K -THEORY OF NON-ARCHIMEDEAN RINGS I
MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME
Abstract. We introduce a variant of homotopy K-theory for Taterings, which we call analytic K-theory. It is homotopy invariant withrespect to the analytic affine line viewed as an ind-object of closed disksof increasing radii. Under a certain regularity assumption we prove ananalytic analog of the Bass fundamental theorem and we compare analytic K-theory with continuous K-theory, which is defined in terms
models. Along the way we also prove some results about the algebraic
K-theory of Tate rings.

1. Introduction
In this note we introduce and study a new version of K-theory for nonarchimedean rings, which takes the topology into account. As a motivation,
we first recall some facts about classical topological K-theory before we
describe our construction in more detail in Subsection 1.2.
1.1. Topological K-theory. Let X be a compact Hausdorff space. The(complex) topological K-theory of X is usually defined as the mapping spectrum
K top (X) = map(Σ∞ X+ , KU ),where KU is the spectrum representing complex K-theory. The connective
covering spectrum of K top (X) is denoted by k top (X). Note that one canrecover K top (X) as(1.1) K top (X) ∼− Ωk top (Σ+ X) →− Ω2 k top (Σ2 X) → · · · )= colim(k top (X) →+Here for a topological space Y we write Σ+ Y = Y+ ∧ S 1 (as unpointed
spaces).
For a ring A let K(A) be the non-connective algebraic K-theory spectrumof A and k(A) be its connective cover. For a simplicial ring A let k(A) be thedegreewise connective algebraic K-theory spectrum of A, i.e. the realizationof the simplicial spectrum [m] 7→ k(Am ). Let C(X) be the ring of complex
continuous functions on the compact Hausdorff space X and let j 7→ ∆jtop
be the cosimplicial space of standard simplices.
Date: February 27, 2018.
The authors are supported by the DFG through CRC 1085 Higher Invariants (Universität Regensburg). The second author thanks the Department of Mathematics at the
University of Regensburg for hospitality while this work was done. He is supported by
JSPS KAKENHI Grant (15H03606).
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MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME

An important observation is that topological K-theory can be obtained
from algebraic K-theory by making it homotopy invariant. Recall that there
is a canonical morphism of spectra k(C(X)) → k top (X). As k top is homotopy
invariant, it is natural to study the induced morphismψ : k(C(X × ∆top )) → k top (X),
where the domain is the realization of the simplicial spectrum given by [m] 7→k(C(X ×∆mtop )). Our definition of non-archimedean analytic K-theory below
is motivated by the fact that ψ is an equivalence of spectra.
1.2. Analytic K-theory and Karoubi–Villamayor K-theory. Thereexist several articles generalizing aspects of either classical homotopy theoryof the real interval [0, 1] ⊂ R or of A1 -homotopy theory from algebraic geometry to the non-archimedean setting, for example [KV71], [Kar71], [Cal85],[Ayo15]. In all those works the approach has been to use the rigid analytic
closed unit ball Bκ over the non-archimedean complete valued field κ (orsome variant of it in the work of Karoubi and Villamayor) as a building
block for homotopy theory.
The novel aspect we suggest in this note is to use the affinoid exhaustionof the whole analytic line
(1.2)

(A1 )anκ = colim Bκ (ρ)ρ

instead of the rigid analytic closed unit ball Bκ = Bκ (1) to construct a homotopy theory, or more precisely a non-archimedean K-theory. Here Bκ (ρ)
is the rigid analytic closed ball of radius ρ centered at the origin. So in somesense this new homotopy theory keeps more information than the previous
approaches mentioned above. Our approach is motivated by the observation that the Picard group of a smooth affinoid algebra is homotopy invariant in our sense, while it is not invariant under Bκ -homotopy in general,see [KST16].
More concretely, we mimic the description of topological K-theory explained in Subsection 1.1 in order to construct a new analytic K-theory ofaffinoid spaces. Instead of actually performing the limit in (1.2) we workwith pro-systems of algebraic K-groups.
Let κ be a complete discretely valued field and let X be an affinoid spaceover κ. For 1 ≤ ρ ∈ |κ× | let ∆mρ be the standard rigid m-simplex of radius
ρ, i.e.
∆m
ρ = Sp κhX0 , . . . , Xm iρ /(X0 + · · · + Xm − 1)
where κhX0 , . . . , Xm iρ is the Tate algebra of formal power series which converge on a closed polydisc of radius ρ.
By varying ρ ≥ 1 we obtain an ind-object ρ 7→ ∆ρ in the category ofcosimplicial affinoid spaces over κ. We suggest to define connective analytic
K-theory of the affinoid space X as the pro-spectrum
k an (X) = “ lim ” k(O(X × ∆ρ )).
ρK -THEORY OF NON-ARCHIMEDEAN RINGS I

3

Here O(Y ) denotes the ring of rigid analytic functions of an affinoid space
Y . There exists also a Karoubi–Villamayor analog of k an (X) defined as the
pro-simplicial set
KV an (X) = “ lim ” BGL(O(X × ∆ρ )).
ρ

For X = Sp A we also write k an (A) for k an (X) and similarly KV an (A) for
KV an (X). We write kian (A) resp. KVian (A) for the homotopy pro-group πiof k an (A) resp. of KV an (A).
This definition of non-archimedean analytic K-theory has very pleasant
properties if A is regular. For technical reasons we also have to assume acertain weak form of resolution of singularities for a ring of definition of A,
denoted (†)A , see Subsection 3.2 for details. Condition (†)A is satisfied forany regular affinoid algebra A if κ has residue characteristic zero by a variantof Hironaka’s resolution of singularities.
Theorem 1.1. For a regular affinoid algebra A which satisfies condition(†)A we obtain:(i) k0an (A) is isomorphic to the constant pro-group K0 (A).
(ii) There are isomorphisms of pro-groups
k1an (A) ∼= “ lim ” K1 (A)/(1 + A(r)) = “ lim ” GL(A)/ GL(A)ρ ,
0<r<1

ρ>1

where GL(A)ρ is the normal subgroup generated by matrices g satisfying the unipotence condition limn k(g − 1)n k ρn = 0 and whereA(r) = {a ∈ A | kak < r} for r > 0.
'(iii) For i > 0 there is an isomorphism KVian (A) −→ kian (A) of pro-groups.
(iv) k an satisfies the analytic analog of the Bass fundamental theorem ofalgebraic K-theory, i.e. for i > 0 there is a canonical exact sequencean
0 → kian (A) → kian (Ahti) × kian (Aht−1 i) → kian (Aht, t−1 i) → ki−1
(A) → 0

together with a canonical splitting of the surjection on the right.
For an affinoid κ-algebra A the choice of a surjection κht1 , . . . tn i1  A
gives rise to a residue norm k.k on A. Up to equivalence, it does not dependon the chosen surjection. This is the norm occurring in (ii). The resultsof Theorem 1.1 are obtained at different places throughout the paper. We
collect the references in Section 1.4 below.
Remark 1.2. Note that (i) and (ii) of Theorem 1.1 are analogous to classicalproperties of topological K-theory, see [Ros05, Sec. 3.2]. We will see ina sequel to this note that (iv) can be interpreted as the Mayer–Vietoris
sequence of k an corresponding to the standard admissible covering of therigid space (P1A )an .
Theorem 1.1(iv) implies that we can use a similar delooping process asin (1.1) to define a non-connective analytic K-theory spectrum K an (A) such

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MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME

that k an (A) is the connective covering of K an (A) and such that the analytic Bass fundamental theorem holds in general, i.e. the analog of the exact
sequence in Theorem 1.1(iv) for Kian exists for all i ∈ Z.
1.3. Continuous K-theory. Let A be an affinoid algebra over the complete
discretely valued field κ. Let κ0 ⊂ κ be the ring of integers and π ∈ κ0 be aprime element. Choose a κ0 -algebra A0 flat and topologically of finite typeover κ0 with A0 [1/π] ∼= A, see [Bos14]. Such an A0 is also called a ring of
definition of A. Morrow [Mor16] defines continuous K-theory of A as
K cont (A) = cofib(K(A0 on (π)) → “ lim ” K(A0 /(π n )),nwhere K stands for the non-connective algebraic K-theory spectrum andwhere the cofibre is taken in the ∞-category of pro-spectra, recalled in Section 2. Using pro-excision one shows that the pro-group Kicont (A) defined as
πi (K cont (A)) for i ∈ Z does not depend on the choice of the ring of definition
A0 up to canonical isomorphism.
In fact, variants of the pro-group Kicont (A) have been studied since theintroduction of algebraic K-theory, for example we have:'(i) The canonical map K0 (A) −→ K0cont (A) is an isomorphism of progroups, see Proposition 5.10. For i ≤ 0 the pro-groups Kicont (A) areconstant and agree with the negative “topological” K-groups studied
in [KV71, Sec. 7] and in [Cal85].
'(ii) The canonical map “ limn” K1 (A)/(1 + π n A0 ) −→ K1cont (A) is an
isomorphism of pro-groups, see Lemma 5.5. In particular, for acomplete discretely valued field κ with ring of integers κ0 we get
K1cont (κ) ∼= “ limn” κ× /(1 + π n κ0 ).
(iii) For a complete discretely valued field κ with ring of integers κ0 the'→ K2cont (κ) is an isocanonical map “ limn” K2 (κ)/{1 + π n κ0 , κ×0} −morphism of pro-groups, see Lemma 5.6. In particular, if the residue
field of κ0 is finite, the norm-residue symbol induces an isomorphism
K2cont (κ) ∼= µ ⊂ κ× , where µ are the roots of unity (Moore’s Theorem), see [Mil71, App.].
The key observation of our note is that for regular affinoid algebras continuous K-theory agrees with analytic K-theory. This unifies the two existing approaches to the K-theory of non-archimedean rings which originate
from [KV71].
Theorem 1.3. For a regular affinoid algebra A which satisfies condition(†)A the pro-spectra K an (A) and K cont (A) are weakly equivalent.
For the definition of weakly equivalent pro-spectra see Subsection 2.2. In
particular, this means that the pro-groups Kian (A) and Kicont (A) are naturally isomorphic. This theorem is proven as Theorem 6.18 below.

K -THEORY OF NON-ARCHIMEDEAN RINGS I

5

1.4. Overview of sections. In Section 2 we collect various facts aboutpro-objects in ∞-categories, and in particular about pro-spectra and prospaces. Most of our results are formulated for adic rings or Tate rings — weuse affinoid algebras in the introduction only for simplicity of presentation.
Basic properties of these non-archimedean rings are discussed in Section 3.
There we also introduce normed rings which are used in the study of the
Karoubi-Villamayor version of our theory in Section 7. In Section 4 wediscuss some general preliminaries about algebraic K-theory and we study
the low degree algebraic K-groups of Tate rings. Continuous K-theory is
introduced in Section 5. Beside the basic properties mentioned in 1.3 above,
we also prove an analytic version of the Bass fundamental theorem and study
homotopy invariance of continuous K-theory. In Section 6 we introduce theanalytic K-theory discussed in 1.2. The main result there is the comparison
with continuous K-theory for a regular affinoid algebra A satisfying (†)A .
We end this overview with the proof of Theorem 1.1. Assertion (i) is Lemma 6.13(ii). By Theorem 1.3 we can identify analytic K-theory with continuous K-theory. Using this identification, Lemma 5.5 gives the first isomorphism of part (ii) and Proposition 5.10(i) gives (iv). Assertion (iii) is
precisely Lemma 7.5. Now Lemma 7.3 gives the second isomorphism in(ii).
Acknowledgment. We would like to thank M. Morrow for helpful discussions on K-theory of non-archimedean rings. In particular, he suggested
the results discussed in Subsection 4.2 and he proposed the definition ofcontinuous K-theory of affinoid algebras.
2. Topological preliminaries
In the study of K-theory of non-archimedean rings, pro-spectra play animportant role. The theory of ∞-categories provides a well developed andflexible framework to deal with homotopy theoretic questions about prospectra. In the following, we therefore collect some facts about pro-objects
in ∞-categories in general and about pro-spectra. General references for
pro-objects are [AM69] or [Isa02]. The ∞-categorical version is discussed forexample in [Lur17b, §A.8.1]. We write Spc for the ∞-category of spaces.
2.1. Pro-objects in ∞-categories. Let C be an ∞-category. For technicalreasons we assume that C is accessible and admits finite limits. A proobject in C is represented by a diagram X : I → C where I is a small
cofiltered ∞-category. We also use the notation X = “ limI”Xi and call thisa level-representation of X. The pro-objects are the objects of an ∞-category
Pro(C). If Y = “ limJ” Yj is a second pro-object, the mapping space is givenby the classical formula(2.1)

