K -THEORY OF NON-ARCHIMEDEAN RINGS II
MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME
Abstract. We study fundamental properties of analytic K-theory of
Tate rings such as homotopy invariance, Bass fundamental theorem,
Milnor excision, and descent for admissible coverings.

Contents
1. Introduction
Notation and conventions
2. Recollections from part I, and complements
2.1. Pro-homotopy algebraic K-theory
2.2. The fundamental fibre sequence
3. Excision for analytic K-theory
4. Descent and applications
4.1. Descent for analytic K -theory
4.2. Globalization
4.3. Properties of analytic K -theory
References

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1. Introduction
This note is a sequel to our note [KST19], and we refer to that paper fora general introduction and motivation. In part I we were mainly concernedwith the construction of what we call analytic K-theory of Tate rings and acomparison with continuous K-theory as proposed by Morrow [Mor16]. The
main result there was a weak equivalence between continuous and analytic
K-theory for regular noetherian Tate rings admitting a ring of definition
satisfying a certain weak resolution of singularities property. Negative continuous K-theory was studied by Dahlhausen in [Dah19].
The goal here is to study more closely the properties of analytic K-theory.
It turns out that analytic K-theory has good formal properties, similar to
Weibel’s homotopy K-theory, without any regularity assumption. However,Date: March 11, 2021.
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).
1

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MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME

we have to restrict to Tate rings admitting a noetherian, finite dimensionalring of definition. In particular, for such Tate rings we establish the analytic
Bass fundamental theorem (Corollary 2.6), pro-homotopy invariance (Corollary 2.7), Milnor excision (Theorem 3.2), and descent for rational coveringsof the associated adic space (Theorem 4.1). The latter allows us to globalize the definition of analytic K-theory to adic spaces satisfying a finitenesscondition.
Our main tool is a fibre sequence relating the continuous K-theory of aring of definition, or more generally a Raynaud type model, with the analytic
K-theory of a Tate ring (Theorems 2.4, 2.8).
Notation and conventions. By a π-adic (or just adic) ring we mean a
topological ring A0 whose topology is the π-adic one for an element π ∈ A0 .
We usually assume that A0 is complete. A topological ring A is called a Tatering if it admits an open subring A0 ⊆ A and an element π ∈ A0 such thatthe subspace topology on A0 is the π-adic one, and π is invertible in A. Suchan A0 is called a ring of definition, and π is called a pseudo-uniformizer. We
have A = A0 [ π1 ]. If A is Tate, then a subring A0 ⊆ A is a ring of definition
if and only if it is open and bounded. the basic reference is [Hub93, Sec. 1].
We denote by Spc and Sp the ∞-categories of spaces and spectra, respectively. Pro(Sp) denotes the ∞-category of pro-spectra, see e.g. [KST19,
Sec. 2]. A map of pro-spectra f is a weak equivalence, if its level-wise truncation τ≤n f is an equivalence of pro-spectra for every integer n. See [KST19,
Sec. 2.2] for a discussion and equivalent characterizations. Similarly, a sequence X → Y → Z of prospectra is a weak fibre sequence if it is weakly
equivalent to a fibre sequence.
The letter K denotes the nonconnective algebraic K-theory functor forrings and quasi-compact, quasi-separated schemes. Its connective covering
is denoted by k. For the following notations, we refer to the corresponding
definitions in [KST19]:- K cont for an adic ring: Definition 5.1.
- k an for an adic ring: Defintion 6.5.
- k an for a Tate ring: Definition 6.11. (Warning: in general, this is not
the connective cover of K an .)- K an for a Tate ring: Definition 6.15.

2. Recollections from part I, and complements
2.1. Pro-homotopy algebraic K-theory. In this subsection we introduce
a variant of Weibel’s homotopy algebraic K-theory, which will be an important tool to compare the analytic K-theory of a Tate ring with the continuous
K-theory of a ring of definition without assuming any regularity condition.
See [KST19, Sec. 6.1] for similar constructions in the setting of complete
Tate rings.

