset{\cong }{\rightarrow } \langle {\mathcal {X}}\rangle \) which extends to an isomorphism
of locally ringed spaces.
6 Main result: affinoid case
Notation
In this section let k be a complete discretely valued field with valuation ring \(k^\circ \) and uniformiser \(\pi \). This implies that the ring \(k^\circ \) is noetherian.
Theorem 6.1
Let A be an affinoid kalgebra of dimension d. Then there is an isomorphism
where \({{\,\textrm{Spa}\,}}(A,A^\circ )\) is the adic spectrum of A with respect to its subring \(A^\circ \) of powerbounded elements [26, §3] and the righthand side is sheaf cohomology.
Before proving the result, we first deduce an immediate consequence.
Corollary 6.2
Let A be an affinoid kalgebra of dimension d. Then there is an isomorphism
where \({{\,\textrm{Spb}\,}}(A)\) is the Berkovich spectrum of A [6, Ch. 1] and the righthand side is sheaf cohomology.
Proof
The category of overconvergent^{Footnote 5} sheaves on an adic spectrum is equivalent to the category of sheaves on the Berkovich spectrum [47, §5, Thm. 6]. The locally constant sheaf \(\textbf{Z}\) is overconvergent and admits a flasque resolution by overconvergent sheaves, hence the claim follows from Theorem 6.1. \(\square \)
Proof of Theorem 6.1
We may assume that A is reduced as the statement is nilinvariant. Let \(A^\circ \) be the subring of A consisting of powerbounded elements of A. Then the pair \((A,A^\circ )\) is a Tate pair [8, §6.2.4, Thm. 1] and \(A^\circ \) is noetherian [8, §6.4.3, Prop. 3 (i)]. For any \(X\in \textrm{Adm}(A^\circ )\) one has \(X_A={{\,\textrm{Spec}\,}}(A)\) and thus by Proposition 2.13 there is a fibre sequence
Passing to the colimit over all admissible models we obtain a fibre sequence of prospectra
For \(i<0\) we have that \(\mathop {\textrm{colim}}\limits _{X\in \textrm{Adm}(A^\circ )}{{\,\textrm{K}\,}}_i(X\,\,\textrm{on}\,\,\pi )=0\) [28, Prop. 7] and hence
Lemma 6.3 below and Theorem A.20 yield
where the last isomorphism uses that \(d=\dim (X/\pi )\) if \(X\in \textrm{Adm}(A^\circ )\) is reduced. Corollary 4.17 says that
The Zariski cohomology is just ordinary sheaf cohomology. The latter one commutes with colimits of coherent and sober spaces with quasicompact transition maps [15, ch. 0, 4.4.1]. Since the admissible ZariskiRiemann space is such a colimit we obtain
where the righthand side is sheaf cohomology. Finally we get that
since the admissible ZariskiRiemann space \(\langle A^\circ \rangle _A\) is homeomorphic to the formal ZariskiRiemann space \(\langle {{\,\textrm{Spf}\,}}(A^\circ )\rangle \) (Lemma 5.7) which is isomorphic to the adic spectrum \({{\,\textrm{Spa}\,}}(A,A^\circ )\) (Theorem 5.8). \(\square \)
Lemma 6.3
Let Y be a noetherian scheme of finite dimension d. Then for \(n\ge d\) we have
Proof
This follows by using the Zariskidescent spectral sequence and nilinvariance of negative algebraic Ktheory for affine schemes. \(\square \)
7 Continous Ktheory for rigid spaces
In this section we see that continuous Ktheory, as defined for algebras in Definition 2.8, satisfies descent and hence defines a sheaf of prospectra for the admissible topology. The result and its proof are due to Morrow [38]; we present here a slightly different argument. For the general theory on rigid kspaces we refer the reader to Bosch’s lecture notes [10, pt. I].
Notation
In this section let k be a complete discretely valued field with valuation ring \(k^\circ \) and uniformiser \(\pi \). This implies that the ring \(k^\circ \) is noetherian. For an affinoid kalgebra A denote by \({{\,\textrm{Spm}\,}}(A)\) its associated affinoid kspace [10, §3.2].^{Footnote 6} Denote by \(\textrm{FSch}_{k^\circ }\) the category of formal schemes over \(k^\circ \) and by \(\textrm{FSch}_{k^\circ }^\textrm{lft}\) its full subcategory of formal schemes that are locally finite type over \(k^\circ \); we consider these as sites equipped with the Zariskitopology.
Lemma 7.1
Let \({\mathcal {X}}\) be a formal scheme over \(k^\circ \) which is assumed to be covered by two open formal subschemes \({\mathcal {X}}_1\) and \({\mathcal {X}}_2\). Setting \({\mathcal {X}}_3 := {\mathcal {X}}_1\cap {\mathcal {X}}_2\) we obtain a cartesian square
in the category \(\textbf{Pro}(\textbf{Sp})\).
Proof
For every \(n\ge 1\), the special fibre \({\mathcal {X}}/\pi ^n\) is covered by \({\mathcal {X}}_1/\pi ^n\) and \({\mathcal {X}}_2/\pi ^n\) with intersection \({\mathcal {X}}_3/\pi ^n\). Applying algebraic Ktheory one obtains cartesian squares by Zariski descent. Now the claim follows as finite limits in the procategory can be computed levelwise (Reminder 2.4). \(\square \)
Corollary 7.2
The presheaf \({{\,\textrm{K}\,}}^\textrm{cont}\) on the site \(\textrm{FSch}_{k^\circ }\) is a sheaf of prospectra and satisfies \({{\,\textrm{K}\,}}^\textrm{cont}({{\,\textrm{Spf}\,}}(A_0)) \simeq {{\,\textrm{K}\,}}^\textrm{cont}({A_0})\) for every \(k^\circ \)algebra \({A_0}\).