Map(“ lim ” Xi , “ lim ” Yj ) ' lim colim Map(Xi , Yj ),
I

J

J

I

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MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME

where the limit and colimit are computed in the ∞-category of spaces. A
precise definition is(2.2)

Pro(C) = Funlex,acc (C, Spc)op

where Funlex,acc (C, Spc) is the full subcategory of Fun(C, Spc) spannedby those functors which are accessible (i.e. preserve small κ-filtered colimits for some regular cardinal κ) and left exact (i.e. commute with finite limits). Here the pro-object X from above corresponds to the functor
colimI Map(Xi , −) : C → Spc, see [Lur17b, Def. A.8.1.1, Rem. A.8.1.5].
Any functor f : C → D induces a functor Pro(f ) : Pro(C) → Pro(D)given by “ limI”Xi 7→ “ limI”f (Xi ), see [Lur17b, Ex. A.8.1.8].
Finite limits and colimits in Pro(C) can be computed level-wise: Let
K be a finite simplicial set. The functor limK : Fun(K, C) → C induces
a functor Pro(Fun(K, C)) → Pro(C), the ‘level-wise limit’. If C admits
finite colimits, we similarly have a level-wise colimit. Any pro-object of
K-diagrams induces a K-diagram of pro-objects via the natural functor
Pro(Fun(K, C)) → Fun(K, Pro(C)).
Lemma 2.1. Let F ∈ Pro(Fun(K, C)). Then the level-wise limit of F is alimit of the induced diagram K → Pro(C). If C admits finite colimits, thesame is true for the colimit.
Proof. This can be checked as in the 1-categorical case (see [AM69, App.,
Prop. 4.1]) using formula (2.1) and the fact that in Spc finite limits commute
with filtered colimits [Lur09, Prop. 5.3.3.3].

We also have level-representations for certain diagrams. In the following Lemma N(A) denotes the nerve of a partially ordered set A viewed as
category.
Lemma 2.2. Let A be a finite partially ordered set. The natural functor
Pro(Fun(N(A), C)) → Fun(N(A), Pro(C))
is essentially surjective.
Proof. If C is small, this follows directly from [Lur09, Prop. 5.3.5.15]. The
general case can be reduced to this as in the proof of [Lur17b, Prop. A.8.1.6].
Lemma 2.3. The ∞-category Pro(C) is complete. If C admits finite resp. allsmall colimits, then so does Pro(C).
Proof. Lemma 2.1 implies in particular that C → Pro(C) commutes withfinite limits and colimits. In particular, Pro(C) has a final object. Combining Lemmas 2.2 and 2.1 we see that Pro(C) admits pullbacks. Now [Lur09,
Cor. 4.4.2.4] implies that Pro(C) has all finite limits. If C admits finite
colimits, then by the same argument Pro(C) has all finite colimits. Since
Pro(C) also has all small cofiltered limits, it follows that Pro(C) is complete by [Lur09, Prop. 4.4.2.6]. In order to show that Pro(C) is cocomplete

K -THEORY OF NON-ARCHIMEDEAN RINGS I

7

if C is, it now suffices to show that Pro(C) has small coproducts [Lur09,
Prop. 4.4.2.6]. This is shown in Lemma 2.4 below.

Recall that an accessible ∞-category [Lur09, Def. 5.4.2.1] which is cocomplete is called presentable [Lur09, Def. 5.5.0.1]. A presentable ∞-category is
automatically complete [Lur09, Cor. 5.5.2.4].
Lemma 2.4. Assume that C is presentable. Let A be a set, and let (X α )α∈A
be a family of objects in Pro(C). Choose level representations
X α = “ lim ” Xiα .
IαQDefine the cofiltered ∞-category K = α∈A Iα . Thena
X = “ lim ”
Xkαα
k∈K

α∈A

is a coproduct of (X α )α∈A .
Proof. (See [Isa02, §8] for the classical situation.) Let Y = “ limJ” Yj be any
pro-object. ThenY
Map(X, Y ) ' lim colim
Map(Xkαα , Yj )
J

K

' lim

Y

J

'

α∈A

Y
α∈A

'

Y

A

colim Map(Xkαα , Yj )Iα(see below)

lim colim Map(Xkαα , Yj )JIαMap(X α , Y ).

α∈A

Q
Q
The second equivalence holds since colimK A ' α∈A colimIα in the ∞category of spaces. Indeed, since products and filtered colimits in spaces
commute with homotopy groups, this follows from the corresponding fact in
the category of sets, where it can be checked directly.
Lemma 2.5. If C is stable, then so is Pro(C).
For the notion of a stable ∞-category see [Lur17a, Def. 1.1.1.9].
Proof. By Lemma 2.1 Pro(C) is pointed. By Lemma 2.3 it has all finite
limits and colimits. Since by Lemma 2.1 again the suspension functor in
Pro(C) can be computed level-wise, it is an equivalence. We conclude using[Lur17a, Cor. 1.4.2.27].

2.2. Weak equivalences of pro-spectra. We let Sp be the presentablestable ∞-category of spectra [Lur17a, §1.4.3]. It carries a t-structure whoseheart is equivalent to the category Ab of abelian groups. In particular, anabelian group A gives a spectrum H(A) with π0 (H(A)) ∼= A, πi (H(A)) = 0
for i 6= 0.

8MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME

We denote by Sp+ ⊆ Sp the full stable subcategory of spectra which arebounded above, and denote the inclusion by ι. By [Lur17b, Ex. A.8.1.8] we
get an adjunctionι∗ : Pro(Sp)  Pro(Sp+ ) : Pro(ι)
where ι∗ is induced using (2.2) by composition with ι, and Pro(ι) is fully
faithful. In other words, Pro(Sp+ ) is a localization of Pro(Sp) in the sense of[Lur09, Def. 5.2.7.2]. In particular, Pro(Sp+ ) also admits all small colimits.
In terms of level representations ι∗ has the following description: Denoteby τ≤n : Sp → Sp≤n the truncation functor. It is left adjoint to the inclusion.
Lemma 2.6. Let “ limI”Xi ∈ Pro(Sp). Then we have a natural equivalenceι∗ (“ lim ” Xi ) ' “ lim ” τ≤n Xi
I

N×I

in Pro(Sp+ ).
Proof. In terms of left exact accessible functors ι∗ (“ limI”Xi ) is given bycolimI Map(Xi , −) ∈ Funlex,acc (Sp+ , Spc) and this colimit can be computed
object-wise (see [Lur09, Cor. 5.1.2.3]). Consider Y ∈ Sp+ . Note that if
Y ∈ Sp≤m then Map(τ≤n Xi , Y ) ' Map(Xi , Y ) for all n ≥ m. Hencecolim Map(Xi , Y ) ' colim Map(τ≤n Xi , Y ),I

N×I

which implies the claim.

Definition 2.7. We call a map of pro-spectra X → Y a weak equivalence if
ι∗ X → ι∗ Y is an equivalence in Pro(Sp+ ). We say that the pro-spectra X, Y
are weakly equivalent if there is an equivalence of ι∗ X and ι∗ Y in Pro(Sp+ ).
We say that X is weakly contractible if it is weakly equivalent to a point.
Finally, we call X → Y → Z a weak fibre sequence if ι∗ X → ι∗ Y → ι∗ Z is afibre sequence in Pro(Sp+ ).
The functor πi : Sp → Ab induces functors
πi : Pro(Sp) → Pro(Ab) and πi : Pro(Sp+ ) → Pro(Ab),given by X = “ limJ” Xj 7→ πi (X) = “ limJ” πi (Xj ), called the i-th homotopy
pro-group. Abusing notation we write τ≤n : Pro(Sp) → Pro(Sp) for thefunctor Pro(τ≤n ).
Lemma 2.8. For a map f : X → Y in Pro(Sp) the following are equivalent:(1) f is a weak equivalence.
(2) τ≤n f is an equivalence for every n ∈ Z.
(3) There exists an m ∈ Z such that τ≤m f is an equivalence and f in∼duces isomorphisms πn (X) −→ πn (Y ) in Pro(Ab) for all n ∈ Z.
Proof. Since the functor τ≤n factors through ι∗ : Pro(Sp) → Pro(Sp+ ), (1)implies (2). Clearly, (2) implies (3). It remains to show that (3) implies

K -THEORY OF NON-ARCHIMEDEAN RINGS I

9(1). Choose a level representation “ limI”Xi → “ limI”Yi . By assumption“ limI”τ≤m Xi → “ limI”τ≤m Yi is an equivalence. Using the fibre sequenceΣn H(πn (Xi )) → τ≤n Xi → τ≤n−1 Xi(similarly for Yi ) and that “ limI”πn (Xi ) → “ limI”πn (Yi ) is an isomorphism,'→ “ limI”τ≤n Yi
we inductively show that we have equivalences “ limI”τ≤n Xi −for all n ∈ Z. Lemma 2.6 now implies that f is a weak equivalence.

2.3. Pro-spaces. Let Spc0∗ denote the ∞-category of pointed connected
spaces.
Definition 2.9. A morphism X → Y in Pro(Spc0∗ ) is called a weak equivalence if it induces an isomorphism between homotopy pro-groups πi (X) →πi (Y ) for every i > 0.
Isaksen [Isa01] has a general notion of weak equivalences between prosimplicial sets. There is a canonical functor from the ∞-category associated to the category of pro-objects in pointed connected simplicial sets toPro(Spc0∗ ) and this functor detects weak equivalences by [Isa01, Cor. 7.5].
Let f : X → Y be a morphism between pro-objects of connected spectra and Ω∞ f : Ω∞ X → Ω∞ Y the induced map on pro-objects of infinite
loop spaces. Then by Lemma 2.8 f is a weak equivalence in the sense of
Definition 2.7 if and only if Ω∞ f is a weak equivalence in Pro(Spc0∗ ).
Lemma 2.10. If X → Y is a weak equivalence in Pro(Spc0∗ ) then the induced map on integral homology pro-groups Hi (X) → Hi (Y ) is an isomorphism in Pro(Ab).
This can be deduced from [Isa01, Thm. 7.3].
The following version of Whitehead’s theorem is only stated for prosystems indexed by a countable category. This suffices for our purposes.
Proposition 2.11. Let f : X → Y be a morphism of pro-systems of pointed
connected nilpotent spaces which can be indexed by a countable category. If
the induced map f∗ : Hi (X) → Hi (Y ) is an isomorphism in Pro(Ab) for
every i > 0, then f is a weak equivalence in Pro(Spc0∗ ).
This can be deduced from [BK72, Prop. III.6.6, Cor. III.5.4(ii)].
3. Non-archimedean rings
3.1. Adic rings. In this note we call a complete topological ring A an adic
ring if a system of neighborhoods of zero is given by the ideals {(π n ) | n > 0}for some element π ∈ A. For an element π ∈ A with this property we also
say that A is π-adic.
For such an adic ring A and element π ∈ A we denote by
Aht1 , . . . , tm i = lim A[t1 , . . . , tm ]/(π n )n

10MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME

the completion of the polynomial ring. Fix a0 ∈ A. By
(3.1)

Ah∆m
a0 i = Aht0 , . . . , tm i/(t0 + · · · + tm − a0 )we denote the standard m-simplex of ‘diameter a0 ’. So Ah∆m
a0 i is the π-adic
completion of
(3.2)

A[∆m
a0 ] = A[t0 , . . . , tm ]/(t0 + · · · + tm − a0 ).

Note that there exists an A-algebra isomorphism'

Ahx1 , . . . , xm i −→ Ah∆ma0 i

xi 7→ ti .

