PROPERTIES OF EXTENDED ROBBA RINGS

arXiv:1709.06221v1 [math.NT] 19 Sep 2017

PETER WEAR

Abstract. We extend the analogy between the extended Robba rings of p-adic Hodge
theory and the one-dimensional affinoid algebras of rigid analytic geometry, proving some
fundamental properties that are well known in the latter case. In particular, we show
that these rings are regular and excellent. The extended Robba rings are of interest as
they are used to build the Fargues-Fontaine curve.

1. Introduction
Since being introduced in [6], the Fargues-Fontaine curve has quickly become an important object in number theory. Given a finite extension K of Qp with Galois group GK ,
an important result from p-adic Hodge theory gives an equivalence of categories between
continuous representations of GK on finite free Zp -modules and étale (φ, Γ)-modules over
the period ring AK (see [2] for an exposition of this result). In [6], Fargues and Fontaine
describe this category in terms of vector bundles on the scheme-theoretic Fargues-Fontaine
curve.
There is also an adic version of the Fargues-Fontaine curve, this is an analytification
satisfying a version of the GAGA principle as seen in [14, §4.7]. Both versions of the curve
parametrize the untilts of characteristic p perfectoid fields. Recently, Fargues has formulated a conjecture using the curve to link p-adic Hodge theory, the geometric Langlands
program and the local Langlands correspondence [5].
The adic version of the curve is built out of extended Robba rings. These rings appear
in p-adic Hodge theory ([13], for example). In [11], Kedlaya proved that they are strongly
noetherian. Kiehl’s theory of coherent sheaves on rigid analytic spaces has been extended
to a similar theory on adic spaces by Kedlaya and Liu in [14] and on rigid geometry by
Fujiwara and Kato in [7]. The strong noetherian property is required to fit the curve into
this theory.
This work suggests an analogy between the extended Robba rings and one-dimensional
affinoid algebras. In [11], Kedlaya established some finer properties of the rings suggested
by this analogy and listed some other expected properties [11, Remark 8.10]. In this paper,
we establish these properties. We hope that the extension of this analogy will help transfer
results from the theory of rigid analytic spaces to the Fargues-Fontaine curve.
We now give an outline of this paper. The extended Robba rings are completions of
rings of generalized Witt vectors. In [10], Kedlaya gave a classification of the points of the
Berkovich space associated to W (R) - the ring of Witt vectors over any perfect Fp -algebra
R. His proof demonstrates a close analogy between W (R) and the polynomial ring R[T ]
1

PROPERTIES OF EXTENDED ROBBA RINGS

2

equipped with the Gauss norm. We first extend this classification to the generalized Witt
vectors by exploiting the many shared functorial properties of the two constructions. We
then consider higher rank valuations to get a description of the corresponding adic space,
again taking advantage of the analogy to R[T ]. Completing, we get the classification for
the extended Robba rings.
Using this explicit classification, we can prove that the rank-1 valuations of these rings
are dense in the constructible topology of the adic spectrum. This allows us to compute
the power bounded elements of a rational localization of these rings. We then extend these
results to the rings defined by étale morphisms of extended Robba rings.
Finally, we prove a form of the Nullstellensatz and use this to prove regularity and
excellence for these rings. In characteristic zero, our proof of excellence is an adaption of
the proof in Matsumura’s book [20, Theorem 101] that the ring of convergent power series
over R or C is excellent. In particular, we work with the derivations of the rings, proving a
Jacobian criterion. In the characteristic p case, excellence follows from a theorem of Kunz
[17, Theorem 2.5].
1.1. Acknowledgements. The author would like to thank Kiran Kedlaya for suggesting
these questions and for many helpful conversations. The author gratefully acknowledges
the support of NSF grant DMS-1502651 and UCSD.
2. Generalized Witt Vectors
Throughout this paper, we will be working in the same setup as [11]. Fix a prime p
and a power q of p. Let L be a perfect field containing Fq , complete for a multiplicative
nonarchimedean norm | • |, and let E be a complete discretely valued field with residue
field containing Fq and uniformizer ̟ ∈ E. Let oL and oE be the corresponding valuation
subrings and write W (oL )E := W (oL ) ⊗W (Fq ) oE . Concretely, each element of W (oL )E can
P
be uniquely written in the form i≥0 ̟i [xi ] with xi ∈ oL .
This ring is treated at length in [6, Sections 5-6] with the notation WoE (oF ). Alternately,
W (oL )E is a ring of generalized Witt vectors as described in [3, Section 2] with the notation
W̟ (oL ). The generalized Witt vectors retain many useful properties of the usual p-typical
Witt vectors. In this section, we briefly go over the results we will need in the rest of the
paper.
Recall that given a perfect Fp -algebra R, we can define W (R) functorially as as the
unique strict p-ring W (R) for which W (R)/(p) ∼
= R. The analogous statement is true for
W (oL )E .
Lemma 2.1. We have W (oL )E /(̟) = oL , and W (oL )E is ̟-torsion-free, ̟-adically
complete and separated.
Proof. [3, Proposition 2.12]



Lemma 2.2. The addition law is given by
X
X
X
[y i ]̟i =
[z i ]̟ i
[xi ]̟i +
i≥0

i≥0

i≥0

PROPERTIES OF EXTENDED ROBBA RINGS
j−i

3

j−i

for j = 0, . . . , i. This polynomial has integer
where z i is a polynomial in xqj , y qj
coefficients and is homogeneous of degree 1 for the weighting in which xj , y j have degree
1. The analogous statement is true for multiplication.
Proof. [6, Remarque 5.14].