K -THEORY OF NON-ARCHIMEDEAN RINGS II

3

Let R be a ring, and let π ∈ R be any element. For non-negative integers
p, j we defineR[∆pπj ] := R[t0 , . . . , tp ]/(t0 + · · · + tp − π j ).
For fixed j and varying p, the R[∆pπj ] form a simplicial ring, which we denoteby R[∆πj ]. Multiplication of the coordinates ti by π induces maps of simplicial rings R[∆πj ] → R[∆πj−1 ], and in this way we obtain the pro-simplicial
ring“ lim ” R[∆πj ].
j

Let SchR denote the category of quasi-compact, quasi-separated (qcqs)
R-schemes. Let F be a functor F : Schop
R → Sp. We define a new func∞j
tor F π : Schop→
Pro(Sp)asfollows.
We let F π (X) be the geometricR
realization of the simplicial spectrum [p] 7→ F (X ⊗R R[∆pπj ]) and define∞jF π (X) = “ lim ” F π (X) ∈ Pro(Sp).
jDefinition 2.1. Let X be a qcqs R-scheme. We define its pro-homotopy
algebraic K-theory (with respect to π ∈ R) as the pro-spectrum∞

K π (X).
Similarly we define a version with support on X/(π) = X ⊗R R/(π):∞

K π (X on (π))
For an R-scheme X we write X[t] = X ⊗R R[t]. By construction, K π
satisfies the following form of pro-homotopy invariance:∞

Lemma 2.2. For X ∈ SchR the canonical map∞∞

K π (X) → “ lim ” K π (X[t])
t7→πt

∞

is an equivalence of pro-spectra. Similarly for K π (− on (π)).
The proof is the same as that of [KST19, Prop. 6.3]. We will use thefollowing easy observation:
Lemma 2.3. If π is invertible on X, there is a natural equivalence∞K π (X) ∼= KH(X),
where KH(X) denotes Weibel’s homotopy K-theory spectrum. In particular,
for any X there is a fibre sequence of pro-spectra∞∞K π (X on (π)) → K π (X) → KH(X[ π1 ]).
Proof. If π is invertible on X, then the transition map R[∆pπj ] → R[∆pπ0 ] =R[∆p ] induces an isomorphism X ⊗R R[∆p ] → X ⊗R R[∆pπj ]. Thus the pro∞system defining K π (X) is constant with value KH(X). The second claim
follows from the first one together with the localization sequence associatedto the identity map X → X[ π1 ].
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MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME

2.2. The fundamental fibre sequence. The following weak fibre sequenceand its generalization to non-affine models in Theorem 2.8 are our fundamental tool to control the analytic K-theory of Tate rings. For a π-adic ringA0 we denote by K cont (A0 ) the pro-spectrum “ lim ” K(A0 /(π n )).
nTheorem 2.4. Let A0 be a complete π-adic ring, and let A = A0 [1/π] bethe associated Tate ring. Assume that A0 is noetherian and of finite Krull
dimension. Then there is a weak fibre sequence of pro-spectra∞K π (A0 on (π)) → K cont (A0 ) → K an (A).
Before entering the proof, we recall the analytic Bass construction, referring to [KST19, Sec. 4.4] for details. For a functor E defined on the categoryof adic or Tate rings and with values in a cocomplete stable ∞-category(such as Sp or Pro(Sp)) we define the functor ΛE by
ΛE(A) = fib(E(Ahti) tE(A) E(Aht−1 i) → E(Aht, t−1 i))
If E has the structure of a module over connective algebraic K-theory, thenthere is a natural map λ : E(A) → ΛE(A) given as the composite`t

λ : E(A) −−→ ΩE(Aht, t−1 i) → ΛE(A).
If λ is an equivalence, then E satisfies an analytic version of the Bass fundamental theorem, see [KST19, Prop. 4.13]. The analytic Bass construction
E B of E is then given byΛ(λ)λ

E B (A) = colim(E(A) −→ ΛE(A) −−−→ Λ2 E(A) → . . . ).
Example 2.5. Let k cont be the functor from adic rings to pro-spectra thatassigns to A0 the connective continuous K-theory
k cont (A0 ) = “ lim ” k(A0 /(π n )).
n

As a rule, we write k for the connective algebraic K-theory τ≥0 K. Thenk cont,B is canonically equivalent to non-connective continuous K-theory K cont .
Indeed, using the isomorphisms A0 hti/(π n ) ∼= A0 /(π n )[t] and similarly for−1t and t , the classical Bass fundamental theorem [TT90, Thm. 6.6] implies that λ : K cont → ΛK cont is an equivalence, hence K cont ' K cont,B .
Moreover, by nilinvariance of negative K-groups, the fibre of the canonicalmap k cont → K cont has values in constant, coconnective pro-spectra. Hence
this fibre vanishes upon applying the analytic Bass construction [KST19,
Lem. 4.15].
Proof of Theorem 2.4. As in [KST19, Sec. 3.1] we denote by A0 h∆pπj i the
π-adic completion of the ring A0 [∆pπj ]. In [KST19, Def. 6.5] we defined the
pro-spectrum
k an (A0 ) = “ lim ” k(A0 h∆πj i)jK -THEORY OF NON-ARCHIMEDEAN RINGS II