For varying m and fixed a0 the Ah∆m
a0 i form in the usual way a simplicial
ring which we denote by Ah∆a0 i. If we choose another element a1 ∈ A we
get a morphism of simplicial A-algebrasΨaa01 : Ah∆a1 a0 i → Ah∆a0 i
by mapping each variable ti to a1 ti . The same applies to the polynomial
rings A[∆ma0 ].
In the following proposition we summarize some facts about excellent
rings. Let κ be a complete discretely valued field with ring of integers κ0 .
We abbreviate Aht1 , . . . , tm i by Ahti.
Proposition 3.1. For a noetherian complete adic ring A we have:(i) For an element π ∈ A as above the ring A is quasi-excellent if andonly if A/(π) is quasi-excellent(ii) If A is quasi-excellent, then Ahti is quasi-excellent, and the ring homomorphism A → Ahti is regular.
(iii) If A is a κ0 -algebra of topologically finite type, it is excellent.
Proof. Part (i) is due to Gabber, see [ILO14, Thm. I.9.2]. Part (ii) follows
immediately from part (i) and [Sta17, 0AH2]. For part (iii) note that a κ0 algebra A which is topologically of finite type is a quotient of a ring κ0 hti,
which is quasi-excellent (by (ii)) and which is universally catenary as it isregular (again by (ii)).

3.2. Tate rings. For the convenience of the reader we recall some facts
about Tate rings in this section, see [Hub93, Sec. 1]. Tate rings are generalizations of affinoid algebras and we are going to state most of our results
about K-theory in the language of Tate rings.
Definition 3.2. A topological ring A is called a Tate ring if there exists
an open subring A0 ⊂ A which is a complete adic ring in the sense of
Subsection 3.1, i.e. it has the π-adic topology for some π ∈ A0 and A =A0 [1/π]. We call such a subring A0 a ring of definition of A.
An element π as in Definition 3.2 is clearly invertible and topologicallynilpotent in A.
Remark 3.3. The following observations about a Tate ring A are easy toprove.

K -THEORY OF NON-ARCHIMEDEAN RINGS I

11(i) For an invertible and topologically nilpotent π ∈ A and a ring of
definition A0 ⊂ A there exists n > 0 such that π n ∈ A0 and A0 has
the π n -adic topology.
(ii) If A0 and A00 are both rings of definition of the Tate ring A thenA0 ∩ A00 and A0 · A00 are also rings of definition of A.
(iii) For a power bounded element a0 ∈ A there exists a ring of definition
A0 ⊂ A with a0 ∈ A0 .
Quite often in this note we need to assume the existence of a ‘nice’ ringof definition of a Tate ring A. We usually consider the following condition
on A:(†)A There exists a ring of definition A0 ⊂ A which is noetherian andthere exists a proper morphism p : X → Spec(A0 ) such that X isregular and such that p is an isomorphism over Spec(A).
There is also the following weaker version of (†)A :(‡)A There exists a ring of definition A0 ⊂ A which is noetherian andquasi-excellent.
Remark 3.4. By [Gro67, Sec. IV.7.9] condition (†)A implies condition (‡)A .
If A is regular and if one has resolution of singularities available, for example
if Q ⊂ A0 [Tem08], then condition (‡)A conversely implies condition (†)A .
We define the Tate algebra over the Tate ring A by
Aht1 , . . . , tm i = (A0 ht1 , . . . , tm i) ⊗A0 A.
It is easy to see that the Tate algebra does not depend on the choice of thering of definition A0 . For a power bounded element a0 ∈ A we define thestandard m-simplex with ‘diameter a0 ’ by the same formula as in (3.1), as
well as its polynomial version (3.2). Given another power bounded a1 ∈ A
there exists a well-defined map
Ψaa01 : Ah∆a1 a0 i → Ah∆a0 i
mapping each variable ti to a1 ti . In these constructions one uses Remark 3.3(iii)
to reduce to the analogous constructions for adic rings.
Lemma 3.5. Consider a Tate ring A which satisfies condition (†)A (resp.
(‡)A ). Then Ahti satisfies condition (†)Ahti (resp. (‡)Ahti ). Moreover the ringhomomorphism A → Ahti is regular.
Here we abbreviate Aht1 , . . . , tm i by Ahti again.
Proof. This is immediately clear from Proposition 3.1. Note that in thesituation of (†)A the scheme X⊗A0 A0 hti is regular as the ring homomorphism
A0 → A0 hti is regular by Proposition 3.1(ii) and X is regular.

Lemma 3.6. If A is an affinoid algebra over a complete discretely valued
field κ, then condition (‡)A holds. If moreover A is regular and the residue
field of κ has characteristic zero, then condition (†)A holds.

12

MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME

Proof. For an affinoid algebra A over κ choose a ring A0 which is flat andtopologically of finite type over κ0 . Then A0 is a ring of definition for A and
it is excellent by Proposition 3.1(iii). For Q ⊂ A0 we know that the scheme
Spec(A0 ) has a resolution of singularities [Tem08].

3.3. Normed rings. When we study the Karoubi–Villamayor analog of ouranalytic K-theory in Section 7 it is convenient to do it for a more generalclass of non-archimedean rings than Tate rings, namely normed rings. This
material is more or less well-known, but for the reader’s convenience we recall
some details in this subsection.
Definition 3.7. Let A be an associative (not necessarily commutative orunital) ring. A non-archimedean norm on A is a map k . k : A → R≥0 satisfying(1) kxk = 0 ⇔ x = 0,(2) k − xk = kxk,
(3) kx + yk ≤ max{kxk, kyk},(4) kxyk ≤ kxkkyk,
for all x, y ∈ A.
In the following, all norms will be non-archimedean, and for simplicity wewill usually drop the adjective ‘non-archimedean’. A ring equipped with anorm is called a normed ring. The normed ring (A, k . k) is called complete
if A is complete for the metric induced by the norm, and it is called unital
if A is unital and k1k = 1.
Example 3.8.
(i) Any associative ring A can be equipped with thetrivial norm k . k given by kak = 1 if a 6= 0, k0k = 0.
e = AoZ(ii) If (A, k . k) is a normed ring, one can equip its unitalization A
with the normk(a, n)k = max{kak, knkZ }e k . k) is a unital normed
where k . kZ is the trivial norm on Z. Then (A,e is isometric. It is complete if A was.
ring and the inclusion A ,→ A(iii) Let A be a Tate ring with a ring of definition A0 ⊂ A and an element
π ∈ A0 which is a topologically nilpotent unit in A. Fix 0 <  < 1.
We define the gauge norm associated to A0 , π, and  byi

kakA0 ,π, = max{i∈Z | a∈π A0 } .
A different choice of A0 , π, and  gives an equivalent norm in thesense defined below.
Definition 3.9. A homomorphism between normed rings φ : (A, k . kA ) →(B, k . kB ) is called s-bounded for some real number s > 0 if there is a C > 0such that kφ(x)kB ≤ CkxksA for all x ∈ A. It is called bounded if it iss-bounded for some s.
Two norms k . k, k . k0 on a ring A are called equivalent if the identities(A, k . k) → (A, k . k0 ) and (A, k . k0 ) → (A, k . k) are bounded.

K -THEORY OF NON-ARCHIMEDEAN RINGS I

13

Let (A, k . k) be a complete normed ring, and let ρ > 0 be a real number.
We define the ring of ρ-convergent power series with variables t1 , . . . , tm andwith coefficients in A
X
|I|→∞
Aht1 , . . . , tm iρ = {aI tI | kaI kρ|I| −−−−→ 0},
I

where we use standard multi-index notation. The variables commute withevery element of the ring. We set
XkaI tI kρ = max{kaI kρ|I| }.
I

This defines a norm k . kρ on Ahtiρ and, equipped with this norm, Ahtiρ
becomes again a complete normed ring. (See [BGR84, Ch. 1, 1.4.1] for thecase of a commutative unital complete normed ring and radius of convergenceρ = 1. Everything carries over to our situation mutatis mutandis.)Let (A, k . k) be a complete normed ring and let ρ ≥ 1. We defineAh∆mρ i = Aht0 , . . . , tm iρ /In,ρ
where In,ρ is the twosided ideal
In,ρ = {t0 f + · · · + tm f − f | f ∈ Aht0 , . . . , tn iρ }.
It is then clear that Ah∆ρ i with the usual face and degeneracy maps is a
simplicial ring. For radii ρ0 ≥ ρ ≥ 1 there is a natural restriction map
Ah∆ρ0 i → Ah∆ρ i of simplicial A-algebras.
Remark 3.10. For a Tate ring A as in Example 3.8(iii) and for ρ = −jwith j ≥ 0 there exists an isomorphism of simplicial A-algebras'Θ : Ah∆πj i −→ Ah∆ρ i,

ti 7→ π j ti .

For ρ0 = −j−1 we obtain a commutative square of simplicial ringsAh∆πj+1 iΘ 'Ah∆ρ0 iΨππj

/ Ah∆ j iπ' Θ/ Ah∆ρ i.

4. Algebraic K-theory of non-archimedean rings
In this section we summarize some background material on algebraic Ktheory and we study algebraic K-theory of non-archimedean rings.
4.1. K-theory spectra and the plus-construction. Let R be a unital
ring. We denote byk(R) ∈ Sp≥0the connective algebraic K-theory spectrum associated to the exact categoryof finitely generated projective (right) R-modules. Later we will also use

14

MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME

the non-connective algebraic K-theory spectrum K(R). Its connected coverτ≥1 k(R) is denoted by K GL (R). We have a natural equivalence
BGL(R)+ ' Ω∞ K GL (R)where the left-hand side is Quillen’s plus construction.
Now let R be a simplicial ring.
Definition 4.1. We write k(R) for the degree-wise K-theory of R, i.e. thegeometric realization of the simplicial spectrum [p] 7→ k(Rp ). Similarly,
we write K GL (R) for the geometric realization of the simplicial spectrum[p] 7→ K GL (Rp ).
Note that in general K GL (R) is not the connected cover of k(R). The
infinite loop space of K GL (R) can again be described as a plus construction.
We write BGL(R) for the diagonal of the bisimplicial set [p] 7→ BGL(Rp ).
Note that in general the diagonal is weakly equivalent to the geometric realization of a bisimplicial set. Let (−)+ : sSet → sSet be the functorial plus
construction as defined in [DGM13, Def. 3.1.1.5].
Lemma 4.2. There are natural weak equivalences
BGL(R)+ ' diag([p] 7→ BGL(Rp )+ )and
BGL(R)+ ' Ω∞ K GL (R).
Proof. The first equivalence follows from [DGM13, Lemma 3.1.3.1]. It implies the second one, since geometric realization commutes with Ω∞ for connective spectra [Lur17a, Prop. 1.4.3.9].
This can be simplified even more for connected simplicial rings:
Lemma 4.3. Assume that the simplicial ring R is connected. Then there isa natural weak equivalence
BGL(R) ' Ω∞ K GL (R).
Proof. Since R is connected, every elementary matrix in GL(R) can be connected with the identity matrix by a 1-simplex in GL(R). This implies that
π1 BGL(R) ∼= π0 GL(R) is abelian, and hence BGL(R) → BGL(R)+ is aweak equivalence. Now the claim follows from Lemma 4.2.

4.2. Pro-A1 -homotopy invariance for algebraic K-theory. The observations of this subsection resulted from a discussion with M. Morrow. Let R
be a commutative noetherian ring and let π ∈ R. Let SchR be the categoryof schemes of finite type over R. In this subsection we study the K-theoretic
properties of the pro-system of R-algebras “ limn∈N” Sn with Sn the polynomial ring R[t] = R[t1 , . . . , tm ] for all n ∈ N and with transition maps
Sn+1 → Sn characterized by ti 7→ πti (1 ≤ i ≤ m). We shall abbreviate thispro-system also as “ limt7”→πt R[t].