We conclude this section with a result on factoring in W (oL )E .
Definition 2.3. An element x ∈ W (oL )E is stable if it has the form
either |xi | = 0 for all i ≥ 0 or |x0 | > p−i |xi | for all i > 0.

P∞

i=0 ̟

i [x

i]

with

Theorem 2.4. Assume that L is algebraically closed. For x ∈ W (oL )E nonzero and not
stable, we can write x = y(̟ − [u1 ]) · · · (̟ − [un ]) for some nonzero stable y ∈ W (oL )E
and some u1 , . . . , un ∈ oL with |u1 |, . . . , |un | ≤ p−1 .
Proof. [6, Théorème 6.46].



3. The Berkovich spectrum of the generalized Witt vectors
Let R be a perfect Fp -algebra, equipped with the trivial norm. In [10, Theorem 8.17],
Kedlaya gives an explicit classification of the points of M(W (R)). In this section, we will
extend this result to M(W (oL )E ). This entire section follows Kedlaya’s paper extremely
closely, the arguments all carry over fairly directly due to the similarities of the rings W (R)
and W (oL )E stated in Section 2. We therefore explain the differences caused by the change
of rings, but when arguments are essentially identical to the original paper we simply give
a sketch and a reference to the original proof.
We first define an analogue of the Gauss seminorm on W (oL )E .
Lemma 3.1. The function λ : W (oL )E → R given by
∞

X

̟i [xi ] = max p−i |xi |
λ
i=0

i

is multiplicative and bounded by the p-adic norm.
Proof. This is the analogue of [10, Lemma 4.1] where the seminorm α (which is | · | in our
case) is assumed to be multiplicative. The proof is identical to the normal Witt vector case,
using the fact that addition and subtraction are defined on the jth Teichmuller component
as homogeneous polynomials of degree q j as in Lemma 2.2.

This acts like the (p−1 )-Gauss seminorm for the generator ̟, and the (r/p)-Gauss
seminorm can be defined by replacing p−i in the above lemma by (r/p)i . We can use this
to build analogues of Gauss seminorms for other generators and weights.
Definition 3.2. Given u ∈ oL with |u| ≤ p−1 , let π = ̟ − [u]. Then given r ∈ [0, 1], we
define the valuation H(u, r) to be the quotient norm on W (oL )E [T ]/(T − π) ∼
= W (oL )E
induced by the (r/p)-Gauss extension of | · | to W (R)[T ].
To show that these valuations are multiplicative and to compute them easily, we define
stable presentations.

PROPERTIES OF EXTENDED ROBBA RINGS

4

Definition
L is complete, so is oL and so W (oL )E is (̟, [u])-adically complete, so
P 3.3. As
i with x ∈ W (o ) converges to some limit x. We say that the sequence
any sum ∞
x
π
i
i
L E
i=0
x0 , x1 , . . . forms a presentation of x with respect to u. If each xi is stable (Definition 2.3),
we call this a stable presentation.
P
i
Lemma 3.4. For any x ∈ W (oL )E , there exists some y = ∞
i=0 ̟ [yi ] ∈ W (oL )E with
x ≡ y (mod π) and |y0 | ≥ |yi | for all i > 0. By definition, y is stable.
Proof. This follows the construction of [10, Lemma 5.5] but is a bit simpler as our π is of
the form ̟ − [u] instead of some general primitive element. For any integer j ≥ 0, we can
write
j
∞
∞
X
X
X
i
i
̟ [xi ] ≡
[u] [xi ] +
̟ i [xi ] (mod π).
x=
i=0
i
i=j+1 ̟ [xi ]

P∞

i=0

i=j+1

goes to zero. So either there exists some N > 0 such that
As j grows,
PN
P
i
i
| i=0 [u] [xi ]| ≥ |xn | for all n > N or the sum ∞
i=0 [u] [xi ] converges. In the first case, we
∞
∞
N
X
X
X
̟ i [xi ], in the second case, we let y =
[u]i [xi ].

[u]i [xi ] +
let y =
i=0

i=N +1

i=0

Lemma 3.5. Every element of W (oL )E admits a stable presentation.
Proof. This is the analogue of [10, Lemma 5.7]. Given x, x0 , x1 , . . . , xi−1 ∈ W (oL )E , apply
P
i
Lemma 3.4 to construct xi congruent to (x − i−1
j=0 xj )/π (mod π). This process yields a
stable presentation of x0 , x1 , . . . of x.

Theorem 3.6. The function H(u, r) is a multiplicative seminorm and bounded by λ. Given
any stable presentation x0 , x1 , . . . of x ∈ W (oL )E , H(u, r) = maxi {(r/p)i λ(xi )}.
Proof. The proof of [10, Theorem 5.11] carries over exactly as all the needed properties of
presentations in W (R) also hold in W (oL )E .