5

where k(A0 h∆πj i) is the geometric realization of the simplicial spectrum[p] 7→ k(A0 h∆pπj i). Similarly, we set1
k an (A0 on (π)) = “ lim ” k(A0 h∆πj i on (π)),
jwhere k(− on (π)) is the connective covering of K(− on (π)). There is anatural map k an (A0 on (π)) → k an (A0 ) and the pro-spectrum k̃ an (A; A0 ) isdefined to fit in the cofibre sequencek an (A0 on (π)) → k an (A0 ) → k̃ an (A; A0 ).
By [KST19, Thm. 6.9], we have a natural weak equivalence k an (A0 ) 'k cont (A0 ), and thus we have a natural weak fibre sequencek an (A0 on (π)) → k cont (A0 ) → k̃ an (A; A0 ).
We claim that applying the analytic Bass construction (−)B yields the asserted weak fibre sequence. Note that (−)B preserves weak equivalences andweak fibre sequences of functors as its construction only involves colimits.
Firstly, by [KST19, Lem. 6.18(i)] there is a natural weak equivalencean(k̃ )B (A; A0 ) ' K an (A). Secondly, by Example 2.5 there is an equivalence k cont,B (A0 ) ∼= K cont (A0 ). It thus remains to prove that the analytic
Bass construction of k an (− on (π)) evaluated on A0 is weakly equivalent to∞K π (A0 on (π)).
Note that by Thomason’s excision [TT90, Prop. 3.19] we have equivalences
K(A0 h∆pπj i on (π)) ' K(A0 [∆pπj ] on (π)),
K(A0 h∆pπj ihti on (π)) ' K(A0 [∆pπj ][t] on (π))
and similarly for t−1 and t, t−1 . By Weibel’s K-dimension conjecture, provenin [KST18], all theses spectra are (−d − 1)-connective for d = dim(A0 ). The
classical Bass fundamental theorem and excision thus imply thatΛn k(A0 h∆pπj i on (π)) ' K(A0 h∆pπj i on (π)) ' K(A0 [∆pπj ] on (π))
for all p and j and every n > d. As finite limits and finite colimits in Pro(Sp)are computed levelwise and as geometric realizations in Sp commute withfinite limits and colimits this shows that∞∞Λn k an (A0 on (π)) ' Λn K π (A0 on (π)) ' K π (A0 on (π))
for n > d. So the diagram in the definition of the analytic Bass construction
for E = k an (− on (π)), evaluated on A0 , is eventually constant with value∞
K π (A0 on (π)). This concludes the proof.
We record the following two applications:
Corollary 2.6 (Bass fundamental theorem for analytic K-theory). Let A
be a complete Tate ring which admits a finite dimensional, noetherian ringof definition. Then the map λ : K an (A) → ΛK an (A) is a weak equivalence
1in [KST19], this pro-spectrum is denoted k̄ an (A ).
0

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MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME

of pro-spectra, and hence the Bass fundamental theorem holds. That is, forevery integer i, we have an exact sequencean
0 → Kian (A) → Kian (Ahti)⊕Kian (Aht−1 i) → Kian (Aht, t−1 i) → Ki−1
(A) → 0and the right-hand map has a splitting given by multiplication with the classof t ∈ K1 (Aht, t−1 i).
Proof. Let A0 be a finite dimensional noetherian ring of definition. As in
Example 2.5, the map λ : K cont (A0 ) → ΛK cont (A0 ) is an equivalence. Using
again Thomason’s excision and the classical Bass fundamental theorem we∞∞see that also λ : K π (A0 on (π)) → ΛK π (A0 on (π)) is an equivalence. By
Theorem 2.4 this implies that λ : K an (A) → ΛK an (A) is a weak equivalence.
The rest now follows from [KST19, Prop. 4.13].