Proof
This is a standard consequence for topologies which are induced by cdstructures [2, Thm. 3.2.5]. \(\square \)
Lemma 7.3
( [38, 3.4]) Let \({{\,\textrm{Spm}\,}}(A)\) be an affinoid kspace which is assumed to be covered by two open affinoid subdomains \({{\,\textrm{Spm}\,}}(A^1)\) and \({{\,\textrm{Spm}\,}}(A^2)\). We set
where \(A_0, A_+^1, A_+^2\) are respective subrings of definition of \(A, A^1, A^2\) and \((A_+^1 \otimes _{A_0} A_+^2)^\wedge \) denotes the \(\pi \)adic completion. Then the square
is weakly cartesian in \(\textbf{Pro}(\textbf{Sp})\), i.e. cartesian in \(\textbf{Pro}(\textbf{Sp}^+)\).
Proof
We note that the definition of the ring \(A^3\) is independent of the choices of the rings of definition \(A_+^1\) and \(A_+^2\) and we forget about these choices. According to Raynaud’s equivalence of categories between quasicompact admissible formal \(k^\circ \)schemes localised by admissible formal blowups and quasicompact and quasiseparated rigid kspaces we find an admissible formal blowup \({\mathcal {X}}\overset{}{\rightarrow }{{\,\textrm{Spf}\,}}(A_0)\) and an open cover \({\mathcal {X}}= {\mathcal {X}}_1 \cup {\mathcal {X}}_2\) whose associated generic fibre is the given cover \({{\,\textrm{Spm}\,}}(A) = {{\,\textrm{Spm}\,}}(A^1) \cup {{\,\textrm{Spm}\,}}(A^2)\) [10, §8.4]. Since every admissible blowup of the algebraic formal scheme \({{\,\textrm{Spf}\,}}(A_0)\) is algebraic (Lemma 5.3), we find an admissible blowup \(X\overset{}{\rightarrow }{{\,\textrm{Spec}\,}}(A_0)\) and an open cover \(X=X_1 \cup X_2\) whose formal completion is the cover \({\mathcal {X}}= {\mathcal {X}}_1 \cup {\mathcal {X}}_2\). We set \(X_3 := X_1 \cap X_2\) and note that for \(i\in \{1,2,3\}\) there exist rings of definition \(A^i_0\) of \(A^i\), respectively, and admissible blowups \(X_i\overset{}{\rightarrow }{{\,\textrm{Spec}\,}}(A^i_0)\). By Zariski descent have two cartesian squares
where the right square is cartesian since it is levelwise cartesian (Reminder 2.4). There is map from the left square to the right square. By Proposition 2.13, the square of cofibres is weakly equivalent to the square (\(\square \)) which is therefore weakly cartesian. \(\square \)
The following statement is a standard result about extending sheaves from local objects to global ones and permits us to extend continuous Ktheory to the category of rigid kspaces.
Proposition 7.4
The inclusion \(\iota :\textrm{Rig}_k^\textrm{aff} \overset{}{\hookrightarrow }\textrm{Rig}_k\) of affinoid kspaces into rigid kspaces induces an equivalence
Moreover, for every \(\infty \)category \({\mathcal {D}}\) which admits small limits, the canonical map
is an equivalence.
Proof
This follows from applying twice an \(\infty \)categorical version of the “comparison lemma” [25, C.3]: first to the inclusion \(\textrm{Rig}_k^\textrm{aff} \overset{}{\hookrightarrow }\textrm{Rig}_k^\textrm{sep}\) of affinoid spaces into separated spaces and secondly to the inclusion \(\textrm{Rig}_k^\textrm{sep} \overset{}{\hookrightarrow }\textrm{Rig}_k\). \(\square \)
Corollary 7.5
( [38, 3.5]) There exists a unique sheaf \({{\,\textrm{K}\,}}^\textrm{cont}\) on the category \(\textrm{Rig}_k\) (equipped with the admissible topology) that has values in \(\textbf{Pro}(\textbf{Sp}^+)\) and satisfies \({{\,\textrm{K}\,}}^\textrm{cont}({{\,\textrm{Spm}\,}}(A)) \simeq {{\,\textrm{K}\,}}^\textrm{cont}(A)\) for every affinoid kalgebra A.
Corollary 7.6
The functor
is a sheaf.
Proof
This follows from the fact that Zariski covers of formal schemes induce on generic fibres admissible covers of rigid spaces. \(\square \)
8 Main result: global case
In this section we conjecture that an analogous version of our main result (Theorem 6.1) for rigid spaces is true. We prove this conjecture in the algebraic case (e.g. affinoid or projective) and in dimension at least two (Theorem 8.8). The constructions in this section are adhoc for our purposes and a full development of the formalism which will be based on adic spaces needs to be examined in future work.
Notation
In this section let k be a complete discretely valued field with valuation ring \(k^\circ \) and uniformiser \(\pi \).
Conjecture 8.1
Let X be a quasicompact and quasiseparated rigid kspace of dimension d. Then there is an isomorphism
of proabelian groups. In particular, the proabelian group \({{\,\textrm{K}\,}}^\textrm{cont}_{d}(X)\) is constant.