The following computation will be useful later.
Corollary 3.7. For u, u′ ∈ oL with |u|, |u′ | ≤ p−1 and r ∈ [0, 1],
H(u, r)(̟ − [u′ ]) = max{r/p, H(u, 0)(̟ − [u′ ])}.
Proof. [10, Lemma 5.13]



Remark 3.8. [10, Remark 5.14] As H(u, 0) is the quotient norm on W (oL )E /(π) induced
by λ, we have H(u, 0)(x) = 0 if and only if x is divisible by π.
Furthermore, any v ∈ M(W (oL )E ) with v(π) = 0 must equal H(u, 0). Given x ∈
W (oL )E , we can construct a stable presentation with respect to π. Then only the first
term of the presentation will affect v(x) as v(π) = 0, so v is exactly H(u, 0) by Theorem
3.6.
Lemma 3.9. For any v ∈ M(W (oL )E ), there exists a perfect overfield L′ of L complete
with respect to a multiplicative nonarchimedean norm extending the one on L and some
u ∈ mL′ \ {0} such that the restriction of H(u, 0) to W (oL )E equals v.

PROPERTIES OF EXTENDED ROBBA RINGS

5

I
defined in 5.1, the exact same
Proof. This is shown in [11, Lemma 6.3] for the rings BL,E
proof will work for W (oL )E . The analogous construction for p-typical Witt vectors is in

[10, Definition 7.5].

This lemma is very important as it allows us to reduce our study of general seminorms
of W (oL )E to those in Definition 3.2.
Definition 3.10. Given v ∈ M(W (oL )E ) and ρ ∈ [0, 1], choose L′ , u as in 3.9 and define
H(v, ρ) to be the restriction of H(u, ρ) to W (oL )E . We define the radius of v to be the
largest ρ ∈ [0, 1] for which H(v, ρ) = v. This is well defined by continuity.
Lemma 3.11. This definition doesn’t depend on L′ or u and defines a continuous map
H : M(W (oL )E ) × [0, 1] → M(W (oL )E ) such that
H(H(v, ρ), σ) = H(v, max{ρ, σ})
Proof. [10, Theorem 7.8]

(v ∈ M(W (oL )E ); ρ, σ ∈ [0, 1]).


Now let L̃ be a completed algebraic closure of L, there is a unique multiplicative extension
of | · | to L̃ so we will continue to call this | · |. Let oL̃ be the valuation ring of L̃ and equip
W (oL̃ ) with the multiplicative norm λ̃.
Definition 3.12. For u ∈ oL̃ with |u| ≤ p−1 and r ∈ [0, 1], let β̃u,r be the valuation H(u, r)
and let βu,r be the restriction of β̃u,r to W (oL )E .
Remark 3.13. There is a natural analogue of β̃u,r in M(K[T ]) where K is an algebraically
closed field. In that case, the seminorm can be identified with the supremum norm over
the closed disc in C of center u and radius r. An analogous statement holds here, Lemma
3.15 implies that β̃u,r dominates the supremum norm. We won’t use or prove this fact, but
it may be helpful for intuition.
We now give a very brief exposition of some useful properties of the valuations β̃u,r and
βu,r that are needed for the classification. All proofs are now identical to those in [10]. By
[10, Lemma 8.3], we have β̃u,r = β̃u′ ,r if and only if r/p ≥ β̃u′ ,0 (̟ − [u]). We can therefore
replace the center u of β̃u,r with a nearby element u′ ∈ oL̃ [10, Corollary 8.4] which we
can choose to be integral over oL [10, Corollary 8.5]. This integrality allows us to move
to βu,r : factoring the minimal polynomial of u reduces computations to checking things of
the form p − [ui ]. This type of argument implies that the radius works as expected, the
radius of βu,r is r [10, Corollary 8.8].
This brings us to the key lemma for our classification.
Lemma 3.14. Given v ∈ M(W (oL )E ) with radius r and s ∈ (r, 1], there exists u ∈ oL̃
with |u| ≤ p−1 for which H(v, s) = βu,s .
Proof. The full proof is given in [10, Lemma 8.10], we will give a sketch. The set of s such
that H(v, s) = βu,s for some u ∈ oL̃ is up-closed and nonempty. Let t be its infimum, we
will check that t ≤ r.

PROPERTIES OF EXTENDED ROBBA RINGS

6

By Lemma 3.9, we can expand L to some L′ and find w ∈ mL′ \ {0} such that v is the
restriction of H(w, 0) to W (oL )E . Taking an algebraic closure L′ lets us identify oL with
a subring of oL′ , so we can compare βw,s with any βu,s ∈ M(W (oL̃ )). If s < t then these
valuations must be distinct, so s/p < βw,0 (p − [u]).
By the computation in Corollary 3.7,
βw,s (p − [u]) = max{s/p, βw,0 (p − [u])} = βw,0 (p − [u])
when s ≤ t, so for these elements the valuation doesn’t depend on s. But because L′
is algebraically closed, by Theorem 2.4 we can factor every element of W (oL )E into the
product of a stable element and finitely many p − [ui ] . As all of the terms of this product
are independent of s, we conclude that βw,s = βw,0 = v for all s ∈ [0, t] and so t ≤ r as
desired.