Corollary 2.7 (Pro-homotopy invariance). Let A be a complete Tate ringwhich admits a finite dimensional, noetherian ring of definition. Then themap
K an (A) → “ lim ” K an (Ahti)
t7→πt

is a weak equivalence.
Proof. Both other terms in the fundamental fibre sequence of Theorem 2.4
satisfy this type of pro-homotopy invariance: See [KST19, Lem. 5.13] for∞
K cont and Lemmas 2.2, 2.3 for K π (− on (π)).
There is a more general version of the fundamental fibre sequence involving a Raynaud model X of the Tate ring A = A0 [ π1 ]. We will use this toprove descent for analytic K-theory in Section 4. Let p : X → Spec(A0 )be an admissible morphism, i.e. a proper morphism of schemes which is an
isomorphism over Spec(A). Let
K cont (X) = “ lim ” K(X ⊗A0 A0 /(π j )).
jTheorem 2.8. Let A0 be a complete π-adic ring which is noetherian andof finite Krull dimension, and let A = A0 [ π1 ]. Let p : X → Spec(A0 ) be anadmissible morphism. Then there is a weak fibre sequence of pro-spectra∞K π (X on (π)) → K cont (X) → K an (A).
Proof. Thanks to Theorem 2.4, it suffices to show that in Pro(Sp) there isa weakly cartesian squareK π (A0 on (π))/ K cont (A0 )/ K cont (X).

∞

(1)
∞

K π (X on (π))

K -THEORY OF NON-ARCHIMEDEAN RINGS II

7

We have a commutative diagram
K π (A0 on (π))/ K π∞ (A0 )/ KH(A)







∞

(2)
∞

/ K π∞ (X)K π (X on (π))∼=/ KH(X[ 1 ])π

where the rows are fibre sequences in Pro(Sp) provided by Lemma 2.3. Since
the right vertical map is an equivalence, the left-hand square is cartesian.
Write X/(π m ) = X ⊗A0 A0 /(π m ). We claim that the squareK π (A0 )/ “ lim ” K(A0 /(π m )[∆ j ])
/ “ lim ” K(X/(π m )[∆ j ])

∞

(3)

π

j,m

∞

K π (X)πj,min Pro(Sp) is weakly cartesian. For this let Q(p, j, m) be the square ofspectra/ K(A0 /(π m )[∆p j ])K(A0 [∆pπj ])π/ K(X/(π m )[∆p j ])K(X[∆pπj ])πand denote by F (p, j, m) its total fibre. So the square (3) is“ lim ” colim
Q(p, j, m).
op
j,m

p∈∆

By “pro cdh decent” [KST18, Thm. A], the square of pro-spectra “ lim ” Q(p, j, m)mis weakly cartesian for every p and every j, i.e. “ lim ” F (p, j, m) ' 0. Notemalso that by Weibel’s K-dimension conjecture [KST18, Thm. B] all spectra in Q(p, j, m) are (−d)-connective for d = dim(A0 ), hence F (p, j, m) is(−d − 2)-connective. The spectral sequence computing the homotopy groupsof the geometric realization of a simplicial spectrum now implies that the
pro-spectrum“ lim ” colim
F (p, j, m)
op
m

p∈∆

is weakly contractible for every j. Taking “ limj” implies that (3) is weakly
cartesian.
Finally, consider the commutative square in Pro(Sp):
K(A0 /(π m ))(4)
K(X/(π m ))'/ “ lim ” K(A0 /(π m )[∆ j ])
π

j

'


/ “ lim ” K(X/(π m )[∆ j ]) .
j

π

8

MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME

The horizontal maps in the above diagram are equivalences by the fact thatthe map A0 /(π m ) → “ limj”A0 /(π m )[∆πj ] is an isomorphism of pro-simplicial
rings and a similar fact with A0 replaced by X.
Composing the left-hand square in (2), (3), and the horizontal inverse of(4), we obtain the desired weakly cartesian square (1).

3. Excision for analytic K-theory
The goal of this section is to establish excision for Milnor squares foranalytic K-theory for Tate rings that admit a finite dimensional, noetherianring of definition.
Recall that a Milnor square is a cartesian square of rings of the form
A
(5)

A

φ

B

B

in which the horizontal maps are surjective. By a Milnor square of complete
Tate rings we mean a Milnor square in which all rings are complete Taterings and all homomorphisms are continuous.
Lemma 3.1. Assume that (5) is a Milnor square of complete Tate rings.
Then there exist rings of definition A0 , B0 , A0 , and B 0 such that (5) restricts
to a Milnor square
A0

A0

φ

B0

B0.