Definition 8.2
For an affine formal scheme \({{\,\textrm{Spf}\,}}({A_0})\) with associated generic fibre \({{\,\textrm{Spm}\,}}(A)\) where \(A={A_0}\otimes _{k^\circ }k\) there is by definition a map \({{\,\textrm{K}\,}}^\textrm{cont}({A_0})\overset{}{\rightarrow }{{\,\textrm{K}\,}}^\textrm{cont}(A)\). This map can be seen as a natural transformation \(\textrm{FSch}^\textrm{aff}\overset{}{\rightarrow }\textbf{Pro}(\textbf{Sp}^+)\) which extends to a natural transformation
For a formal scheme \({\mathcal {X}}\) locally of finite type over \(k^\circ \) we define
where \({\mathcal {X}}_\eta \) is the associated generic fibre. By construction and by Corollary 7.2 and Corollary 7.6 the induced functor
is a sheaf.
Lemma 8.3
Let X be a \(k^\circ \)scheme locally of finite type. Then there is a canonical equivalence
In particular, \({{\,\textrm{K}\,}}^\textrm{cont}({\hat{X}}\,\,\textrm{on}\,\,\pi )\) is equivalent to a constant prospectrum.
Proof
If \(X={{\,\textrm{Spec}\,}}({A_0})\) is affine we have by Definition 2.8 a pushout square
Since the category \(\textbf{Sp}\) is stable, this also holds for \(\textbf{Pro}(\textbf{Sp})\). Thus the square is also a pullback and we have an equivalence \({{\,\textrm{K}\,}}({A_0}\,\,\textrm{on}\,\,\pi )\simeq {{\,\textrm{K}\,}}^\textrm{cont}({A_0}\,\,\textrm{on}\,\,\pi )\) of the horizontal fibres. If X is quasicompact and separated, choose a finite affine cover \((U_i)_i\) which yields a commutative diagram
where \({\check{U}}_\bullet \) and \(\check{{\hat{U}}}_\bullet \) are the \(\check{\textrm{C}}\)ech nerves of the cover \((U_i)_i\) of X respectively the induced cover \(({\hat{U}}_i)_i\) of \({\hat{X}}\). Thus the horizontal maps are equivalences. By the affine case the right vertical map is an equivalence, hence also the left vertical map as desired. For the quasiseparated case we reduce analogously to the separated case.
\(\square \)
Corollary 8.4
Let X be a \(k^\circ \)scheme locally of finite type. Then the square
is cartesian in \(\textbf{Pro}(\textbf{Sp}^+)\) where \(X_k := X \times _{{{\,\textrm{Spec}\,}}(k^\circ )} {{\,\textrm{Spec}\,}}(k)\).
Proof
By design there is a commutative diagram of fibre sequences
where the left vertical map is an equivalence due to Lemma 8.3. \(\square \)
Corollary 8.5
Let X be a reduced \(k^\circ \)scheme locally of finite type. For \(n\ge 1\) we have
where \({\mathcal {X}}'\) runs over all admissible formal blowups of \({\hat{X}}\).^{Footnote 7}
Proof
Since every admissible formal blowup of an algebraic formal scheme is algebraic (Lemma 5.3), due to Lemma 8.3, and by [28, Prop. 7] we have
where the latter two colimits are indexed by all \(X_k\)admissible blowups of and where \(X_k := X \times _{{{\,\textrm{Spec}\,}}(k^\circ )} {{\,\textrm{Spec}\,}}(k)\). \(\square \)
Lemma 8.6
Let \({\mathcal {X}}\) be a quasicompact admissible formal scheme. For \(n\ge 2\) have
where \({\mathcal {X}}'\) runs over all admissible formal blowups of \({\mathcal {X}}\).
Proof
Let \(\alpha \in {{\,\textrm{K}\,}}^\textrm{cont}_{n}({\mathcal {X}}\,\,\textrm{on}\,\,\pi )\). We choose a finite affine cover \(({\mathcal {U}}_i)_{i\in I}\) of \({\mathcal {X}}\) where \(I=\{1,\ldots ,k\}\). By the affine case, we find for every \(i\in I\) an admissible formal blowup \({\mathcal {U}}'_i\overset{}{\rightarrow }{\mathcal {U}}_i\) such that the map \({{\,\textrm{K}\,}}^\textrm{cont}_{n}({\mathcal {U}}_i)\overset{}{\rightarrow }{{\,\textrm{K}\,}}^\textrm{cont}_{n}({\mathcal {U}}'_i)\) sends \(\alpha _{{\mathcal {U}}_i}\) to zero. There exists an admissible formal blowup \({\mathcal {X}}'\overset{}{\rightarrow }{\mathcal {X}}\) locally dominating these local blowups, i.e. for every \(i\in I\) the pullback \({\mathcal {X}}'\times _{{\mathcal {X}}}{\mathcal {U}}_i \overset{}{\rightarrow }{\mathcal {U}}_i\) factors over \({\mathcal {U}}'\overset{}{\rightarrow }{\mathcal {U}}\) [10, 8.2, Prop. 14]. We may assume that \({\mathcal {X}}'\times _{\mathcal {X}}{\mathcal {U}}_i = {\mathcal {U}}'_i\). Setting \({\mathcal {V}}:= {\mathcal {U}}_2 \cup \ldots \cup {\mathcal {U}}_k\) one obtains a commutative diagram
of MayerVietoris sequences. By the affine case and by induction on the cardinality of the affine cover, \(\alpha \) maps to zero in \({{\,\textrm{K}\,}}^\textrm{cont}_{n}({\mathcal {U}}'_1) \oplus {{\,\textrm{K}\,}}^\textrm{cont}_{n}({\mathcal {V}}')\). Hence its image in \({{\,\textrm{K}\,}}^\textrm{cont}_{n}({\mathcal {X}}'\,\,\textrm{on}\,\,\pi )\) comes from an element \(\alpha '\) in \({{\,\textrm{K}\,}}^\textrm{cont}_{n+1}({\mathcal {U}}'_1\cap {\mathcal {V}}')\). As an admissible formal blowup of the quasiaffine admissible formal scheme \({\mathcal {U}}_1\cap {\mathcal {V}}\), the formal scheme \({\mathcal {U}}_1'\cap {\mathcal {V}}'\) is algebraic according to Lemma 5.5. Hence there exists an admissible formal blowup of \({\mathcal {U}}_1'\cap {\mathcal {V}}'\) where \(\alpha '\) vanishes. As above this can be dominated by an admissible formal blowup \({\mathcal {X}}''\overset{}{\rightarrow }{\mathcal {X}}'\) so that the image of \(\alpha \) in \({{\,\textrm{K}\,}}^\textrm{cont}_{n}({\mathcal {X}}''\,\,\textrm{on}\,\,\pi )\) vanishes. \(\square \)
Next we do another similar reduction to the affinoid case.