Lemma 3.15. For u ∈ oL̃ with |u| ≤ p−1 and r ∈ [0, 1], let D(u, r) be the set of βv,0
dominated by βu,r . Then for r, s ∈ [0, 1], D(u, r) = D(u, s) if and only if r = s.
Proof. [10, Lemma 8.16]



Theorem 3.16. Each element of M(W (oL )E ) is of exactly one of the following four types.
(1) A point of the form βu,0 for some u ∈ oL̃ with |u| ≤ p−1 . Such a point has radius
0.
(2) A point of the form βu,r for some u ∈ oL̃ with |u| ≤ p−1 and some r ∈ (0, 1] such
that t/p is the norm of an element of oL̃ . Such a point has radius r.
(3) A point of the form βu,r for some u ∈ oL̃ with |u| ≤ p−1 and some r ∈ (0, 1) such
that t/p is the not norm of an element of oL̃ . Such a point has radius r.
(4) The infimum of a sequence βui ,ri for which the sequence D(ui , ri ) is decreasing with
empty intersection. Such a point has radius inf ri > 0.
i

Proof. This is [10, Lemma 8.17], we again give a sketch. Types (i), (ii), (iii) are distinct
as they have different radii, and βu,r cannot be type (iv) because βu,0 would be in each
D(ui , ti ). So the four types of points are distinct and we must check that any β not of the
form βu,s is type (iv).
Let r be the radius of β, choose a sequence 1 ≥ t1 > t2 > · · · with infimum r. Then by
3.14 we have H(β, ti ) = βui ,ti for some ui ∈ oL̃ . Then βu1 ,t1 , βu2 ,t2 , . . . is decreasing with
infimum β, so the sequence D(ui , ti ) is also decreasing. Any u in the intersection would
allow us to write β = βu,r , so the D(ui , ti ) must have empty intersection. The radius must
be nonzero because any decreasing sequence of balls with empty intersection must have
radii bounded below by a nonzero number.

4. The points of the adic spectrum
We will now add in a classification of the higher rank valuations, giving a complete description of the adic spectrum. To move to Spa(W (oL )E , W (oL )E ◦ ), we must also consider
higher rank valuations. In the lecture notes [4, Lecture 11], Conrad gives an explicit description of the points of the adic unit disk Spa(khti, k◦ hti) over a non-archimedean field k.

PROPERTIES OF EXTENDED ROBBA RINGS

7

In this section, we adapt this proof to show that all the higher rank points are the natural
equivalent of the type 5 points of the adic unit disk.
Define the abelian group Γ := R>0 × Z with the lexicographical order, the group action
given by (t1 , m1 )(t2 , m2 ) = (t1 t2 , m1 + m2 ). We define 1− := (1, −1), 1+ := (1, 1) = 1/1− ,
r − = r1− , and r + = r1+ . Intuitively, 1− is infinitesimally less than 1.
Definition 4.1. Given u ∈ oL̃ with |u| ≤ p−1 and r ∈ (0, 1], we define β̃u,r+ : W (oL̃ )E →
Γ ∪ {0} by

β̃u,r+ (x) = max (r + /p)i |xi |}
i

for any stable presentation x0 , x1 , . . . of x with respect to u. We define βu,r+ to be the
restriction of β̃u,r+ to W (oL )E and we define β̃u,r− and βu,r− analogously. We call these
the type 5 valuations.
We remark that this definition doesn’t depend on the choice of stable presentation by
the argument of [10, Theorem 5.11], this was already used in Theorem 3.6. One can check
that these valuations are continuous. If r/p isn’t the norm of an element of W (oL̃ )E then
×
βx,r+ = βx,r− = βx,r . We will therefore assume from now on that |r/p| ∈ |W (oL̃ )×
E | = |oL |.
Remark 4.2. Unlike the rank one case, the maximum in the definition of β̃u,r+ and β̃u,r−
is attained by a unique element. This is clear as any two terms have different powers of
1+ .
Theorem 4.3. All of the points of Spa(W (oL )E , W (oL )E ◦ ) \ M(W (oL )E ) are of type 5.
Proof. The argument is essentially that of [4, Theorem 11.3.13], we give a sketch pointing
out the differences. We assume that L is algebraically closed; as before the general case
follows by restricting valuations from the algebraic closure. Let v be a valuation of W (oL )E
with rank greater than 1. Then there is some x′ ∈ W (oL )E with v(x′ ) 6∈ |o×
L |. As L is
algebraically closed, we can use Theorem 2.4 to factor x′ = y(p − [u1 ]) · · · (p − [un ]) for
y ∈ W (oL )E stable and ui ∈ oL with |ui | ≤ p−1 . We have v(y) ∈ |o×
L |, so we must have
×
some ui with v(p − [ui ]) 6∈ |oL |. For simplicity we will call this u and define π = p − [u], γ =
v(p − [ui ]). We will show that u must act as our center.
m 6∈ |o× | for any non-zero integer m.
As L is algebraically closed, |o×
L
L | is divisible, so γ
P
i
Then given any x ∈ W (oL )E , if we construct a stable presentation x = ∞
i=0 xi π we have
v(x) = max v(xi π i ) = max γ i |xi |
i