If A and B admit finite dimensional noetherian rings of definition, one canchoose these rings of definition to be noetherian and finite dimensional as
well.
Proof. First note that if A admits the noetherian ring of definition A0 and
A1 is another ring of definition containing A0 , then A1 is finite over A0 andin particular noetherian with dim(A0 ) = dim(A1 ). Indeed, if π ∈ A0 is a
pseudo-uniformizer, then by boundedness A1 is contained in the free A0 module π −n A0 for n big enough. Keeping this in mind, the second claim
will be clear from the following construction.
Choose rings of definition A0 and B0 of A and B, respectively. Replacing
B0 by B0 · φ(A0 ) if necessary, we may assume that φ restricts to a map
A0 → B0 . Now let A0 be the image of A0 in A and let B 0 be the image of
B0 in B. Define A1 ⊆ A as the pullback
A1 = B0 ×B 0 A0 .
By Banach’s open mapping theorem (see [Hen14] for a version for Tate rings)
the continuous isomorphism A → B ×B A is a homeomorphism. As B0 ⊆ B

K -THEORY OF NON-ARCHIMEDEAN RINGS II

9and A0 ⊆ A are bounded, this implies that A1 ⊆ A is bounded. As it also
contains A0 , it is open and hence a ring of definition. Replacing A0 by A1
we have constructed the desired rings of definition.

Theorem 3.2. Let (5) be a Milnor square of complete Tate rings. Assume
that A and B admit noetherian rings of definition that are finite dimensional.
Then the induced square of pro-spectra
K an (A)K an (A)K an (B)K an (B)( an )

is weakly cartesian.
Proof. Using nilinvariance of K an , we will reduce this to pro-excision for∞
K cont and K π of adic rings, using Theorem 2.4.
Choose noetherian, finite dimensional rings of definition as in Lemma 3.1,
and let I0 = ker(A0 → A0 ), J0 = ker(B0 → B 0 ), so that I0 ∼= J0 via φ. For
every m ≥ 1(A0 /I m )[ 1 ] → (A0 /I0 )[ 1 ] ∼
=A
0

π

π

is then a nilpotent extension of Tate rings and hence
K an ((A0 /I0m )[ π1 ]) → K an (A)is a weak equivalence by [KST19, Prop. 6.17]. From Theorem 2.4 we therefore
deduce a weak fibre sequence∞“ lim ” K π (A0 /I0m on (π)) → “ lim ” K cont (A0 /I0m ) → K an (A)mmand similarly for B0 , J0 . So we get a weak fibre sequence of squares ofpro-spectra
π ∞ → cont → an .
The theorem thus follows from the following two pro-excision Claims 3.3,
3.4.
Claim 3.3. The square of pro-spectra
K cont (A0 )( cont )“ lim ” K cont (A0 /I0m )m

φK cont (B0 )is weakly cartesian.

“ lim ” K cont (B0 /J0m ).
m

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MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME

Claim 3.4. The square of pro-spectra∞∞K π (A0 on (π))( π∞ )“ lim ” K π (A0 /I0m on (π))
m

φ
∞

∞

K π (B0 on (π))“ lim ” K π (B0 /J0m on (π)).
mis weakly cartesian.
∼∼→ J0 via φ, also I0m −→ J0m . As π is a nonProof of Claim 3.3. As I0 −zero divisor on A0 /I0 , it is also a non-zero divisor on A0 /I0m for all m, andsimilarly on B0 /J0m . It follows from these two observations that for every
pair of positive integers n, m we have a Milnor square
A0 /(π n )(6)A0 /(π n , I0m )φ

B0 /(π n )B0 /(π n , J0m ).

Pro-excision for noetherian rings [Mor18, Cor. 0.4] (see also [LT19, Sec. 2.4]
for more general statements) implies that the square of K-theory pro-spectra“ lim ” K((6))
mis weakly contractible for every n. Taking the limit over n gives the claim. 
Proof of Claim 3.4. From pro-excision for noetherian rings, applied to the
Milnor squares
A0 [∆pπj ](7)

A0 /I0m [∆pπj ]φ

B0 [∆pπj ]B0 /J0m [∆pπj ],we deduce that for fixed p and j, the square of pro-spectra“ lim ” K((7) on (π))
mis weakly cartesian. As in the proof of Theorem 2.8, using finite dimensionality and Weibel’s conjecture we deduce that we can pass to geometricrealizations, and deduce the claim.