Lemma 8.7
For every quasicompact admissible formal scheme \({\mathcal {X}}\) and every constant rhsheaf F the canonical map
is an isomorphism.
Proof
This is similar to the proof of Proposition 8.6. By a MayerVietoris argument and by induction on the number of affine formal schemes needed to cover \({\mathcal {X}}\), we can reduce to one degree less. Fortunately, this also works in degree 0 due to the sheaf condition. \(\square \)
We now prove Conjecture 8.1 in almost all cases.
Theorem 8.8
Let X be a quasicompact and quasiseparated rigid kspace of dimension d. Assume that \(d\ge 2\) or that there exists a formal model which is algebraic (Definition 5.1, e.g. X is affinoid or projective). Then there is an isomorphism
where the righthand side is sheaf cohomology with respect to the admissible topology on the category of rigid kspaces.
Proof
Let \({\mathcal {X}}\) be an admissible formal model of X. By Definition 8.2 there is a fibre sequence
If \(\mathop {\textrm{colim}}\limits _{{\mathcal {X}}'} {{\,\textrm{K}\,}}^\textrm{cont}_{n}({\mathcal {X}}'\,\,\textrm{on}\,\,\pi ) = 0\) for \(n\in \{d1,d\}\), then the induced map
is an isomorphism; this is the case if \({\mathcal {X}}\) is algebraic (Corollary 8.5; note that the reducedness assumption does not harm due to Lemma 6.3) or if \(\dim (X)\ge 2\) (Lemma 8.6). By Lemma 6.3 and Theorem A.20 we conclude
By Lemma 8.7 we have that
Since every formal scheme is homeomorphic to its special fibre, the latter one identifies with \(\mathop {\textrm{colim}}\limits _{{\mathcal {X}}'}\,{{\,\textrm{H}\,}}^d({\mathcal {X}}';\textbf{Z})\) as sheaf cohomology only depends on the topology. Since the formal ZariskiRiemann space \(\langle {\mathcal {X}}\rangle = \lim _{{\mathcal {X}}'}{\mathcal {X}}'\) is a colimit of coherent and sober spaces with quasicompact transition maps, it commutes with cohomology [15, ch. 0, 4.4.1]. Hence we conclude that
by using Theorem 5.8 for the middle isomorphism. \(\square \)
Remark 8.9
The cases of Conjecture 8.1 which are not covered by Theorem 8.8 are curves which are not algebraic. In particular, they must not be affine nor projective nor smooth proper (cf. [37, 1.8.1]).
As the constant sheaf \(\textbf{Z}\) is overconvergent we infer the following.
Corollary 8.10
Let X be a quasicompact and quasiseparated rigid analytic space of dimension d over a discretely valued field. Assume that \(d\ge 2\) or that there exists a formal model of X which is algebraic (e.g. X is affinoid or projective). Then there is an isomorphism
where \(X^\textrm{berk}\) is the Berkovich space associated with X.
Remark 8.11
If \(X^\textrm{berk}\) is smooth over k or the completion of a kscheme of finite type, then there is an isomorphism [11, III.1.1]
with singular cohomology since the Berkovich space \(X^\textrm{berk}\) is locally contractible. For smooth Berkovich spaces this is a result of Berkovich [7, 9.1] and for completions of kschemes of finite type this was proven by HrushovskiLoeser [24].
Finally, we prove vanishing and homotopy invariance of continous Ktheory in low degrees. The corresponding statement for affinoid algebras was proven by Kerz [28, Thm. 12].
Theorem 8.12
Let k be a complete discretely valued field and let X be a quasicompact and quasiseparated rigid kspace of dimension d. Then:

(i)
\({{\,\textrm{K}\,}}^\textrm{cont}_{i}(X)=0\) for \(i>d\).

(ii)
The canonical map \({{\,\textrm{K}\,}}^\textrm{cont}_{i}(X) \overset{}{\rightarrow }{{\,\textrm{K}\,}}^\textrm{cont}_{i}(X\times {\textbf{B}}_k^n)\) is an isomorphism for \(i\ge d\) and \(n\ge 1\) where \({\textbf{B}}_k^n:={{\,\textrm{Spm}\,}}(k\langle t_1,\ldots ,t_n\rangle )\) is the rigid unit disc.