i

as the valuations of the terms are pairwise distinct. The rest of the proof now follows
exactly as in [4]. One checks that γ must be infinitesimally close to some r ∈ |o×
L | (if this
weren’t the case we could construct an order-preserving homomorphism from Γ → R>0 ,
contradicting that Γ is higher rank). Then either v = βu,r+ or βu,r− depending on if γ > r
or γ < r in Γ.


PROPERTIES OF EXTENDED ROBBA RINGS

8

5. The extended Robba rings
We now define the rings used in the construction of the Fargues Fontaine curve and
extend the classification of 3.16 and 4.3 to them.
Definition 5.1. Following [11, Definition 2.2], we define
AL,E = W (oL )E [[x] : x ∈ L],

BL,E = AL,E ⊗oE E.

P
Each element of AL,E (resp. BL,E ) can be written uniquely in the form i∈Z ̟ i [xi ] for
some xi ∈ L which are zero for i < 0 (resp. for i sufficiently small) and bounded for i
large. The valuations H(0, r) for r ∈ (0, 1] (defined in 3.2) therefore extend naturally to
these rings, allowing us to make the following definition.
Definition 5.2. Let ArL,E be the completion of AL,E with respect to H(0, r) and define
r
BL,E
analogously. Given a closed subinterval I = [s, r] of (0, ∞), let λI = max{H(0, s), H(0, r)},
I
this is a power multiplicative norm as the H(0, r) are multiplicative. Let BL,E
be the completion of BL,E with respect to λI .
r
I
are called B b and the rings BL,E
are called BI . In
Remark 5.3. In [6] the rings BL,E
[11] the Gauss norms used to obtain these rings are written in a different form but are
equivalent.

Proposition 5.4. The points of adic spectra of all the rings defined in 5.1 and 5.2 can be
classified into types 1-5 as in 3.16 and 4.3.
◦ )) and an element
Proof. Given a valuation v ∈ Spa(AL,E , A◦L,E ) (resp. Spa(BL,E , BL,E
P
x = i∈Z ̟i [xi ] in AL,E or BL,E , there exists some y ∈ oL and k ≥ 0 such that ̟ k x[y −1 ] ∈
W (oL )E . As v is by definition multiplicative, we have v(x) = v(̟ k x[y −1 ])v([y])v(̟)−k , so
v(x) is uniquely determined by the restriction of v to W (oL )E . Taking completions won’t
r , and
add points to the adic spectrum, so the desired result also follows for ArL,E , BL,E
I .
BL,E


6. Rational Localizations
I , B I,◦ )
We can now use this explicit classification to determine some properties of Spa(BL,E
L,E
I ). We first set some notation. Let (B I , B I,◦ ) → (C, C + )
that can be checked on M(BL,E
L,E
L,E
be a rational localization, so there exist elements f1 , . . . , fn , g ∈ BL,E generating the unit
I
ideal in BL,E
such that
I,◦
I
) : v(fi ) ≤ v(g) 6= 0 (i = 1, . . . , n)}.
Spa(C, C + ) = {v ∈ Spa(BL,E
, BL,E

By [13, Lemma 2.4.13a] we have
I
C = BL,E
{T1 /ρ1 , . . . , Tn /ρn }/(gT1 − f1 , . . . , gTn − fn )
I
(as BL,E
is strongly noetherian [11, Theorem 4.10], this ideal is already closed so we don’t
need to take the closure) and by definition

C + = {x ∈ C : v(x) ≤ 1 (v ∈ Spa(C, C + ))}.

PROPERTIES OF EXTENDED ROBBA RINGS

9

Remark 6.1. By [13, Remark 2.4.7], we can choose the defining elements f1 , . . . , fn , g of
our rational localization to be elements of BL,E . This is convenient as it can P
be difficult to
I
work with general elements of BL,E
- they aren’t necessarily all of the form i∈Z ̟i [xi ].
We start with a useful computation, showing that inequalities coming from a type-5
valuation continue to be true near that valuation.

I , B I,+ ),
Lemma 6.2. Given elements x and y in BL,E and a type-5 valuation βu,r+ in Spa(BL,E
L,E
if βu,r+ (x) ≤ βu,r+ (y) then there exists some real number s > r such that for every
r ′ ∈ (r, s), βu,r′ (x) ≤ βu,r′ (y). If we instead choose βu,r− such that βu,r− (x) ≤ βu,r− (y),
then there exists some real number s < r such that for every r ′ ∈ (s, r), βu,r′ (x) ≤ βu,r′ (y).