4. Descent and applications
4.1. Descent for analytic K -theory. Let (A, A+ ) be an affinoid ring inthe sense of [Hub93, Sec. 3], and let X = Spa(A, A+ ). We assume that A
is Tate and that it admits a noetherian, finite dimensional ring of definition
A0 . If U ⊆ X is a rational subdomain, then the same conditions hold for theaffinoid ring (OX (U ), OX (U )+ ). We fix a pseudo-uniformizer π of A. Then

K -THEORY OF NON-ARCHIMEDEAN RINGS II

11(the image of) π is also a pseudo-unifomizer for every Tate ring OX (U ). Forease of notation we set
K an (U ) := K an (OX (U )).
Let f1 , . . . , fn ∈ A be elements generating A as an ideal. These give rise to
the standard rational covering U = (Ui )1≤i≤n of X where
Ui = X(

f1 , . . . , fn
).
fi

Write
K an (|U• |) := lim K an (Up ),

(8)

p∈∆

where U• is the Čech nerve of the covering U.
Theorem 4.1. In the above situation, the natural map
K an (A) → K an (|U• |)is a weak equivalence of pro-spectra.
Proof. Fix a finite dimensional, noetherian ring of definition A0 of A. Multiplying by a power of π, we may assume that the fi belong to A0 . Let I ⊆ A0
be the ideal generated by f1 , . . . , fn , and let
M
Y = Proj(Ik)
k≥0

be the blowup of Spec(A0 ) along the ideal I. Then Y → Spec(A0 ) is anadmissible morphism. Let Vi = D+ (fi ) ⊆ Y . Then V = (Vi )1≤i≤n is anaffine open covering of Y . The π-adic completion OY (Vi )∧ can be canonically
identified with a ring of definition of the Tate ring OX (Ui ), and similarly forthe higher intersections Vi0 ∩ · · · ∩ Vip .
Consider the following diagram of pro-spectraK π (Y on (π))/ K cont (Y )/ K an (A)/ K cont (|V• |)
/ K an (|U• |) ,

∞

(9)
∞

K π (|V• | on (π))∞

where K cont (|V• |) and K π (|V• | on (π)) are defined as (8). The top line is aweak fibre sequence by Theorem 2.8. Note that the coverings U and V consistof n open subsets of X and Y , respectively, and thus their Čech nerves are
n-skeletal. This implies that the limit
K cont (|V• |) = lim K cont (Vp )[p]∈∆in Pro(Sp) can be computed level-wise (see e.g. [KST19, Lem. 2.1]) and∞similarly for K π (|V• | on (π)). From Zariski descent for non-connective Ktheory [TT90, Thm. 8.1] we thus deduce that the left two vertical maps in(9) are equivalences of pro-spectra.

12

MORITZ KERZ, SHUJI SAITO, AND GEORG TAMMELet
Vi0 = Spec(OY (Vi )∧ )and similarly for the higher intersections. By Thomason’s excision [TT90,
Prop. 3.19] we have an equivalence∞∞K π (|V• | on (π)) ' K π (|V•0 | on (π)),
and we clearly also have K cont (|V• |) ' K cont (|V•0 |). Hence we deduce from
Theorem 2.4 that the lower horizontal line in (9) is also a fibre sequence.
We conclude that the right vertical map is a weak equivalence, as was to beshown.

4.2. Globalization. In order to globalize the construction of analytic Ktheory, we first recall some sheaf theory. Let C be a small category. We
denote byPSh(C) = Fun(C op , Spc)
the ∞-category of presheaves of spaces on C and by y : C → PSh(C) the
Yoneda embedding. If F and U are presheaves, we write
F (U ) = MapPSh(C) (U, F ).
Note that by the Yoneda lemma, we have F (X) = F (y(X)) for every object
X of C.
If D is any complete ∞-category, then composition with the Yoneda embedding induces an equivalence(10)'