Proof
Let \(i\ge 1\). We have an exact sequence
where \({\mathcal {X}}\) runs over all admissible formal models of X. The last term in the sequence vanishes due to Lemma 8.6. For \(i>d\), we have \({{\,\textrm{K}\,}}^\textrm{cont}_{i}({\mathcal {X}}) = {{\,\textrm{K}\,}}_{i}({\mathcal {X}}/\pi ) = 0\) by Lemma 6.3 and vanishing of algebraic Ktheory as \(\dim ({\mathcal {X}}/\pi ) = d\). This shows (i). For (ii) this works analogously for \(N^\textrm{cont}_{i,n}(X) := {{\,\textrm{coker}\,}}\bigl ( {{\,\textrm{K}\,}}^\textrm{cont}_{i}(X) \overset{}{\rightarrow }{{\,\textrm{K}\,}}^\textrm{cont}_{i}(X\times {\textbf{B}}_k^n) \bigr )\) which is enough as the map in question is injective. \(\square \)
Notes
Actually, Morrow only talks about affinoid algebras.
One should not confuse our notion of a Tate pair with the notion of an affinoid Tate ring \((A,A^+)\), i.e. a Tate ring A together with an open subring \(A^+\) of the powerbounded elements of A which is integrally closed in A. The latter one is used in the context of adic spaces.
Unfortunately, KaroubiVillamayor call these groups “positive”.
Cf. the proof of Lemma A.13.
Cf. [47, §4, p. 94] for the definition of overconvergent sheaves.
Bosch uses the notation \(\textrm{Sp}(A)\).
There is no trouble with this colimit since the prospectrum in question is constant by Lemma 8.3. In general, colimits in procategories are hard to compute.
Even though most parts of Temkin’s article [45] deal with characteristic zero, this is not the case for the mentioned result.
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Acknowledgements
This article is a condensed version of my PhD thesis. I am thankful to my advisors Moritz Kerz and Georg Tamme for their advise and their constant support. I thank Johann Haas for many helpful discussions on the subject and Florent Martin for explaining to me Berkovich spaces and their skeleta. Concerning the expositon, I thank my thesis’ referees Moritz Kerz and Matthew Morrow for their feedback as well as Georg Tamme for helpful comments on a draft of the present paper. The author was supported by the RTG/GRK 1692 “Curvature, Cycles, and Cohomology” and the CRC/SFB 1085 “Higher Invariants” (both Universität Regensburg) funded by the DFG and the Swiss National Science Foundation (grant number 184613). Further thanks go to the Hausdorff Research Institute for Mathematics in Bonn for generous hospitality during the Hausdorff Trimester Program on “Ktheory and related fields” during the months of May and June in 2017.
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Appendix A The rhtopology
Appendix A The rhtopology
In this section we examine the rhtopology introduced by GoodwillieLichtenbaum [20, 1.2]. We use a different definition in terms of abstract blowup squares and show that both definitions agree (Corollary A.9). In the end, we will prove some rhversions of known results for the cdhtopology; most importantly, that the rhsheafification of Ktheory is KHtheory (Theorem A.15).
Notation
Every scheme in this section is noetherian of finite dimension. Under these circumstances, a birational morphism is an isomorphism over a dense open subset of the target [42, Tag 09YP].
Definition A.1
An abstract blowup square is a cartesian diagram of schemes
where \(Z\overset{}{\rightarrow }X\) is a closed immersion, \({{\tilde{X}}}\overset{}{\rightarrow }X\) is proper, and the induced morphism \({{\tilde{X}}} \setminus E \overset{}{\rightarrow }X \setminus Z\) is an isomorphism.
Definition A.2
Let S be a noetherian scheme. The rhtopology on \(\textrm{Sch}_S\) is the topology generated by Zariski squares and covers \(\{Z\overset{}{\rightarrow }X, {\tilde{X}}\overset{}{\rightarrow }X\}\) for every abstract blowup square (abs) as well as the empty cover of the empty scheme. The cdhtopology (completely decomposed htopology) is the topology generated by Nisnevich squares and abstract blowup squares as well as the empty cover of the empty scheme.
Remark A.3
The cdhtopology relates to the Nisnevich topology in the same way as the rhtopology relates to the Zariksi topology. Thus a lot of results in the literature concerning the cdhtopology are also valid for the rhtopology. Possible occurences of the Nisnevich topology may be substituted by the Zariski topology. Hence the same proofs apply almost verbatim by exchanging only the terms “cdh” by “rh”, “Nisnevich” by “Zariksi”, and “étale morphism” by “open immersion”.
Remark A.4
Our definition of an abstract blowup square coincides with the one given in [31]. Other authors demand instead the weaker condition that the induced morphism \(({{\tilde{X}}}\setminus E)_\textrm{red}\overset{}{\rightarrow }(X\setminus Z)_\textrm{red}\) on the associated reduced schemes is an isomorphism, e.g. [39, Def. 12.21]. Indeed, both notions turn out to yield the same topology. To see this, first note that the map \(X_\textrm{red}\overset{}{\rightarrow }X\) is a rhcover since
is an abstract blowup square in the sense of Definition A.1 since \(X_\textrm{red}\overset{}{\rightarrow }X\) is a closed immersion, \(\emptyset \overset{}{\rightarrow }X\) is proper, and the induced map on the complements \(\emptyset \overset{}{\rightarrow }\emptyset \) is an isomorphism. Now we consider the following situation as indicated in the diagram
where \(U := X\setminus Z\) and \({{\tilde{X}}}_U := {{\tilde{X}}} \times _X U\). The morphism \(Z \sqcup {{\tilde{X}}}\overset{}{\rightarrow }X\) is a cover in the sense of [39] but not a priori in the sense of Definition A.2. However, it can be refined by the composition \(Z_\textrm{red}\sqcup {{\tilde{X}}}_\textrm{red}\overset{}{\rightarrow }X_\textrm{red}\overset{}{\rightarrow }X\) in which both maps are rhcovers.