Proof. We will just prove the first statement, the other statement follows identically. We
can assume that L is algebraically closed as the other case follows by restricting the valuations. Let π = ̟ − [u] and fix stable presentations x0 , x1 , . . . and y0 , y1 , . . . of x and y, so
βu,r+ (x) = maxi {(r + /p)i λI (xi )} and βu,r+ (y) = maxi {(r + /p)i λI (yi )}. Let j be the unique
(by Remark 4.2) index such that (r + /p)j λI (yj ) = βu,r+ (y).
For any term xi , we have (r + /p)i λI (xi ) ≤ (r + /p)j λI (yj ) by assumption. If i ≤ j then
for all r ′ > r we have (r ′ /p)i λI (xi ) ≤ (r ′ /p)j λI (yj ) so any choice of s will retain the desired
inequality in this case. If i > j, we must have a strict inequality (r/p)i λI (xi ) < (r/p)j λI (yj )
as if this were an equality moving from r to r + would increase the left side more than the
right side. For any fixed i, there is some interval (r, si ) where this inequality remains strict.
It is therefore enough to show that we only need to consider finitely many terms. But there
are only finitely many i such that (1/p)i λI (xi ) > (r/p)j λI (yj ) and these are the only terms
of our presentation of x that could ever pass the leading term of y.

I ) are dense in Spa(B I , B I,◦ ) in the
Proposition 6.3. The rank one valuations M(BL,E
L,E
L,E
constructible topology.
I ) is dense in Spa(B I , B I,◦ ) in
Proof. It is known (eg [13, Definition 2.4.8]) that M(BL,E
L,E
L,E
I , B I,◦ ) that is locally
the standard topology. We must show that any subset of Spa(BL,E
L,E
I ). As we have a
closed in the standard topology has nonempty intersection with M(BL,E
basis of rational subsets, it is enough to show that if we have rational subsets Spa(C, C + )
I , B I,◦ ) and we have some semivaluation v ∈ Spa(C, C + ) \
and Spa(D, D + ) of Spa(BL,E
L,E
I ).
Spa(D, D + ), then there is some semivaluation v ′ ∈ (Spa(C, C + ) \ Spa(D, D + )) ∩ M(BL,E
I ) there is nothing to check, so we may assume that v has rank greater than
If v ∈ M(BL,E
1. By the above classification, v must be a type-5 point. We will assume that v = βu,r+ , a
similar argument will take care of the other option of v = βu,r− . Then if the corresponding
type-2 point βu,r is in Spa(C, C + ) \ Spa(D, D + ) we are done, so we can assume this is not
the case. Then we will use 6.2 to show that there is some s > r such that for all r ′ ∈ (r, s),
βu,r′ is in Spa(C, C + ) \ Spa(D, D + ).
As βu,r 6∈ Spa(C, C + ) \ Spa(D, D + ), we either have βu,r ∈ Spa(D, D + ) or βu,r 6∈
Spa(C, C + ). If we are in the first case, then by definition there are finitely many elements (possibly not all the defining elements of the rational subset) fi , g ∈ BL,E such that

PROPERTIES OF EXTENDED ROBBA RINGS

10

βu,r (fi ) ≤ βu,r (g) 6= 0 and βu,r+ (fi ) > βu,r+ (g). Then for each i, by Lemma 6.2 we have
some interval (r, si ) such that for all r ′ ∈ (r, si ), βu,r′ (fi ) > βu,r′ (g) 6= 0. As we are assumI,+
I,◦
I , B I,◦ ) for all
ing that BL,E
= BL,E
, there is some interval (r, s′ ) such that βu,r′ ∈ Spa(BL,E
L,E
′
′
r ∈ (r, s ). The intersection of this finite set of intervals is nonempty, giving valuations of
types 2 and 3 that are also in Spa(C, C + ) \ Spa(D, D + ) as desired. The other case follows
similarly.

I , B I,◦ ) forms a covCorollary 6.4. A finite collection of rational subspaces of Spa(BL,E
L,E
I ) do so.
ering if and only if the intersections with M(BL,E
I , B I,◦ ) is determined by its intersection
Corollary 6.5. A rational subspace of Spa(BL,E
L,E
I ).
with M(BL,E
I,◦
I,+
I , then
, the ring of power-bounded elements of BL,E
equals BL,E
Proposition 6.6. If BL,E
I , B I,+ ) → (C, C + ), one also has C + = C ◦ .
for any rational localization (BL,E
L,E