K -THEORY OF NON-ARCHIMEDEAN RINGS I

15Let F be a contravariant functor from SchR to Pro(Sp). We define a new
functor N π F from SchR to Pro(Sp) byp∗

N π F (X) = cofib(F (X) −→ “ lim ” F (X ⊗R R[t])),

(4.1)

t7→πt

where p : X ⊗R R[t] → X is the canonical map. Note that this p∗ has a left
inverse given by the zero section.
We are interested in criteria, which guarantee that N π F (X) is contractible
resp. weakly contractible, see Definition 2.7. For X in SchR we write Xn forthe scheme X ⊗R R/(π n ).
Lemma 4.4. Let X be in SchR such that there exists n > 0 with X = Xn .
Then N π F (X) is contractible.
Proof. This is clear as the map X ⊗R R[t] given by t 7→ π n t factors throughX.

A morphism f : X̃ → X in SchR is called admissible if f is proper andan isomorphism over X ⊗R R[1/π]. Consider the following conditions on thefunctor F :(P1)F For any admissible morphism f : X̃ → X the functor F satisfies
pro-descent, i.e. we have a weak equivalence“ lim ” F (X, Xn ) → “ lim ” F (X̃, X̃n ).
nn(P2)F For any regular scheme X in SchR the pro-spectrum N π F (X) is
weakly contractible.
Here for a morphism of schemes Y 0 → Y in SchR we use the notation
F (Y, Y 0 ) = fib(F (Y ) → F (Y 0 )).
Lemma 4.5. Let F be a functor satisfying condition (P1)F and let p : X̃ →X be an admissible morphism in SchR . Then p induces a weak equivalence'

N π F (X) −→ N π F (X̃).
Proof. Consider the commutative diagram of fibre sequences of pro-spectra“ lim ” N π F (X, Xn )'n

/ N π F (X)/ “ lim ” N π F (Xn )/ N π F (X̃)

/ “ lim ” N π F (X̃n )

n

'



“ lim ” N π F (X̃, X̃n )n'n

The pro-spectra on the right are both contractible by Lemma 4.4, so the left
horizontal maps are equivalences. The left vertical map is a weak equivalenceby condition (P1)F .

Proposition 4.6. Let F be a functor satisfying the conditions (P1)F and(P2)F . Then for any X in SchR such that there exists an admissible morphism X̃ → X with X̃ regular the pro-spectrum N π F (X) is weakly contractible.

16

MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME

Proof. By Lemma 4.5 the map N π F (X) → N π F (X̃) is a weak equivalenceof pro-spectra. As N π F (X̃) is weakly contractible by (P2)F the same is true
for N π F (X).
Using that non-connective algebraic K-theory spectrum K satisfies theconditions (P1)F resp. (P2)F by [KST18, Thm. A] resp. [Wei13, Thm. V.6.3]
we get:Corollary 4.7. For X as in Proposition 4.6 we get a weak equivalence of
pro-spectra'
K(X) −→ “ lim ” K(X ⊗R R[t]).
t7→πtWe finish this subsection with a variant of the above observations. Let
now R be a commutative not necessarily noetherian ring and π ∈ R. Fora covariant functor F from R-algebras to Pro(Sp) we define by the same
method as in (4.1) a new functor N π F from R-algebras to Pro(Sp).
We consider the following excision property for F :(E)F For any extension of π-torsion free R-algebras S1 ⊂ S2 such that
there exists m > 0 with π m S2 ⊂ S1 we have a weak equivalence'“ lim ” F (S1 , S1 /(π n )) −→ “ lim ” F (S2 , S2 /(π n )).
nn

The same argument as in the proof of Lemma 4.5, replacing condition(P1)F by condition (E)F gives:
Lemma 4.8. Let F satisfy condition (E)F and let S1 ⊂ S2 be an extension
as in condition (E)F . Then the map of pro-spectra'

N π F (S1 ) −→ N π F (S2 )is a weak equivalence.
Remark 4.9.
(i) Algebraic K-theory satisfies the condition (E)K by
work of Suslin, Geisser–Hesselholt and Morrow, see [Mor14].
(ii) Combining (i) and Lemma 4.8 we get a weak equivalence of prospectra'
N π K(S1 ) −→ N π K(S2 ).
4.3. K0 and K1 of non-archimedean rings. In this subsection we discuss
some results about K0 and K1 of non-archimedean rings.
We define a continuous version of K1 of a Tate ring A as the pro-group
K1cont (A) = “ lim ” K1 (A)/(1 + π n A0 ).
n

Here A0 is a ring of definition and π ∈ A0 is a topologically nilpotent unit in
A. Following Morrow [Mor16] we shall in Section 5 give a definition of thecontinuous K-theory pro-group Kicont (A) for any i. Note that the sequence
0 → 1 + π n A0 → K1 (A0 ) → K1 (A0 /(π n )) → 0
is exact [Wei13, Lem. III.2.4].

K -THEORY OF NON-ARCHIMEDEAN RINGS I

17Let F be a covariant functor from the category of Tate rings to an abeliancategory. Using a Bass type construction, compare [Wei13, Sec. III.4], we
define a new functor ΣF on the same categories by
ΣF (A) = coker(F (Ahti) × F (Aht−1 i) → F (Aht, t−1 i)),
Lemma 4.10. For any Tate ring A the natural map ΣK1 (A) → ΣK1cont (A)is an isomorphism of pro-groups.
Proof. We are going to show that the product map(Čn )(1 + π n A0 hti) × (1 + π n A0 ht−1 i) → (1 + π n A0 ht, t−1 i)is surjective for all n > 0. The cokernel of Čn is the Čech cohomology group
H 1 (U, 1+π n O), where U is the standard covering of the formal π-adic scheme(P1A0 )∧ and O denotes its structure sheaf. Together with the vanishing of
H 1 (U, O/πO) ∼), see [Gro67, Prop. III.2.1.12], the long
= H 1 (P1A0 /(π) , OP1
A0 /(π)exact sequence of Čech cohomology corresponding to the exact sequence
0 → 1 + π n+1 O → 1 + π n O → O/πO → 0
implies that the map
coker(Čn+1 ) → coker(Čn )is surjective. A simple convergence argument, as in the final part of the proofof [KST16, Prop. 18], shows however that the limn coker(Čn ) vanishes. In analogy with [Gru68, p. 77] we define a canonical homomorphisms : K0 (A) → K1 (C),t[P ] 7→ [P ⊗A C →− P ⊗A C].

Here and in the following we let C = Aht, t−1 i.
Proposition 4.11. For any Tate ring A the map s induces an isomorphism∼s̄ : K0 (A) −→ ΣK1 (A).
Proof. Gruson constructs a left inverse r to s̄, see [Gru68, Sec. IV.4]. Thismap r has the following geometric description: The adic space (P1A )an has theopen covering U = {Sp Ahti, Sp Aht−1 i}. The set of analytic vector bundles
up to isomorphism Vec((P1A )an ) is in bijection with the set of algebraic vector
bundles up to isomorphism Vec(P1A ) by GAGA [Hub96].
Consider an element ξ ∈ ΣK1 (A) induced by a matrix M ∈ GLn (C). One
defines r(ξ) as the composition of the maps
GLn (C) → H 1 (U, GLn ) → Vec((P1 )an ) ∼= Vec(P1 ) →A(p∗ ,([O(1)]−[O(0)])·p∗ )

A

pr

2
K0 (P1A ) ←−−−−−−−−−−−−−− K0 (A) × K0 (A) −−→
K0 (A),'

where pr2 is the projection to the second component.
In order to finish the proof of Proposition 4.11 we thus have to show that s̄
is surjective. The argument is parallel to the proof of [Gru68, Prop. IV.6]. We
start with an element ξ ∈ ΣK1 (A) represented by a matrix M ∈ GLn (C).
Using the density of Laurent polynomials in C and Lemma 4.10, we can

18

MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME

assume without loss of generality that the entries of M lie in A[t, t−1 ]. For
any integer N > 0 the matrix tN M represents the element ξ + s̄(AnN ) ∈ΣK1 (A). So after making this replacement, we can assume without loss ofgenerality that the entries of M lie in A[t].
As explained in parts c. and d. of the proof of [Gru68, Prop. IV.6] we
can furthermore assume without loss of generality that M = b0 + b1 t withb0 , b1 ∈ M(n × n; A) and b0 + b1 = 1.
By [Gru68, Lem. IV.6] there exists a decomposition of A-modules An =P0 ⊕ Q0 , which is stable under the action of b0 and b1 and such that b1 is
topologically nilpotent on P0 and b0 is topologically nilpotent on Q0 . Let
P = P0 ⊗A C, Q = Q0 ⊗A C and let M 0 be the matrix which acts by M
on P and by t−1 M on Q. Then the element induced by M 0 |P in K1 (C)comes from K1 (Ahti) and the element induced by M 0 |Q in K1 (C) comesfrom K1 (Aht−1 i), so M 0 represents zero in ΣK1 (A). On the other hand M 0
represents the element ξ − s̄([Q0 ]) ∈ ΣK1 (A).

Remark 4.12. The non-archimedean negative K-groups Σi K0 (A) are studtop
ied by Calvo [Cal85], she denotes them K−i(A). Karoubi established a result analogous to Proposition 4.11 using a different convergence condition
in [Kar71].
4.4. Analytic Bass delooping. In this subsection we study an abstract
spectral version of the analytic Bass construction which we used in Proposition 4.11. It is characterized by a universal property (see Lemma 4.15).
Consider a functor E from adic rings or from Tate rings to Pro(Sp). We
defineΓE(A) = E(Ahti) tE(A) E(Aht−1 i)andΛE(A) := fib(ΓE(A) → E(Aht, t−1 i)).
Similarly, for a functor F from adic rings or from Tate rings to an abeliancategory we writeΓF (A) := coker F (A) → F (Ahti) ⊕ F (Aht−1 i) .
For E as above we then have
πi (ΓE(A)) ∼= Γ(πi E)(A)and a long exact sequence(4.2)· · · → Γ(πi+1 E)(A) → πi+1 (E(Aht, t−1 i)) → πi (ΛE(A)) → Γ(πi E)(A) → . . .
Connective algebraic K-theory refines to a functor from commutative rings
to E∞ -ring spectra, and we can consider functors E as above which carry a
K-module structure. Fix a map from the sphere spectrum to ΩK(Z[t, t−1 ])representing the class of t in K1 (Z[t, t−1 ]). If E carries a K-module structure,K -THEORY OF NON-ARCHIMEDEAN RINGS I

19

we use that map to construct a map λ : E → ΛE given on an adic or Tatering A by the composition`t∂

λ : E(A) −−→ ΩE(Aht, t−1 i) −→ ΛE(A),
where the second map comes from the definition of ΛE(A) as a fibre.
Definition 4.13. We write M for the ∞-category of functors from adic or
Tate rings to Pro(Sp) equipped with a K-module structure and MB ⊆ M
for the full subcategory spanned by those functors E for which λ : E → ΛE
is an equivalence.
Note that the ∞-category MB does not depend on the particular map
representing t, since a different choice yields a homotopic λ. The functors in
MB satisfy an analytic version of the Bass fundamental theorem:
Proposition 4.14 (Bass fundamental theorem). For E ∈ MB we have exact
sequences`t0/ Ei (A)/ Ei (Ahti) ⊕ Ei (Aht−1 i)z/ Ei (Aht, t−1 i)/ Ei−1 (A)/0where the split is induced by cup-product with t and, as usual, Ei (A) =πi (E(A)), etc. In particular, there is an isomorphism λ : Ei (A) → ΣEi+1 (A)of pro-groups.
Here Σ is as defined in Subsection 4.3.
Proof. Since the map A → Ahti is split, we get short exact sequences(4.3)

0 → Ei (A) → Ei (Ahti) ⊕ Ei (Aht−1 i) → ΓEi (A) → 0.

On the other hand, we have a long exact sequence∂· · · → ΛEi (A) → ΓEi (A) → Ei (Aht, t−1 i) −→ ΛEi−1 (A) → . . .
`t∂Since E ∈ MB , the composition Ei−1 (A) −−→ Ei (Aht, t−1 i) −→ ΛEi−1 (A) is
an isomorphism. Hence the long exact sequence yields short exact sequences
(4.4)

∂

0 → ΓEi (A) → Ei (Aht, t−1 i) −→ Ei−1 (A) → 0

split by ` t. Splicing the exact sequences (4.3) and (4.4) together gives theresult.