Funlim (PSh(C)op , D) −→ Fun(C op , D)

where Funlim (PSh(C)op , D) ⊆ Fun(PSh(C)op , D) denotes the full subcategory spanned by the limit preserving functors (see [Lur09, Thm. 5.1.5.6]).
We now assume that C is equipped with a (Grothendieck) topology. Thereis a bijection between sieves on an object X ∈ C and subobjects U ,→ y(X)of y(X) in PSh(C) [Lur09, Prop. 6.2.2.5]. For simplicity, we call the latter
sieves, too. By definition, a presheaf F ∈ PSh(C) is a sheaf if and only if
the natural map
F (X) → F (U )is an equivalence for every X ∈ C and every covering sieve U ,→ y(X) of X.
We denote by
Sh(C) ⊆ PSh(C)the full subcategory consisting of sheaves. The inclusion Sh(C) ,→ PSh(C)admits a left adjoint L : PSh(C) → Sh(C) called sheafification [Lur09,
Prop. 5.5.4.15, Lemma 6.2.2.7]. We say that a functor F ∈ Fun(C op , D)
is a (D-valued) sheaf if F (X) → F (U ) is an equivalence in D for every
X and covering sieve U ,→ y(X) as above. Here F (U ) is defined using (10)above. By [Lur09, Prop. 5.5.4.20] composition with L induces a fully faithful
functorFunlim (Sh(C)op , D) → Funlim (PSh(C)op , D) ' Fun(C op , D)

K -THEORY OF NON-ARCHIMEDEAN RINGS II

13whose essential image consists precisely of the D-valued sheaves.
We now return to analytic K-theory of Tate rings. Fix a complete affinoid
Tate ring (R, R+ ) such that R admits a finite dimensional, noetherian ringof definition R0 , and let S = Spa(R, R+ ). We denote by
AdSaff,ft ⊆ Adsep,lft⊆ AdlftSS
the categories of affinoid adic spaces of finite type over S, of adic spaces
separated and locally of finite type over S, and of adic spaces locally offinite type over S, respectively. See [Hub94, Sec. 3] and [Hub96, Sec. 1.2,
1.3] for these notions. Note that for any X ∈ Adlft
S and any open affinoid
U = Spa(A, A+ ) ⊆ X, A admits a finite dimensional, noetherian ring of
definition. Indeed, by [Hub94, Prop. 3.6] the map (R, R+ ) → (A, A+ ) is
topologically of finite type. By Lemma 3.3(iii) there, R → A factors througha continuous open surjection Rhx1 , . . . xn i → A, and we can take the imageof R0 hx1 , . . . , xn i in A as a ring of definition.
We equip the categories of adic spaces above with the usual topologies.
We then have:⊆ Adlft
Lemma 4.2. The inclusion Adaff,ft
S induces an equivalenceSaff,ft).
Sh(Adlft
S ) ' Sh(AdS

Proof. This follows by applying [Hoy14, Lemma C.3] to both inclusions
lft⊆ Adlft⊆ Adsep,lft
Adaff,ft
S . Here we use that every X ∈ AdS has aSScovering {Ui → X} such that all intersections Ui0 ∩ · · · ∩ Uin are separated,
and any separated X has such a covering with all intersections affinoid. 
Corollary 4.3. The presheaf→ Pro(Sp+ ),K an : Adaff,ft,op
S

Spa(A, A+ ) 7→ K an (A),is a sheaf. In particular, it extends essentially uniquely to a sheaf with values
in Pro(Sp+ ) on the category Adlft
S .
In particular, we can now make sense of the analytic K-theory K an (X)for every adic space X locally of finite type over the base S.
Proof. In view of the general discussion at the beginning of this subsection
the second assertion follows from the first one and Lemma 4.2. Now we prove
the first one. For each affinoid X we define a family of sieves τ (X) on X as
follows: A sieve U ,→ y(X) belongs to τ (X) if and only if the natural map
K an (X 0 ) → K an (U ×y(X) y(X 0 ))is an equivalence in Pro(Sp+ ) for every morphism X 0 → X in AdSaff,ft .
Then the association X
τ (X) defines a topology on AdSaff,ft , see [Hoy14,
Prop. C.1]. Since the topology on AdSaff,ft is generated by rational coverings,
it suffices to show that the sieve generated by a rational covering of X belongs

14

MORITZ KERZ, SHUJI SAITO, AND GEORG TAMME

to τ (X). Since rational coverings are stable under pullback, it suffices toshow that
K an (X) → K an (|U• |)is an equivalence, where |U• | = colim[p]∈∆op y(U[p] ) is the realization of theČech nerve of a rational covering of X. Note here that |U• | ,→ y(X) is precisely the sieve generated by that covering. This equivalence is a consequenceof Theorem 4.1.