Proof. By definition we have C + ⊂ C ◦ , so we must show that C ◦ ⊂ C + . By the description
of C + at the start of the section, this can be done by showing that for any x ∈ C ◦ and
v ∈ Spa(C, C + ), v(x) ≤ 1. We claim that it is enough to check this for v ∈ Spa(C, C + ) ∩
I ), i.e. when v is a rank-one valuation. Assume the contrary: that C + 6= C ◦ but
M(BL,E
I ) = Spa(C, C ◦ ) ∩ M(B I ). Then we can choose some f ∈ C ◦ \ C +
Spa(C, C + ) ∩ M(BL,E
L,E
I , B I,+ ) →
and add the condition that v(f g) ≤ v(g) to the defining inequalities of (BL,E
L,E
I , B I,+ ) → (C ′ , C ′+ ) such that
(C, C + ). This will give a new rational localization (BL,E
L,E
Spa(C ′ , C ′+ ) ( Spa(C, C + ) as we are simply enforcing an extra nontrivial inequality. But
by our assumption, this inequality was already satisfied by all the elements of Spa(C, C + )∩
I ), so we have Spa(C, C + ) ∩ M(B I ) = Spa(C ′ , C ′+ ) ∩ M(B I ). This contradicts
M(BL,E
L,E
L,E
Corollary 6.5, so we must have C ◦ = C + as desired. We remark that Spa(C, C ◦ ) need not
a priori be a rational localization, this is why we had to construct Spa(C ′ , C ′+ ).
I ). By [11, Lemma 6.3], v is the restriction of
Now fix some v ∈ Spa(C, C + ) ∩ M(BL,E
a norm of the form H(u, 0) on some perfect overfield L′ of L (compare with Lemma 3.9).
I,◦
I
I
The inclusion L → L′ gives an inclusion BL,E
→ BLI ′ ,E such that BLI,◦
′ ,E ∩ BL,E = BL,E . Let
I,◦
I , B I,◦ ) → (B I
(C ′ , C ′+ ) denote the base extension of (C, C + ) along (BL,E
L′ ,E , BL′ ,E ), so
L,E

C ′ = BLI ′ ,E {T1 /ρ1 , . . . , Tn /ρn }/(gT1 − f1 , . . . , gTn − fn )
I
where f1 , . . . , fn , g are simply the corresponding elements of BL,E
viewed as elements of
′◦
◦
′
′
′+
I
BL′ ,E . As C ∩ C = C and v ∈ Spa(C , C ) restricts to v on C, checking that v ′ (x) ≤ 1
for all x ∈ C ′◦ will give our result.
By [11, Remark 5.14], the norm v ′ on BLI ′ ,E is just the quotient norm on BLI ′ ,E /(̟−[u]) =
H(v ′ ) (compare with 3.8). So extending v ′ to C ′ , we get the quotient norm on C ′ /(̟ − [u]).
By [11, Lemma 7.3], the map BLI ′ ,E /(̟ − [u]) → C ′ /(̟ − [u]) is an isomorphism, so we
have reduced to looking at a multiplicative norm on a field. In this case, it is clear that
power bounded elements have norm at most 1, so we are done.


PROPERTIES OF EXTENDED ROBBA RINGS

11

7. Étale Morphisms
Étale morphisms of adic spaces were first defined and studied by Huber in [9]. In [11,
Section 8], Kedlaya gives some results on étale morphisms of extended Robba rings. In
this section, we recall the setup of Huber and some of Kedlaya’s results. We then extend
the results of the previous sections to étale morphisms.
I , B I,+ ) → (C, C + ) be a morphism of adic Banach rings which
Hypothesis 7.1. Let (BL,E
L,E
is étale in the sense of Huber [9, Definition 1.6.5]. In particular, C is a quotient of
I {T /ρ , . . . , T /ρ } for some n, so it is strongly noetherian by [11, Theorem 4.10].
BL,E
1 1
n n
I , B I,+ ) and M(C) →
By definition, we get induced morphisms Spa(C, C + ) → Spa(BL,E
L,E
I ).
M(BL,E

Lemma 7.2. There exist finitely many rational localizations {(C, C + ) → (Di , Di+ )}i such
I , B I,+ ) → (D , D + ) factors as a
that ∪i Spa(Di , Di+ ) = Spa(C, C + ) and for each i, (BL,E
i
i
L,E
I , B I,+ ) → (C , C + ) followed by a finite étale morphism
connected rational localization (BL,E
i
i
L,E
(Ci , Ci+ ) → (Di , Di+ ) with Di also connected.

Proof. [9, Lemma 2.2.8]



The above construction commutes with base extension, so when we extend from L to L′
as in Lemma 3.9 the above result is retained.
Proposition 7.3. The rings Ci are all principal ideal domains and the rings Di are all
Dedekind domains.
Proof. See [11, Theorem 7.11(c)] and [11, Theorem 8.3(b)].



We now start working towards a classification of the points of Spa(C, C + ). We have a
I , B I,+ ) and a good understanding of the points of Spa(B I , B I,+ )
map Spa(C, C + ) → Spa(BL,E
L,E
L,E
L,E
I , B I,+ ) we will look at the preimage
from Proposition 5.4, so given a point v ∈ Spa(BL,E
L,E
{wj }j∈J . We will show that this is a finite set of valuations with the same rank and radius
as v.
I , B I,+ ),
Proposition 7.4. Given a valuation w ∈ Spa(C, C + ) mapping to v ∈ Spa(BL,E
L,E
the rank of w is the same as the rank of v.
I
→ C is a finite map of rings, so C is a
Proof. By Lemma 7.2 we can assume that BL,E
I
finitely generated BL,E -module. Let the value group of v be H and the value group of w
be G, then [G : H] is finite so the groups must have the same rank.

I , B I,+ ), the preimage {γ }
Proposition 7.5. Given a valuation β ∈ Spa(BL,E
j j∈J is a finite
L,E
set.