Lemma 4.15. The inclusion MB ⊆ M admits a left adjoint which we denoteby E 7→ E B . Explicitly, it can be described by
(4.5)

λ

Λ(λ)

E B (A) ' colim(E(A) −→ ΛE(A) −−−→ Λ2 E(A) → . . . ).

Proof. Define a functor (−)B : M → MB by (4.5). It comes with an obvious
natural transformation α : id → (−)B . By construction, the map λ : E B →ΛE B , which is induced by the system of maps λ : Λk E → Λk+1 E, is anequivalence. Hence E B ∈ MB . Similarly, both maps αE B , (αE )B : E B →(E B )B are equivalences. The recognition principle [Lur09, Prop. 5.2.7.4]
implies that (−)B is indeed a left adjoint of the inclusion MB ⊆ M.
20

MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME

In Section 6 we will use the following simple observation.
Lemma 4.16. Assume that there is an integer N such that Ei (A) = 0 for
every i > N and every adic resp. Tate ring A and that for any A the prospectrum E(A) is constant, i.e. is given by an ordinary spectrum. ThenE B (A) ' 0 for every such A.
Proof. From the long exact sequence (4.2) we deduce that also ΛEi (A) = 0
for i > N and every A. Since E has values in spectra, we deduce from (4.5)
that
EiB (A) ' colim Λk Ei (A) = 0
kfor i > N and every A. But by Proposition 4.14 the spectrum EiB (A) is a
B (Aht, t−1 i). By a downwards induction we conclude
direct summand of Ei+1
B
that Ei (A) = 0 for all i ∈ Z.

5. Continuous K-theory
5.1. Construction. Let A0 be a complete π-adic ring for some π ∈ A0 .
Definition 5.1. Continuous K-theory of A0 is defined as the pro-spectrum
K cont (A0 ) = “ lim ” K(A0 /(π n )).
n

Clearly, K cont (A0 ) is independent of the choice of π up to canonical equivalence of pro-spectra.
Lemma 5.2. We have
” K1 (A0 )/(1 + π n A0 ),“ limnKicont (A0 ) = K0 (A0 ) = K0 (A0 /(π)),

Ki (A0 /(π)),

for i = 1,
for i = 0,
for i < 0.

Proof. The first line is [Wei13, Lem. III.2.4], the second and third lines area consequence of [Wei13, Lemma II.2.2].
For a topological ring R whose topology is generated by the powers ofsome ideal I ⊂ R we write(5.1)

K̄(R) = K(R on I)

for the non-connective K-theory spectrum with support on I, see [TT90,
Def. 6.4]. By k̄(R) = τ≥0 K̄(R) we denote the connective cover of K̄(R).
Let now A be a Tate ring with a ring of definition A0 ⊂ A and let π ∈ A0
be a topologically nilpotent unit in A.
Definition 5.3. Continuous K-theory of A (with respect to the ring of
definition A0 ) is defined as the pro-spectrum
K cont (A; A0 ) = cofib(K̄(A0 ) → K cont (A0 )).
We also set
K cont (A) = colim K cont (A; A0 ),A0K -THEORY OF NON-ARCHIMEDEAN RINGS I

21

where the colimit is over the partially ordered set of rings of definition A0 ⊂A.
Note that for a fixed Tate ring A the partially ordered set of rings ofdefinition A0 ⊂ A is directed, see Remark 3.3(ii).
We are going to show in Proposition 5.4, following [Mor16], that
K cont (A; A0 ) ' K cont (A)depends only on A up to weak equivalence. For this we need some preparations. By the localization theorem [TT90, Thm. 7.4] we have a fibre sequenceof spectraK̄(A0 ) → K(A0 ) → K(A),(5.2)

so there exists a canonical morphism K(A) → K cont (A). Moreover, thepro-spectrum K cont (A; A0 ) is part of a cartesian square(5.3)

K(A0 )/ K(A)


/ K cont (A; A0 ).

K cont (A0 )Proposition 5.4. For a Tate ring A with ring of definition A0 the map∼
K cont (A; A0 ) −→ K cont (A) is a weak equivalence.
Proof. By Remark 3.3 it is sufficient to show that for A0 ⊂ A00 two rings ofdefinition of A the map K cont (A; A0 ) → K cont (A; A00 ) is a weak equivalenceof pro-spectra. For this consider the commutative cube of pro-spectra(5.4)/ K(A)

K(A0 )'/ K(A)

K(A00 )




/ K cont (A; A0 )K cont (A0 )
'

α




)
/ K cont (A; A0 ).

K cont (A00 )0

The back square is the same as the cartesian square (5.3) and the front squareis the corresponding cartesian square with A0 replaced by A00 . The left squareis weakly cartesian by Remark 4.9(i). So the right square is weakly cartesianas well or in other words α is a weak equivalence of pro-spectra.

In Subsection 4.3 we gave another definition of the pro-group K1cont (A)for a Tate ring A. In fact both definitions are compatible.
Lemma 5.5. The natural map of pro-groups'“ lim ” K1 (A)/(1 + π n A0 ) −→ K1cont (A)nis an isomorphism.

22

MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME

Proof. This follows from the five lemma and the commutative diagram withexact columns
K̄1 (A0 )K̄1 (A0 )“ lim ” K1 (A0 )/(1 + π n A0 )n“ lim ” K1 (A)/(1 + π n A0 )n'
/ K cont (A0 )1/ K cont (A)1K̄0 (A0 )K̄0 (A0 )

K0 (A0 )'/ K0 (A0 /(π))

Here the second and the fifth horizontal arrows are isomorphisms by Lemma 5.2.
An argument parallel to the proof of Lemma 5.5, using Theorems 2.5
and 2.13 from [Ste73], shows:
Lemma 5.6. For a Tate ring which has a local ring of definition A0 ⊂ A
the map'→ K2cont (A)“ lim ” K2 (A)/{1 + π n A0 , A×0}−n

is an isomorphism of pro-groups.
Example 5.7. Assume that the Tate ring A admits a ring of definition A0
which is noetherian and regular. Then K̄i (A0 ) = 0 for i < 0. By definitionof K cont (A) and Lemma 5.2 we thus get isomorphisms'

Ki (A0 /(π)) −→ Kicont (A)for i < 0.

In particular, the negative continuous K-groups need not vanish even if A
admits a regular ring of definition.
A concrete example is the following. Let κ be a discretely valued field withring of integers κ0 and uniformizer π. Set A0 = κ0 hx, yi/(y 2 − x3 + x2 − π)and A = A0 [1/π], a thickened version of a nodal curve. Then A0 is regular,cont (A) ∼ K (A /(π)) ∼ Z (see [Wei13, Exc. III.4.12]).
but K−1= −1 0=Continuous K-theory of a Tate ring A can also be calculated in termsof a Raynaud type geometric model of A. By this we mean the following:
Assume that there exists a noetherian ring of definition A0 ⊂ A and let
π ∈ A0 be a topologically nilpotent unit in A. Assume given an “admissible
blow-up” or more precisely a proper morphism p : X → Spec(A0 ) which is an

K -THEORY OF NON-ARCHIMEDEAN RINGS I

23

isomorphism over Spec(A). For n > 0 set Xn = X ⊗A0 A0 /(π n ) and define
continuous K-theory of X as the pro-spectrum
K cont (X) = “ lim ” K(Xn ).
n

Proposition 5.8. There is a weak fibre sequence of pro-spectra.
K̄(X) → K cont (X) → K cont (A).
Here K̄(X) stands for the algebraic K-theory spectrum of X with supportin X1 .
Proof. The proof is very similar to the proof of Proposition 5.4. Let for the
moment W be the cofibre of K̄(X) → K cont (X). Then we get a commutative
cube of pro-spectra(5.5)/ K(A)

K(A0 )'/ K(A)K(X)/ K cont (A; A0 )K cont (A0 )'αK cont (X)

' 
/ W.

The back square is the same as the cartesian square (5.3). The front squareis cartesian by the same reasoning. The left square is weakly cartesianby [KST18, Thm. A]. So the right square is weakly cartesian as well orin other words α is a weak equivalence of pro-spectra.

Remark 5.9. Proposition 5.8 can be used to show that continuous K-theory
satisfies Mayer–Vietories for finite affinoid coverings of affinoid spaces as
indicated in [Mor16].
5.2. Bass fundamental theorem. Let A be a Tate ring with a ring of
definition A0 and let π ∈ A0 be a topologically nilpotent unit in A. As bothK̄(A0 ) and K cont (A0 ) carry a module structure over K(A0 ) and K cont (A)carries a compatible module structure over K(A) the techniques of Subsection 4.4 apply. In particular we obtain a commutative diagram of fibre
sequencesK̄(A0 )λ1ΛK̄(A0 )/ K cont (A0 )λ2/ ΛK cont (A0 )/ K cont (A; A0 )λ3/ ΛK cont (A; A0 )in which the vertical maps are equivalences of pro-spectra. Indeed, in orderto show that λ1 and λ2 are equivalences one can replace in the definitionof Λ the rings A0 ht± i and A0 ht, t−1 i by their dense subrings A0 [t± ] andA0 [t, t−1 ] in view of [TT90, Prop. 3.19], so these equivalences λ1 and λ2

24

MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME

are a consequence of [TT90, Thm. 6.6]. This means the canonical map
K cont (A) → (K cont )B (A) is an equivalence, so we deduce that the Bass
fundamental theorem holds for K cont , see Proposition 4.14.
Proposition 5.10. Continuous K-theory of a Tate ring A satisfies:
cont (A) is an isomorphism of pro-groups.
(i) λ : Kicont (A) → ΣKi+1(ii) The canonical map K0 (A) → K0cont (A) is an isomorphism of progroups.
(iii) The pro-groups Kicont (A) are constant for i ≤ 0 and agree with thegroups Kitop (A) defined and studied in [Cal85].
Proof. Part (i) is clear from what is explained above. Part (ii) follows frompart (i), Lemma 4.10 and Proposition 4.11. Part (iii) is clear in view of part(ii) and the Bass fundamental theorem for K cont .
Remark 5.11. Combining Proposition 5.10(ii) and Lemma 5.5 with theexact sequence in Ab
0 → lim1 K1cont (A) → π0 lim K cont (A) → lim0 K0cont (A) → 0,'

we deduce that K0 (A) −→ π0 lim K cont (A) is an isomorphism. Here lim0
and lim1 are the usual limit functors from the category Pro(Ab) to Ab andlim : Pro(Sp) → Sp is the right adjoint to the inclusion.
Using Gabber’s rigidity theorem we can compute continuous K-theoryof Tate rings with finite coefficients. Let ` be a positive integer. If F is afunctor from adic rings or from Tate rings to spectra or pro-spectra, we write
F (A; Z/`) for the (level-wise) smash product of F (A) with the mod-` Moore
spectrum S/`.
Let A be a Tate ring with a ring of definition A0 , and let ` be a positive
integer which is invertible in A0 .
Lemma 5.12. The canonical map K(A) → K cont (A) induces isomorphisms''
Ki (A; Z/`) −→ Kicont (A; Z/`) and K0 (A)/` −→ K0cont (A)/` for i ≥ 1.
Proof. The statement about K0 follows directly from Proposition 5.10(ii).
Since smashing with S/` is an exact functor, the cartesian square (5.3) and
Proposition 5.4 yield the weakly cartesian square
K(A0 ; Z/`)

/ K(A; Z/`)
/ K cont (A; Z/`)

K cont (A0 ; Z/`)

Since A0 is π-adically complete (where π generates the topology on A0 ),Gabber’s rigidity theorem [Gab92, Thm. 1] implies that the left vertical
map induces an isomorphism on πi for every i > 0. This implies that
Ki (A; Z/`) → Kicont (A; Z/`) is an isomorphism for i > 1 and injective for
i = 1. In order to show that it is also surjective for i = 1, we consider theK -THEORY OF NON-ARCHIMEDEAN RINGS I

25

following diagram in which the exact rows come from the Bockstein sequenceand [`] denotes the `-torsion subgroup:0/ K1 (A)/`/ K1 (A; Z/`)

/ K0 (A)[`]

/0

0


/ K cont (A)/`/ K cont (A; Z/`)
/ K cont (A)[`]

/0

1

1

0

The right vertical map is an isomorphism by Proposition 5.10(ii). The left
vertical map is surjective by Lemma 5.5. Hence the middle vertical map issurjective, too.