Definition A.5
A morphism of schemes \({\tilde{X}} \overset{}{\rightarrow }X\) satisfies the Nisnevich lifting property iff every point \(x\in X\) has a preimage \({\tilde{x}}\in {\tilde{X}}\) such that the induced morphism \(\kappa (x)\overset{}{\rightarrow }\kappa ({\tilde{x}})\) on residue fields is an isomorphism.
Lemma A.6
Let \(p:{\tilde{X}}\overset{}{\rightarrow }X\) be a proper map satisfying the Nisnevich lifting property and assume X to be reduced. Then there exists a closed subscheme \(X'\) of \({\tilde{X}}\) such that the restricted map \(p_{X'} :X'\overset{}{\rightarrow }X\) is birational.
Proof
Let \(\eta \) be a generic point of X. By assumption, there exists a point \({\tilde{\eta }}\) of \({\tilde{X}}\) mapping to \(\eta \). Since p is a closed map, we have \(p\bigl (\,\overline{\{{\tilde{\eta }}\}}\,\bigr ) \supset \overline{p(\{{\tilde{\eta }}\})} = \overline{\{\eta \}}\) and hence equality holds. Thus the restriction \(\overline{\{{\tilde{\eta }}\}}\overset{}{\rightarrow }\overline{\{\eta \}}\) is a morphism between reduced and irreducible schemes inducing an isomorphism on the stalks of the generic points, hence it is birational. Thus setting \(X'\) to be the (finite) union of all \(\overline{\{{\tilde{\eta }}\}}\) for all generic points \(\eta \) of X does the job. \(\square \)
Lemma A.7
( [49, 2.18]) A proper map is an rhcover if and only if it satisfies the Nisnevich lifting property.
Definition A.8
A proper rhcover is a proper map which is also an rhcover, i.e. a proper map satisfying the Nisnevich lifting property. By Lemma A.6, every proper rhcover of a reduced scheme has a refinement by a proper birational rhcover.
Corollary A.9
The rhtopology equals the topology which is generated by Zariski covers and by proper rhcovers.
Definition A.10
We say that a setvalued presheaf F satisifes rhexcision iff for every abstract blowup square (abs) as in Definition A.1 the induces square
is a pullback square.
Proposition A.11
A Zariski sheaf is an rhsheaf if and only if it satisfies rhexcision.
Proof
The proof is analogous to the proof of the corresponding statement for the Nisnevich topology [39, 12.7], cf. Remark A.3. \(\square \)
Definition A.12
The hrhtopology (honest rhtopology) (resp. the hcdhtopology) is the topology generated by honest blowup squares
and Zariski squares (resp. Nisnevich squares) as well as the empty cover of the empty scheme.
Lemma A.13
Let S be a noetherian scheme. On \(\textrm{Sch}_S\) the hrhtopology equals the rhtopology and the hcdhtopology equals the cdhtopology.
Proof
Every hrhcover is an rhcover. We have to show conversely that every rhcover can be refined by an hrhcover. Let \(X\in \textrm{Sch}_S\). It suffices to show that a cover coming from an abstract blowup square over X can be refined by an hrhcover. As \({{\,\textrm{Bl}\,}}_{X_\textrm{red}}(X)=\emptyset \), the map \(X_\textrm{red}\overset{}{\rightarrow }X\) is an hrhcover. Hence we can assume that X is reduced since pullbacks of abstract blowup squares are abstract blowup squares again. Let \(X = X_1 \cup \ldots \cup X_n\) be the decomposition into irreducible components. For a closed subscheme Z of X one has
If \(Z=X_n\) , then \({{\,\textrm{Bl}\,}}_{X_n}(X_n)=\emptyset \) and \({{\,\textrm{Bl}\,}}_{X_n}(X_i)\) is irreducible for \(i\in \{1,\ldots ,n1\}\) [21, Cor. 13.97]. By iteratively blowing up along the irreducible components, we can hence reduce to the case where X is irreducible. Let
be an abstract blowup square. As X is irreducible, the complement \(U:=X\setminus Z\) is schematically dense in X. As \(p:{{\tilde{X}}}\overset{}{\rightarrow }X\) is proper and birational, also \({{\tilde{X}}}\) is irreducible and \(p^{1}(U)\) is schematically dense in \({{\tilde{X}}}\). Thus p is a Umodification and a result of Temkin tells us that there exists a Uadmissible blowup factoring over \({{\tilde{X}}}\) [45, Lem. 2.1.5].^{Footnote 8} That means that there exists a closed subscheme \(Z'\) of X which lies in \(X\setminus U=Z\) such that \({{\,\textrm{Bl}\,}}_{Z'}(X)\overset{}{\rightarrow }X\) factors over \({{\tilde{X}}}\overset{}{\rightarrow }X\). Thus the rhcover \(\{Z\overset{}{\rightarrow }X,{{\tilde{X}}}\overset{}{\rightarrow }X\}\) can be refined by the hrhcover \(\{Z'\overset{}{\rightarrow }X,{{\,\textrm{Bl}\,}}_{Z'}(X)\overset{}{\rightarrow }X\}\) which was to be shown. The second part has the same proof. \(\square \)
Proposition A.14
Let S be a noetherian scheme and let \(X\in \textrm{Sch}_S\). Every rhcover of X admits a refinement of the form \(U \overset{f}{\rightarrow } {{\tilde{X}}} \overset{p}{\rightarrow } X\) where f is a Zariski cover and p is a proper rhcover.