4.3. Properties of analytic K -theory. In this subsection we record some
immediate consequences. We keep the assumptions on the affinoid adic basespace S as in the previous subsection.
Proposition 4.4 (A1 -invariance). Let X be an adic space locally of finite
type over S, and let A1X denote the adic affine line over X. Then the naturalmap
K an (X) → K an (A1X )is an equivalence in Pro(Sp+ ).
Proof. It suffices to check this when X = Spa(A, A+ ) is affinoid. Let
Xhti = Spa(Ahti, A+ hti). The sheaf (of spaces) y(A1X ) is equivalent tocolimt7→πt y(Xhti). HenceK an (A1X ) ' lim K an (Xhti) ' K an (X)t7→πtwhere the second equivalence follows from Corollary 2.7.



Recall that the K-module structure of K an as a functor on affinoids provides a natural transformation ` t : K an → K an ( ht, t−1 i). It induces anatural transformation on the extension of K an to all adic spaces in Adlft
S .
Proposition 4.5 (Projective line). Let X be an adic space locally of finite
type over S, and denote by P1X the adic projective line over X. We have anatural weak equivalence
K an (P1X ) ' K an (X) ⊕ K an (X).
Proof. We may again assume that X is affinoid. Consider the covering of
P1X given by Xhti, Xht−1 i. Denote the projection P1X → X by p and let
q := p|Xht,t−1 i . Since K an is a sheaf, the upper line in the following diagram
is a fibre sequence.
ΩK an (Xht, t−1 i)Oq∗ΩΓK an (X)/ K an (P1 )O X
p∗/ K an (X)/ K an (Xhti) ⊕ K an (Xht−1 i)O'/ K an (Xhti) ⊕ K an (Xht−1 i)HereΓK an (X) = K an (Xhti) tK an (X) K an (Xht−1 i)
The lower line is a fibre sequence by definition of ΓK an (X). Since p∗ issplit, we have an equivalence K an (P1X ) ' K an (X) ⊕ cofib(p∗ ). From theK -THEORY OF NON-ARCHIMEDEAN RINGS II

15

diagram we get a weak equivalence cofib(q ∗ ) ' cofib(p∗ ). By definition ofΛK an (see Subsection 2.2) we have an equivalence cofib(q ∗ ) ' ΛK an (A). By
Corollary 2.6 the composite`t

K an (X) −−→ ΩK an (Xht, t−1 i) → ΛK an (X) ' cofib(q ∗ ) ' cofib(p∗ )is a weak equivalence. Thus this map together with p∗ induces the desired'
weak equivalence K an (X) ⊕ K an (X) −→ K an (P1X ).
We can globalize Theorem 2.8 as follows. Recall that (R, R+ ) denotes anaffinoid Tate ring such that R admits a noetherian, finite dimensional ring of
definition R0 , π ∈ R0 is a pseudo-uniformizer, and S = Spa(R, R+ ). Denoteby Schft
R0 the category of R0 -schemes of finite type. There is a functor
lft
Schft
R0 → AdS ,X 7→ X ad ,1
which on affine schemes is given by Spec(A0 ) 7→ Spa(A, A◦ ) with A = A∧
0 [π]◦and A ⊆ A the subring of powerbounded elements (compose the π-adic completion functor, the functor t from [Hub94, Prop. 4.1], which sends Spf(A∧0)∧∧
to Spa(A0 , A0 ), and the functor Y 7→ Y (π 6= 0)).

Theorem 4.6. For X a scheme of finite type over R0 , there is a weak fibre
sequence of pro-spectra∞K π (X on (π)) → K cont (X) → K an (X ad ).
e → X with Xe regular, then there is
If there exists an admissible morphism X
a weak fibre sequenceK(X on (π)) → K cont (X) → K an (X ad ).
Proof. We equip Schft
R0 with the Zariski topology. Considered as functors
ft,opon SchR0 , all pro-spectra appearing in the statement of the Theorem are∞sheaves (see the proof of Theorem 4.1 for the cases of K π ( on (π)) andK cont ). Since every (regular) X ∈ Schft
R0 is a colimit of (regular) affine
schemes in the category of sheaves Sh(Schft
R0 ), it suffices to check the firststatement in the affine case. But this is Theorem 2.4.
e → X with Xe regular, thenIf there exists an admissible morphism X[KST19, Cor. 4.7] implies that for every p, the map
K(X on (π)) → “ lim ” K(X ⊗R0 R0 [∆pπj ] on (π))
jis a weak equivalence of pro-spectra. Using arguments similar to those in∞
the proof of Theorem 2.4, Weibel’s conjecture that K π (X on (π)) is weakly
equivalent to K(X on (π)). This establishes the second claim.

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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