5.3. Analytic pro-homotopy invariance. Let A be an adic ring resp. a
Tate ring and let π ∈ A be an element generating the topology on A resp.
a topologically nilpotent unit. Let CA be the comma category of adic rings
resp. Tate rings over A. In the following we fix an integer m > 0 and wewrite Ahti = Aht1 , . . . , tm i. For a functor F from CA to Pro(Sp) we define
π F on the same categories by
the new functor N̂ π F = N̂mN̂ π F (A) = cofib(F (A) → “ lim ” F (Ahti)).
t7→πtRecall that the polynomial version N π of N̂ π has been studied in Subsection 4.2.
Let A be a Tate ring, A0 ⊂ A be a ring of definition and π ∈ A0 anelement which is a topologically nilpotent unit in A.
Lemma 5.13. We have:(i) N̂ π K cont (A0 ) is a contractible pro-spectrum.
If moreover A is regular and satisfies condition (†)A we have:(ii) N̂ π K̄(A0 ) is a weakly contractible pro-spectrum.
(iii) There is a natural weak equivalenceN̂ π K(A) ' cofib K(A0 ) → “ lim ” K(A0 [[t]]) .
t7→πt(iv) In the case of one variable, i.e. m = 1, there is a natural isomorphismof pro-abelian groups
π1 (N̂ π K(A)) ∼= “ lim ” W (A0 ),[π]

where W (A0 ) = (1 + tA0 [[t]])× is the ring of big Witt vectors and theright hand side denotes the pro-system
W (A0 ) o[π]

W (A0 ) o[π]with [π] = (1 − πt)−1 ∈ W (A0 ).

W (A0 ) o

[π]

···

26

MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME

Proof. Part (i) is clear as for fixed j > 0 the map
A0 /(π j ) → “ lim ” Ahti/(π j )t7→πt

is an isomorphism of pro-rings. For (ii) we use that the canonical map'
N π K̄(A0 ) −→ N̂ π K̄(A0 ) is an isomorphism by excision [TT90, Thm. 7.1],
where the left side is as defined in Subsection 4.2. From localization theory [TT90, Thm. 7.4] we get a fibre sequence(5.6)

N π K̄(A0 ) → N π K(A0 ) → N π K(A).

The right pro-spectrum in (5.6) is contractible as A is regular and K-theoryof regular rings is A1 -homotopy invariant [Wei13, Thm. V.6.3]. Up to weakequivalence the pro-spectrum in the middle of (5.6) is independent of the
choice of the ring of definition A0 by Remark 4.9. So we can assume bycondition (†)A that A0 is noetherian and that there exists a proper morphismX̃ → Spec(A0 ) which is an isomorphism over Spec(A) and such that X̃ isregular. Corollary 4.7 finally implies that N π K(A0 ) is weakly contractible.
Together this implies that also the pro-spectrum on the left of (5.6) is weakly
contractible proving (ii). For (iii), note that the natural map
N̂ π K(A0 ) → N̂ π K(A)is a weak equivalence by (ii). From the commutative diagram
A0 htit→πt

z

A0 hti/ A0 [[t]]
t7→πt



t7→πt

/ A0 [[t]]we obtain an isomorphism of pro-simplicial rings“ lim ” A0 hti ' “ lim ” A0 [[t]].
t7→πt

t7→πt

This proves (iii). Part (iv) follows from part (iii) and the isomorphism
K1 (A0 [[t]])/K1 (A0 ) ' (1 + tA0 [[t]])× ,see [Wei13, Lem. III.2.4].
From the fibre sequenceN̂ π K̄(A0 ) → N̂ π K cont (A0 ) → N̂ π K cont (A)and Lemma 5.13 we immediately deduce:

Proposition 5.14. For a regular Tate ring A which satisfies condition (†)Athe canonical map'

K cont (A) −→ “ lim ” K cont (Ahti)
t7→πt

is a weak equivalence of pro-spectra.

K -THEORY OF NON-ARCHIMEDEAN RINGS I

27

As a corollary to Proposition 5.14 and Proposition 5.10(ii) we recover thefollowing pro-homotopy invariance result for K0 , which has already beenshown in [KST16] using a more direct approach.
Corollary 5.15. For a regular Tate ring A which satisfies condition (†)Athe canonical map'
K0 (A) −→ “ lim ” K0 (Ahti)
t7→πt

is an isomorphism of pro-groups.
For K1 we at least have the following weak pro-homotopy invariance result:
Proposition 5.16. If A is regular and satisfies condition (†)A , thenRi lim π1 N̂ π K(A) = 0

for i ≥ 0.

t→πt

Proof. We have a commutative diagram with exact rows
/ W (A0 )0/ W (A0 )[π]/ W (A0 )0[π]n+1[π]n=/ W (A0 )/ W (A0 )/[π]n+1/0
/ W (A0 )/[π]n/0.

From this we get\ker(W (A0 ) → W (A0 )[π] ),\Ri lim W (A0 ) ∼= coker(W (A0 ) → W(A0 )[π] ),
t→πt


0,

for i = 0,
for i = 1,otherwise,\
where W
(A0 )[π] is the [π]-adic completion of W (A0 ). Hence the assertion
follows from Lemma 5.13 and the following lemma.
Lemma 5.17. The ring W (A0 ) is [π]-adically complete.
Proof. We havef ∈ [π]n · W (A0 ) for f = 1 +Xai ti ∈ W (A0 )i≥1

if and only if there are ci ∈ A0 , i ≥ 1, such thatai = π ni ci for i ≥ 1.
The lemma follows from this by direct computations.
6. Analytic K-theory

6.1. Basic constructions. Let A be a π-adic ring for some π ∈ A. In orderto define analytic K-theory we consider the system of simplicial rings
(6.1)

···

/ Ah∆ 3 i
π

Ψπ
π

2

/ Ah∆ 2 i
π

Ψπ
π

1

/ Ah∆ 1 i
π

Ψπ
π

0

/ Ah∆1 i,

28

MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME

see Subsection 3.1. We would like to see that the pro-system of simplicial
rings (6.1) does not depend on the choice of π up to canonical isomorphism.
Indeed, let $ ∈ A be another element generating the topology of A andconsider the analog of the pro-system (6.1) with π replaced by $. We have$n = aπ and that π n = b$ for some positive integer n and a, b ∈ A. In
view of the commutative squarenj

Ah∆$n(j+1) i
j+1
Ψπj+1
a

Ψ$
$n

/ Ah∆

$nj i
j



Ah∆πj+1 i

Ψπ
π

Ψπj


/ Ah∆ j iajπ

and the analogous square with π and $ interchanged and a replaced byb we get a canonical isomorphism of pro-simplicial rings Φπ$ such that the
diagramΦπ$“ lim ” Ah∆πj i/ “ lim ” Ah∆

'

j

&

j

Ah∆1 i$j iwcommutes.
If A is a Tate ring and π ∈ A is a topologically nilpotent unit, one similarly
constructs a pro-simplicial ring “ limj”Ah∆πj i and one shows that it does notdepend on the choice of π up to canonical isomorphism.
In the remainder of this subsection we assume that A is either an adic
ring or a Tate ring. Let F be a functor from the category of adic rings orfrom the category of Tate rings to connective spectra. We set
F (j) (A) = F (Ah∆πj i)and define a new functor F an to the ∞-category of pro-spectra by
F an (A) = “ lim ” F (j) (A).
jHere F (Ah∆πj i) denotes the geometric realization of the simplicial spectrum[p] 7→ F (Ah∆pπj i).
Lemma 6.1. For any A and any functor F as above, there is a convergent
spectral sequence of pro-abelian groups
1
Epq= “ lim ” πq F (Ah∆pπj i) =⇒ πp+q F an (A)jwhich yields an isomorphism of pro-groups∂ −∂1
π0 F an (A) = “ lim ” coker(π0 F (Ah∆1πj i) −−0−−→π0 F (A)).

j

Proof. For fixed j we have the classical convergent E 1 -spectral sequence ofa simplicial spectrum (see e.g. [Lur17a, Prop. 1.2.4.5])
πq F (Ah∆pπj i) =⇒ πp+q F (j) (A).

K -THEORY OF NON-ARCHIMEDEAN RINGS I

29

Since F has valued in connective spectra, these are bounded uniformly in j.
They hence yield the desired convergent spectral sequence in the abeliancategory Pro(Ab). The asserted isomorphism is an immediate consequence.

Lemma 6.2. For fixed j, the maps of simplicial rings Ahtih∆πj i → Ahtih∆πj iinduced by t 7→ π j t and t 7→ 0, respectively, are simplicially homotopic.
Proof. For fixed n and 0 ≤ ` ≤ n we have ring homomorphismsh` : Ahtih∆nπj i → Ahtih∆n+1iπjinduced by h` (t) = t(t0 + · · · + t` ) and h` (ti ) = s` (ti ), where s` is the `th degeneracy map and the ti are the coordinates on ∆nπj . These give thedesired homotopy.
Given variables t = (t1 , . . . , tm ), we letπN̂m
F (A) = cofib(F (A) → “ lim ” F (Ahti))
t7→πt

as in Subsection 5.3.
Proposition 6.3. For each j, the natural map
F (j) (A) → “ lim ” F (j) (Ahti)
t7→πt

π F an (A)is an equivalence of pro-spectra. In particular, the pro-spectrum N̂mis contractible for any m ≥ 0.

Proof. The map Ahti → A, t 7→ 0, induces a left inverse. As any functor
preserves simplicial homotopies, Lemma 6.2 implies that each j-fold composition of transition maps F (j) (Ahti) → F (j) (Ahti) factors through F (j) (A)up to homotopy. This implies the first claim. Taking “ lim ” over j implies
that'
F an (A) −→ “ lim ” F an (Ahti)
t7→πt

is also an equivalence. By induction on m this gives the second assertion. 
If G → F is the connected covering of F , we consider the fibre sequence
G → F → F0
applied to the pro-simplicial ring “ limj”Ah∆πj i, where F0 = H(π0 F ) denotes
the Eilenberg–MacLane spectrum living in homotopy degree 0.
Lemma 6.4. For an adic ring resp. Tate ring A there is a fibre sequence of
pro-spectraGan (A) → F an (A) → “ lim ” F0 (Ah∆πj i).
jπ F (A) is contractible for any number of variables m ≥ 0, then there is
If N̂m0a weak fibre sequence of pro-spectra
Gan → F an → F0 (A).

30

MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME

π F (A) is contractible if and only if the natural map F (A) →Note that N̂m00m“ limj”F0 (Ah∆πj i) is an equivalence.

Proof. For every j and m there is a fibre sequence of spectramm
G(Ah∆mπ j i) → F (Ah∆π j i) → F0 (Ah∆π j i).

Since fibre sequences of spectra are preserved by geometric realization, thisinduces a fibre sequence
G(j) (A) → F (j) (A) → F0 (Ah∆πj i)for every j. The first claim follows by taking “ lim ”over j.
For the second claim we use that the homotopy pro-groups of the prospectrum “ limj”F0 (Ah∆πj i) are computed as the homotopy pro-groups ofthe pro-simplicial abelian group “ limj”(π0 F )(Ah∆πj i).