Proof
The proof is analogous to the corresponding result for the cdhtopology due to SuslinVoevodsky [43, 5.9] or the proof given in [39, 12.27,12.28], cf. Remark A.3. \(\square \)
The following theorem and its proof are just rhvariants of the corresponding statement for the cdhtopology by KerzStrunkTamme [31, 6.3]. The theorem goes back to Haesemeyer [22]. Another recent proof for the cdhtopology which also works for the rhtopology was recently given by KellyMorrow [29, 3.4].
Notation
Let S be a scheme and let \(\textrm{Sh}_\textrm{Ab}(\textrm{Sch}_S^\textrm{rh})\) be the category of rhsheaves on \(\textrm{Sch}_S\) with values in abelian groups. Its inclusion into the category \(\textrm{PSh}_\textrm{Ab}(\textrm{Sch}_S)\) of presheaves on \(\textrm{Sch}_S\) with values in abelian groups admits an exact left adjoint \({{\,\textrm{a}\,}}_\textrm{rh}\). Similarly, the inclusion \(\textbf{Sh}_\textbf{Sp}(\textrm{Sch}_S^\textrm{rh}) \overset{}{\hookrightarrow }\textbf{PSh}_\textbf{Sp}(\textrm{Sch}_S)\) of rhsheaves on \(\textrm{Sch}_S\) with values in the \(\infty \)category of spectra \(\textbf{Sp}\) admits an exact left adjoint \({{\,\textrm{L}\,}}_\textrm{rh}\).
Theorem A.15
Let S be a finitedimensional noetherian scheme. Then the canonical maps of rhsheaves with values in spectra on \(\textrm{Sch}_S\)
are equivalences.
In the proof of the theorem we will make use of the following lemma.
Lemma A.16
Let S be a finite dimensional noetherian scheme and let F be a presheaf of abelian groups on \(\textrm{Sch}_S\). Assume that

(i)
for every reduced affine scheme X and every element \(\alpha \in F(X)\) there exists a proper birational morphism \(X'\overset{}{\rightarrow }X\) such that \(F(X)\overset{}{\rightarrow }F(X')\) maps \(\alpha \) to zero and that

(ii)
\(F(Z)=0\) if \(\dim (Z)=0\) and Z is reduced.
Then \({{\,\textrm{a}\,}}_\textrm{rh}F =0\).
Proof
It suffices to show that for every affine scheme X the map
vanishes as this implies that the rhstalks are zero. Consider the diagram
As \(X_\textrm{red}\overset{}{\rightarrow }X\) is an rhcover, the right vertical map is an isomorphism. Thus we can assume that X is reduced. For any \(\alpha \in F(X)\) there exists, by condition (i), a proper birational morphism \(f:X'\overset{}{\rightarrow }X\) such that \(\alpha \) maps to zero in \(F(X')\). Let U be an open dense subscheme of X over which f is an isomorphism and set \(Z := (X\setminus U)_\textrm{red}\). Then \(\dim (Z) < \dim (X)\) as U is dense. By condition (ii) we can argue by induction that \(\alpha \) maps to zero in \({{\,\textrm{a}\,}}_\textrm{rh}F(Z)\). Since \(X'\sqcup Z \overset{}{\rightarrow }X\) is an rhcover by construction, \(\alpha \) vanishes on some rhcover of Y. Thus the map \(F(X) \overset{}{\rightarrow }{{\,\textrm{a}\,}}_\textrm{rh}F(X)\) maps every element to zero which finishes the proof. \(\square \)
Example A.17
(i) For \(i<0\), the functor \(F={{\,\textrm{K}\,}}_i\) satisfies the conditions of Lemma A.16. Zerodimensional reduced schemes are regular and hence their negative Ktheory vanishes, and condition (i) was proven by KerzStrunk [30, Prop. 5].
(ii) Another example is the functor \(F = N{{\,\textrm{K}\,}}_i\) for \(i\in \textbf{Z}\) which is defined by \(N{{\,\textrm{K}\,}}_i(X) := {{\,\textrm{K}\,}}_i({\textbf{A}}^1_X) / {{\,\textrm{K}\,}}_i(X)\). The Ktheory of regular schemes is homotopy invariant, and condition (i) was proven by KerzStrunkTamme [31, Prop. 6.4].
The following proposition is just a recollection from the literature which will be used in the proof of Theorem A.15.
Proposition A.18
(Voevodsky, AsokHoyoisWendt) Let S be a noetherian scheme of finite dimension. Then:

(i)
The \(\infty \)topos \(\textbf{Sh}^\textrm{rh}(\textrm{Sch}_S)\) of spacevalued rhsheaves on \(\textrm{Sch}_S\) is hypercomplete.

(ii)
The \(\infty \)category \(\textbf{Sh}^\textrm{rh}_\textbf{Sp}(\textrm{Sch}_S)\) of spectrumvalued rhsheaves on \(\textrm{Sch}_S\) is leftcomplete.