6.2. Analytic K-theory of adic rings. We apply the construction of theprevious subsection to the connective K-theory of adic rings. Let A be anadic ring, and let π ∈ A generate the topology of A.
Definition 6.5. The analytic K-theory of A is defined as the pro-spectrum
k an (A) = “ lim ” k(Ah∆πj i).
j

By the discussion in Subsection 6.1 this is independent of the choice ofπ up to canonical equivalence. In the following, we also use the connective
covering
k cont (A) = “ lim ” k(A/(π n ))ncontof the continuous K-theory K(A) of A. As a bridge between continuousand analytic K-theory we further define the ‘mixed’ version
k cont,an = “ lim ” k(Ah∆πj i/(π n )).
j,nLemma 6.6. The natural map'

k cont (A) −→ k cont,an (A)is an equivalence of pro-spectra.
Proof. For fixed n the map A/(π n ) → “ limj”Ah∆πj i/(π n ) is an isomorphismof pro-simplicial rings. Hence it induces an equivalence of pro-spectra'k(A/(π n )) −→ “ lim ” k(Ah∆πj i/(π n ))j

The claim follows by taking the limit over n.
Proposition 6.7. The map
k an (A) → k cont,an (A)is a weak equivalence of pro-spectra.

K -THEORY OF NON-ARCHIMEDEAN RINGS I

31

Proof. We first reduce to connected K-theory. For fixed j, n, and m wehave, as in the discussion preceding Lemma 6.4, a map of fibre sequences ofspectra/ k(Ah∆mj i)

K GL (Ah∆mi)πj/ K0 (Ah∆mj i)

π



π

∼
=


/ k(Ah∆mj i/(π n ))K GL (Ah∆mi/(π n ))πj/ K0 (Ah∆mj i/(π n )).

ππ

The right vertical map is an isomorphism because Ah∆m
i is π-adically comπj
plete, see [Wei13, Lemma II.2.2].
It is now enough to show that for each j the map
K GL (Ah∆πj i) → “ lim ” K GL (Ah∆πj i/(π n ))n

is a weak equivalence of pro-spectra. We check this on the associated infinite
loop spaces. By Lemma 4.2 this means we have to show that the bottom
horizontal map in the commutative diagram
BGL(Ah∆πj i)

BGL(Ah∆πj i)+φ/ “ lim ” BGL(Ah∆ j i/(π n ))
π

n

φ+


/ “ lim ” BGL(Ah∆ j i/(π n ))+πnis a weak equivalence in Pro(Spc0∗ ) (see Subsection 2.3). Since the map to
the plus-construction is acyclic, the vertical maps induce isomorphisms on
integral homology.
Claim 6.8. The map φ is a weak equivalence in Pro(Spc0∗ ).
By Lemma 2.10 this claim implies that φ+ induces an isomorphism on
integral homology. Since φ+ is a morphism between countable pro-systems ofconnected H-spaces, Proposition 2.11 implies that φ+ is a weak equivalence.
Proof of the Claim. We have to show that φ induces an isomorphism on homotopy pro-groups. For this we use that
πi (BGL(Ah∆ j i)) ∼= πi−1 (GL(Ah∆ j i)) for i ≥ 1.
ππ

For each n, the natural map
GL(Ah∆πj i) → GL(Ah∆πj i/(π n ))is a surjective homomorphism of simplicial groups and hence a Kan fibration.
Its fibre is given by the simplicial group of finite matrices of arbitrary size
1 + π n M(Ah∆πj i)
and it suffices to show that the inclusion
1 + π n+j M(Ah∆πj i) ,→ 1 + π n M(Ah∆πj i)

32

MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME

is null-homotopic. We let ∆[1] be the simplicial 1-simplex, and we consider
t1 ∈ Ah∆1πj i as a map of simplicial sets
t1 : ∆[1] → Ah∆πj isatisfying t1 (0) = 0, t1 (1) = π j . A contracting homotopy is then given by
1 + π n+j M(Ah∆πj i) × ∆[1] → 1 + π n M(Ah∆πj i),(1 + a, s) 7→ 1 + t(s) · (π −j a).

'

Since by Lemma 6.6 the map of pro-spectra k cont (A) −→ k cont,an (A) is anequivalence, it has an essentially unique inverse. Composing this with the
weak equivalence k an (A) → k cont,an (A) of Proposition 6.7 gives an essentiallyunique weak equivalence of pro-spectra k an (A) → k cont (A) which makes thediagram/ k cont (A)'k an (A)
'

&

w

'

k cont,an (A)commutative. Summarizing we have:
Theorem 6.9. For any π-adic ring A, there is a natural weak equivalenceof pro-spectra'

k an (A) −→ k cont (A).
Remark 6.10. In fact, this result holds more generally. We needed thecompleteness of a π-adic ring only in the proof of Proposition 6.7. Moreprecisely, we used that for a π-adically complete ring A the map K0 (A) →K0 (A/(π)) is an isomorphism and GL(A) → GL(A/(π)) is surjective withkernel 1 + πM(A). But these two facts also hold if A is only assumed to behenselian with respect to (π). Hence we get the same result if we work withhenselian rings and replace the simplicial ring Ah∆πj i in the construction ofanalytic K-theory with the henselisation of A[∆πj ] along the ideal (π).
6.3. Analytic K-theory of Tate rings. Let now A be a Tate ring andπ ∈ A a topologically nilpotent unit.
Definition 6.11. The connective analytic K-theory of A is defined as the
pro-spectrum
k an (A) = “ lim ” k(Ah∆πj i).
jAgain, this is well-defined up to canonical equivalence. As in the previous
subsection we will compare analytic K-theory and continuous K-theory. For
this we need some auxiliary constructions. If A0 ⊆ A is a ring of definition

K -THEORY OF NON-ARCHIMEDEAN RINGS I

33

with π ∈ A0 , we setk̄ an (A0 ) = “ lim ” k̄(A0 h∆πj i),jk̃(A; A0 ) = cofib(k̄(A0 ) → k(A0 )),k̃ cont (A; A0 ) = cofib(k̄(A0 ) → k cont (A0 )),k̃ an (A; A0 ) = cofib(k̄ an (A0 ) → k an (A0 )),k̃ cont,an (A; A0 ) = cofib(k̄ an (A0 ) → k cont,an (A0 ))(see (5.1) for the definition of k̄). Note that the above spectra are all connective and that τ≥1 k̃(A; A0 ) ' τ≥1 k(A) ' K GL (A). There is a natural
map k̃ cont (A; A0 ) → K cont (A; A0 ). It is immediate from the constructions
that this map induces an equivalence on connected covers τ≥1 k̃ cont (A; A0 ) →τ≥1 K cont (A; A0 ). In view of Proposition 5.4 we have a natural weak equivalence'τ≥1 k̃ cont (A; A0 ) −→ τ≥1 K cont (A).
Recall condition (†)A from subsection 3.2 (page 11).
Lemma 6.12. Assume that A is regular and satisfies condition (†)A . Then
there is a natural zig-zag of weak equivalences of pro-spectra''k̃ cont (A; A0 ) −→ k̃ cont,an (A; A0 ) ←− k̃ an (A; A0 ).
Proof. Lemma 5.13(ii) and the convergent spectral sequence from Lemma 6.1
1
an
Epq= “ lim ” k̄q (A0 h∆pπj i) =⇒ k̄p+q(A0 )j

imply that the natural map k̄(A0 ) → k̄ an (A0 ) is a weak equivalence. Combining this with Lemma 6.6 and the definitions of the tilde versions yields
the left weak equivalence of the assertion. The right-hand weak equivalence
follows from Proposition 6.7.
Lemma 6.13. Let A, A0 be as above.
(i) There is a natural isomorphism k̃0 (A; A0 ) ∼= k̃0an (A; A0 ).
(ii) If A is regular and satisfies condition (†)A , then we have a natural
isomorphism
K0 (A) ∼= k0an (A)and a natural weak equivalenceτ≥1 k̃ an (A; A0 ) ' τ≥1 k an (A).
Proof. Recall that k and k̃ have common connected cover K GL . For fixed m
we have a commutative diagram of pro-groups
K0 (A0 )

k̃0 (A; A0 )∼=/ “ lim ” K0 (A0 h∆mj i)πj
/ “ lim ” k̃0 (Ah∆mj i, A0 h∆mj i).
j

π

π

34

MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME

The top horizontal map is an isomorphism by Lemmas 5.2 and 5.13(i). The
vertical maps are epimorphisms since they are level-wise surjective by definition of k̃. Hence the bottom horizontal map is an epimorphism, too. Since
it has a left inverse, it is an isomorphism. Now Lemma 6.4 implies that wehave a weak fibre sequence(6.2)

K GL,an (A) → k̃ an (A; A0 ) → k̃0 (A; A0 ).

Since K GL,an (A) is connected, this gives (i).
Now assume that A satisfies (†)A . By Corollary 5.15 the map K0 (A) →“ limj”K0 (Ah∆mi) is an isomorphism for each m. By Lemma 6.4 again, we
πjhave a weak fibre sequence(6.3)

K GL,an (A) → k an (A) → K0 (A).

This gives the first statement of (ii).
There is a natural map from (6.2) to (6.3). By what we have shown so'far, it induces a weak equivalence τ≥1 k̃ an (A; A0 ) −→ τ≥1 k an (A).

The continuous K-theory K cont (A) of A can be recovered from k̃ cont (A; A0 )via the Bass construction:
Lemma 6.14. There is a natural equivalence of pro-spectra'(k̃ cont )B (A; A0 ) −→ K cont (A; A0 ).
Proof. Since K cont (A; A0 ) (viewed as functor in A0 ) satisfies the Bass fundamental theorem (see first paragraph of Subsection 5.2), the natural map
K cont (A; A0 ) → (K cont )B (A; A0 ) is an equivalence and we have an inducedmap (k̃ cont )B (A; A0 ) → K cont (A; A0 ). The cofibre cofib(k̃ cont (A; A0 ) →K cont (A; A0 )) is equivalent to cofib(τ≤−1 k̄ B (A0 ) → τ≤−1 K cont (A0 )). By nilinvariance of negative algebraic K-theory of rings, the latter is a constantpro-spectrum, which is moreover 0-truncated. Now Lemma 4.16 implies thatthe Bass construction applied to the cofibre (again viewed as functor in A0 )vanishes, whence (k̃ cont )B (A; A0 ) → K cont (A; A0 ) is an equivalence.

Definition 6.15. The non-connective analytic K-theory of the Tate ring A
is defined by the analytic Bass construction applied to K GL,an :K an (A) = (K GL,an )B (A).
Remark 6.16. Another candidate for the non-connective analytic K-theory
would be the pro-spectrum (k an )B (A). If A is regular and satisfies condition(†)A , both constructions yield weakly equivalent pro-spectra by the lemma
below. In general, it seems however that K an (A) has better formal properties. For example, one can prove that it has descent for admissible coverings,
as we will explain in a sequel to this note.
Choose a ring of definition A0 ⊂ A.

K -THEORY OF NON-ARCHIMEDEAN RINGS I

35

Lemma 6.17.
(i) The natural map
K an (A) = (K GL,an )B (A) → (k̃ an )B (A; A0 )is a weak equivalence.
(ii) If A is regular and satisfies (†)A , then the natural map(k̃ an )B (A; A0 ) → (k an )B (A)is an equivalence.
Proof. Apply the analytic Bass construction to the weak fibre sequence (6.2).
Then Lemma 4.16 implies (i). In the situation of (ii), Lemma 6.13(ii) implies
that the cofibre of k̃ an (A; A0 ) → k an (A) is a constant pro-spectrum concentrated in homotopy degree 0. Lemma 4.16 again implies the assertion.

Theorem 6.18. If the Tate ring A is regular and satisfies (†)A , there is anatural zig-zag of weak equivalences''

K cont (A) −→ K cont,an (A) ←− K an (A).
Here K cont,an (A) is defined by first applying the Bass construction (4.5) tok̃ cont,an (A; A0 ) and then taking the colimit over the poset of rings of definition
A0 ⊂ A as in Definition 5.3.
Proof. Fix a ring of definition A0 ⊂ A. We apply the Bass construction(4.5) to the zig-zag of Lemma 6.12. By the previous Lemma K an (A) →(k̃ an )B (A; A0 ) is a weak equivalence of pro-spectra. Combining this with
Lemma 6.14 we get the natural zig-zag of weak equivalences''

K cont (A; A0 ) −→ (k̃ cont,an )B (A; A0 ) ←− K an (A).
Taking the colimit over the rings of definition A0 ⊂ A gives the result.



Corollary 6.19. Assume that A is regular and satisfies (†)A . Then the prospectra k an (A) and τ≥0 K an (A) are weakly equivalent. In particular, we have
an isomorphism of pro-groups
K0 (A) ∼= K an (A).
0

Proof. By Lemma 6.17 the canonical map K an (A) → (k an )B (A) is a weakequivalence and we claim that k an (A) → (k an )B (A) induces a weak equivalence on connective covers. We first consider connected covers. For thiswe choose a ring of definition A0 ⊂ A. By Theorem 6.18 the pro-spectraτ≥1 K an (A) and τ≥1 K cont (A) are weakly equivalent. The latter is clearly
equivalent to τ≥1 k̃ cont (A; A0 ), which is weakly equivalent to τ≥1 k̃ an (A; A0 )by Lemma 6.12. Lemma 6.13 finally gives the desired weak equivalence withτ≥1 k an (A). One checks that all the weak equivalences involved are compatible with the canonical maps K an (A) → (k an )B (A) ← k an (A). This gives
the claim on connected covers. For the π0 statement we look at the com∼ cont (A). The first map is an
position K0 (A) → k0an (A) → (k an )B
0 (A) = K0

36

MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME

isomorphism by Lemma 6.13, and the composition is an isomorphism by
Proposition 5.10. Hence the second map is an isomorphism, as desired. 
Remark 6.20. Let A be a Tate ring with ring of definition A0 and ` apositive integer which is invertible in A0 . A theorem of Weibel [TT90,
Thm. 9.5] says that K-theory of Z[1/`]-algebras with Z/`-coefficients is homotopy invariant. Using this one can show that the analogs of Lemmas 6.12,
6.13, and 6.17 with Z/`-coefficients are true without assuming that A isregular. Hence the pro-spectra K cont (A; Z/`) and K an (A; Z/`) are weakly
equivalent. Combining this with Lemma 5.12, we also get isomorphisms
Ki (A; Z/`) ∼= Kian (A; Z/`) for i > 0.
7. Analytic KV -theory
In this section we briefly explain the main points of the KV -analog of
our analytic K-theory. Here A = (A, k . k) is a complete normed ring as in
Subsection 3.3. In particular, A need not be commutative nor unital. If A is
e → GL(Z)) using the unitalization A.
e
not unital, we set GL(A) := ker(GL(A)As in Subsection 6.1 we get a pro-simplicial ring
“ lim ” Ah∆ρ i.
ρ

Definition 7.1. The analytic KV -theory of A is defined as the pro-simplicial
set
KV an (A) = “ lim ” BGL(Ah∆ρ i).
ρ

Its i-th homotopy pro-group is denoted by KVian (A).
We may also view KV an (A) as a pro-space, i.e. as an object of the ∞category Pro(Spc0∗ ). Directly from the definition we get:Lemma 7.2. There are natural isomorphisms
KV an (A) ∼= “ lim ” πi−1 GL(Ah∆ρ i).
i

ρ

If a = (aij ) is a matrix with coefficients in A, we set kak = max{kaij k}.
This defines a norm on the ring of matrices M(A).
Lemma 7.3. There is a natural isomorphism
KV an (A) ∼= “ lim ” GL(A)/ GL(A)ρ ,1

ρ

where GL(A)ρ is the subgroup of GL(A) generated by matrices g such thatlimn→∞ k(g − 1)n kρn = 0.
Proof. By the previous Lemma 7.2, it suffices to show that π0 GL(Ah∆ρ i) ∼=GL(A)/ GL(A)ρ . We can identify π0 GL(Ah∆ρ i) = GL(A)/G where G is
the subgroup consisting of matrices g such that there exists a matrix g(t) ∈GL(Ahtiρ ) with g(0) = 1 and g(1) = g. The same proof as in [KV71, App. 3]
shows that G = GL(A)ρ .


K -THEORY OF NON-ARCHIMEDEAN RINGS I

37

Remark 7.4. An s-bounded homomorphism between complete normed rings
φ : A → B (see Definition 3.9) induces a map of simplicial rings Ah∆ρ1/s i →Bh∆ρ i and hence a map of pro-spaces KV an (A) → KV an (B). This implies
that, up to natural equivalence, KV an (A) depends only on the equivalenceclass of the norm. Thus by Example 3.8(iii), the analytic KV -theory of a
Tate ring is well defined.
Lemma 7.5. If A is a regular Tate ring satisfying (†)A , then we have natural
weak equivalences
KV an (A) ' Ω∞ τ≥1 k an (A) ' Ω∞ τ≥1 K an (A).
Proof. Since for each ρ ≥ 1 the simplicial ring Ah∆ρ i is connected, Lemma 4.3
implies that KV an (A) is naturally equivalent to “ limρ”Ω∞ K GL (Ah∆ρ i). By
Remark 3.10 the latter is equivalent to Ω∞ K GL,an (A). Hence the claim
follows from (6.3) and Corollary 6.19.
The same proof as that of Proposition 6.3 shows:
Proposition 7.6. The inclusions A ⊂ Ahtiσ induce an equivalence of prospaces∼
KV an (A) −→ “ lim ” KV an (Ahtiσ ).
σ

We write GL(Aht1 , . . . , tn iρ )0 for the subgroup consisting of matrices fsuch that f (0, . . . , 0) = 1.
Definition 7.7. A bounded homomorphism φ : A → B of complete normedrings is called a pro-GL-fibration if for every n the induced map“ lim ” GL(Aht1 , . . . , tn iρ )0 → “ lim ” GL(Bht1 , . . . , tn iρ )0ρρis an epimorphism of pro-sets.
Remarks 7.8.
(i) A pro-GL-fibration is surjective. (For b ∈ B consider

1 t1 b .)0 1(ii) If I ⊂ A is a closed ideal, we equip A/I with the residue norm. If
I is moreover nilpotent, then A → A/I is a pro-GL-fibration. (EachIh∆nρ i is nilpotent, hence GL(Ah∆nρ i) → GL(A/Ih∆nρ i) is surjective.)(j)

Let X = “ limj”X (j) be a pro-simplicial set. Write Xn = “ limj”Xn forthe pro-set of its n-simplices. We call X weakly discrete if, for every n, theunique degeneracy map s := sn0 : X0 → Xn is an isomorphism of pro-sets.
Lemma 7.9. If X → Y is a map of pro-simplicial sets such that each
Xn → Yn is an epimorphism of pro-sets, and if X is weakly discrete, then so
is Y .
Proof. Consider the commutative diagram of pro-sets
X0


Y0

s
∼
=

/ Xn

s


/ Yn .

38

MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME

Since X0 → Xn is an isomorphism and Xn → Yn is an epimorphism, s : Y0 →Yn is also an epimorphism. Since any composition of face maps Yn → Y0 is
a left-inverse, s is indeed an isomorphism.
Lemma 7.10. If X is a pointed weakly discrete pro-simplicial set whichis level-wise Kan, then π0 (X) ∼= X0 and the higher homotopy pro-groups
vanish.
Proof. The analog of Lemma 7.10 for truncated pro-simplicial sets is clear
as they are weakly discrete if and only if they are equivalent to pro-systemsof discrete simplicial sets. As the map X → X \ to the Postnikov tower,
see [AM69, Ch. 4], is a weak equivalence under our assumption on X, one
immediately reduces the lemma to the truncated case.
Analytic KV -theory satisfies excision for pro-GL-fibrations:
Proposition 7.11. If A → B is a pro-GL-fibration of complete normedrings with kernel I, there is a long exact sequence of pro-abelian groupsan· · · → KVian (I) → KVian (A) → KVian (B) → KVi−1(I) → . . .

ending in KV1an (B) → K0 (I) → K0 (A) → K0 (B).
Proof. Let Gρ denote the image of the simplicial group GL(Ah∆ρ i) in GL(Bh∆ρ i).
For each ρ we have a short exact sequence of simplicial groups
1 → GL(Ih∆ρ i) → GL(Ah∆ρ i) → Gρ → 1.

(7.1)

Since A → B is a pro-GL-fibration,“ lim ” GL(Ah∆nρ i) × GL(B) → “ lim ” GL(Bh∆nρ i)ρρis an epimorphism of pro-sets. This implies that
GL(B) → “ lim ” GL(Bh∆nρ i)/Gρ,nρ

is an epimorphism, too. Hence the pro-simplicial set “ limρ” GL(Bh∆ρ i)/Gρ
is weakly discrete by Lemma 7.9. For each ρ, the projection
GL(Bh∆ρ i) → GL(Bh∆ρ i)/Gρ
is a Kan fibration between Kan complexes with fibre Gρ . Together with
Lemma 7.10 this implies that“ lim ” πn (Gρ ) ∼= “ lim ” πn (GL(Bh∆ρ i)) = KV an (B)
ρ

n+1

ρ

for n > 0 and that we have an exact sequence(7.2) 0 → “ lim ” π0 (Gρ ) → “ lim ” π0 (GL(Bh∆ρ i)) →ρρ→ “ lim ” GL(Bh∆0ρ i)/Gρ,0 → 0.
ρNote that Gρ,0 = im(GL(A) → GL(B)) is a normal subgroup of GL(Bh∆0ρ i) =GL(B) as A → B is surjective by Remark 7.8 and hence every elementaryK -THEORY OF NON-ARCHIMEDEAN RINGS I

39

matrix in GL(B) can be lifted to GL(A). By [Wei13, Exc. II.2.3] we have anexact sequence(7.3)

0 → GL(B)/ im(GL(A) → GL(B)) → K0 (I) → K0 (A) → K0 (B).

Splicing together the long exact sequence of homotopy groups associated to(7.1), the sequence (7.2), and (7.3), we get the desired result.

Corollary 7.12. If I ⊂ A is a closed, nilpotent ideal, then KVian (A) ∼=KVian (A/I) for all i ≥ 1.
Proof. Since I is nilpotent, K0 (I) = 0. By Remark 7.8(iii) andProposition
7.11, it suffices to proof that GL(Ih∆ρ i) is contractible for every ρ. Since
Ih∆nρ i is nilpotent for each n and ρ, GL(Ih∆ρ i) ∼= 1 + M(Ih∆ρ i) and thecontractibility is shown as in the proof of Claim 6.8.
s7→0Since the maps Ahsiρ −−−→ A have a section, they are pro-GL-fibrations.
By Propositions 7.11 and 7.6 it follows that“ lim ” KVian (sAhsiρ ) = 0

(7.4)

ρ

for every i ≥ 1.
s7→1

Lemma 7.13. For ρ > 1, the map sAhsiρ −−−→ A is a pro-GL-fibration.
Proof. Write t = (t1 , . . . , tn ). Fix ρ0 ≥ 1. There is a unique continuous
homomorphism Ahtiρρ0 → Ahsiρ htiρ0 sending ti to sti . Given a matrix g =g(t) ∈ GL(Ahtiρρ0 ) with g(0) = 1, we get a matrix g(st) ∈ GL(sAhsiρ htiρ0 )whose image in GL(Ahtiρ0 ) under the map s 7→ 1 coincides with that of g,
concluding the proof.
For ρ > 1 we set
s7→1Ωρ A := ker(sAhsiρ −−−→ A) = s(s − 1)Ahsiρ .
Corollary 7.14. We have natural isomorphisms
an
KVian (A) ∼(Ωρ A)= “ lim ” KVi−1
ρfor i ≥ 2, andKV1an (A) ∼= “ lim ” ker(K0 (Ωρ A) → K0 (sAhsiρ )).
ρProof. For every ρ > 1 we have a long exact excision sequenceanan· · · → KVian (sAhsiρ ) → KVian (A) → KVi−1(Ωρ A) → KVi−1(sAhsiρ ) →· · · → KV1an (A) → K0 (Ωρ A) → K0 (sAhsiρ ) → K0 (A).
Now apply (7.4).

40

MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME

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Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
E-mail address: moritz.kerz@ur.deInteractive Research Center of Science, Graduate School of Scienceand Engineering, Tokyo Institute of Technology, 2-12-1 Okayama, Meguro,
Tokyo 152-8551, Japan
E-mail address: sshuji@msb.biglobe.ne.jp
Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany
E-mail address: georg.tamme@ur.de