(ii)
A map in \(\textbf{Sh}^\textrm{rh}_\textbf{Sp}(\textrm{Sch}_S)\) is an equivalence if and only if it is an equivalence on stalks.
Proof
The rhtopology is induced by a cdstructure [48, Def. 2.1] which is complete, regular, and bounded [49, Thm. 2.2]. Hence a spacevalued presheaf is a hypercomplete rhsheaf if and only if it is rhexcisive; this follows from [48, Lem. 3.5]. On the other hand, as the cdstructure is complete and regular, a spacevalued presheaf is an rhsheaf if and only if it is rhexcisive; this follows from [2, Thm. 3.2.5]. Together this implies (i), cf. [2, Rem. 3.2.6]. The \(\infty \)category \(\textbf{Sh}^\textrm{rh}_\textbf{Sp}(\textrm{Sch}_S)\) is equivalent to the \(\infty \)category \(\textbf{Sh}^\textrm{rh}_\textbf{Sp}(\textbf{Sh}^\textrm{rh}(\textrm{Sch}_S))\) of sheaves of spectra on the \(\infty \)topos \(\textbf{Sh}^\textrm{rh}(\textrm{Sch}_S)\) [36, 1.3.1.7]. Hence the \(\infty \)topos \(\textbf{Sh}^\textrm{rh}_\textbf{Sp}(\textrm{Sch}_S)\) of connective objects is Postnikov complete. As we can write every object F as the colimit \(\mathop {\textrm{colim}}\limits _{n\in \textbf{N}} F_{\ge n}\) of objects which are (up to a shift) connective, this implies (ii) and (iii). \(\square \)
Proof of Theorem A.15
As the \(\infty \)topos of spacevalued sheaves on \(\textrm{Sch}_S^\textrm{rh}\) is hypercomplete, we can test the desired equivalences on stalks (Proposition A.18). Since spheres are compact, taking homotopy groups commutes with filtered colimits and we can check on the sheaves of homotopy groups of the stalks whether the maps are equivalences. Thus the first equivalence follows directly by applying Lemma A.16 and Example A.17 (i) and since the connective cover has isomorphic nonnegative homotopy groups.
For the second equivalence we assume for a moment the existence of a weakly convergent spectral sequence
in \(\textrm{Sh}_\textrm{Ab}(\textrm{Sch}_S^\textrm{rh})\). It suffices to show that \({{\,\textrm{a}\,}}_\textrm{rh}N^p{{\,\textrm{K}\,}}_q =0\) for \(p\ge 1\) which follows from Lemma A.16 and Example A.17 (ii). Thus the proof is finished by the following lemma. \(\square \)
Lemma A.19
There is a weakly convergent spectral sequence
of rhsheaves of abelian groups on \(\textrm{Sch}_X\).
Proof
For every ring R there is a weakly convergent spectral sequence [54, IV.12.3]
This yields a spectral sequence \(E^1_{p,q}=N^p{{\,\textrm{K}\,}}_q\) on the associated presheaves of abelian groups on \(\textrm{Sch}_X\) and hence a spectral sequence \(E^1_{p,q} = {{\,\textrm{a}\,}}_\textrm{rh}N^p{{\,\textrm{K}\,}}_q\) of the associated rhsheafifications. We have to check that the latter one converges to \({{\,\textrm{a}\,}}_\textrm{rh}{{\,\textrm{KH}\,}}_{p+q}\). This can be tested on rhstalks which are filtered colimits of the weakly convergent spectral sequence above. As filtered colimits commute with colimits and finite limits, a filtered colimit of weakly convergent spectral sequence yields a weakly convergent spectral sequence. Hence we are done. \(\square \)
Theorem A.20
Let X be a ddimensional noetherian scheme. Then there exists a canonical isomorphism
Proof
As \({{\,\textrm{KH}\,}}\) is an rhsheaf, the Zariski descent spectral sequence appears as
We know the following:

\(E_2^{p,q}=0\) for \(p>d\) as the rhcohomological dimension is bounded by the dimension [49, 2.27].

\({{\,\textrm{a}\,}}_\textrm{rh}({{\,\textrm{K}\,}}_{\ge 0})_0 = \textbf{Z}\) since \(({{\,\textrm{K}\,}}_{\ge 0})_0(R) = {{\,\textrm{K}\,}}_0(R)=\textbf{Z}\) for any local ring R.

\({{\,\textrm{K}\,}}_{d}(X) \cong {{\,\textrm{KH}\,}}_{d}(X)\) by \({{\,\textrm{K}\,}}_{d}\)regularity and the vanishing of \({{\,\textrm{K}\,}}_{i}\) for \(i>\dim (X)\) [31, Thm. B] together with the spectral sequence relating Ktheory and KHtheory [54, IV.12.3].

\({{\,\textrm{a}\,}}_\textrm{rh}{{\,\textrm{K}\,}}_{q} = 0\) for \(q>0\) by Lemma A.16 and Example A.17.
This implies that in the 2page only the term \(E_2^{d,0} = {{\,\textrm{H}\,}}_{\textrm{rh}}^d(X;\textbf{Z})\) contributes on the line \(pq=d\) and that already \(E_2^{d,0} = E_\infty ^{d,0}\) since all differentials of all \(E_i^{d,0}\) for \(i\ge 2\) come from or go to zero. Hence the theorem follows. \(\square \)
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Dahlhausen, C. Continuous Ktheory and cohomology of rigid spaces. manuscripta math. 173, 119–153 (2024). https://doi.org/10.1007/s0022902301470x
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DOI: https://doi.org/10.1007/s0022902301470x