Proof. By Lemma 7.2, we can choose a neighborhood of β so that we are working with a
finite étale morphism. This now follows from [9, Lemma 1.5.2c].


PROPERTIES OF EXTENDED ROBBA RINGS

12

We can now extend the definition of radius to valuations in M(C). Retaining the
notation of the proof of 7.5, the radius of β is the maximal r ∈ [0, 1] such that the restriction
I
of βu,r to BL,E
is β. It is therefore natural to define the radius rj of γj to be the maximal
rj ∈ [0, 1] such that some element of the preimage of βu,r restricts to γj on C.
I ) ⊂ Spa(B I , B I,+ ) with radius r and γ in the
Proposition 7.6. Given β ∈ M(BL,E
j
L,E
L,E
preimage of β with radius rj , we have r = rj .
I
Proof. If s > r, βu,s doesn’t restrict to β on BL,E
so the preimage of βu,s won’t contain
γj . We therefore have r ≥ rj . To show that r ≤ rj , we note that extending the radius is
continuous and that the preimage of βu,s in Spa(C ′ , C ′+ ) maps to a subset of the finite set
{γj } ⊂ Spa(C, C + ) for all s ∈ [0, r]. So by continuity, every γj in the preimage of β is the
restriction of some element of the preimage of βu,r as desired.


We finally extend to higher rank valuations.
Proposition 7.7. Let βu,r± be a type 5 valuation as in Definition 4.1. Then the preimage
of βu,r± is in bijection with the preimage of βu,r .
Proof. This follows from continuity and the fact that the size of the fibers is locally constant.

I , B I,◦ ), the rank one valuations M(C) are
Proposition 7.8. If (C, C + ) is étale over (BL,E
L,E
dense in Spa(C, C + ) in the constructible topology.
I , B I,◦ ) is locally the composition of open immerProof. The map Spa(C, C + ) → Spa(BL,E
L,E
sions and finite étale maps. Étale maps are smooth and therefore open by [9, Proposition
1.7.8], and finite maps are closed by [9, Lemma 1.4.5], so finite étale maps send locally
closed subsets to locally closed subsets. The same is certainly true of open immersions, so
any subset U in Spa(C, C + ) that is locally closed under the standard topology is mapped
I , B I,◦ ). By Proposition 6.3, there is some rank
to a locally closed subset V in Spa(BL,E
L,E
one valuation v ∈ V , and by Proposition 7.4 the preimage of v is made up of rank one
valuations.

I , B I,◦ ), a finite collection of rational subCorollary 7.9. If (C, C + ) is étale over (BL,E
L,E
spaces of Spa(C, C + ) forms a covering if and only if the intersections with M(C) do so.
I , B I,◦ ), a rational subspace of Spa(C, C + )
Corollary 7.10. If (C, C + ) is étale over (BL,E
L,E
is determined by its intersection with M(C).

Proposition 7.11. If C + equals C ◦ , then for any rational localization (C, C + ) → (D, D + ),
one also has D + = D ◦ .
Proof. The proof of 6.6 carries over. By 7.8, we can again reduce to checking specific
inequalities on rank one valuations. We can therefore use Lemma 7.2 and Proposition 7.3
to assume that C is a Dedekind domain. By extending L to some L′ , we reduce to checking
norm with nonempty kernel. As C is a Dedekind domain, the kernel is a maximal ideal.

PROPERTIES OF EXTENDED ROBBA RINGS

13

We are therefore again dealing with power bounded elements in a field with a multiplicative
norm, where the desired inequalities are clear.

8. Consequences of the strong noetherian property
In [11, Theorem 3.2], Kedlaya gives a proof that the ring ArL,E is strongly noetherian, i.e. that ArL,E {T1 /ρ1 , . . . , Tn /ρn } is noetherian for any nonnegative integer n and
ρ1 , . . . , ρn > 0. This is done very explicitly, using the theory of Gröbner bases to construct generators for a given ideal. In this section, we adapt this proof to give a version
of the Nullstellensatz for the rings A{T1 /ρ1 , . . . , Tn /ρn }. We then use the Nullstellensatz
to prove that these rings are regular. In the next section, we will use regularity to prove
that the rings A{T1 /ρ1 , . . . , Tn /ρn } are excellent. We also show that ArL,E is also strictly
I
and rings C
noetherian. As in [11], these arguments can be generalized to the rings BL,E
I
coming from étale extensions of BL,E as in Definition 7.1.
Definition 8.1. Given a ring R and a subring A, we say that the pair (R, A) satisfies the
Nullstellensatz condition if every maximal ideal of R restricts to a maximal ideal of A.
Remark 8.2. We use this name because Munshi proved and then used this property for
(F [x1 , . . . , xn ], F [x1 ]) for F a field to give a proof of Hilbert’s Nullstellensatz, his proof is
the subject of [19].
Theorem 8.3. Let A be a nonarchimedean Banach ring with a multiplicative norm | · |.
Assume further that A is a strongly noetherian Euclidean domain; in particular, this holds
for A = ArL,E . Let n be a positive integer and ρ = (ρ1 , . . . , ρn ) an n-tuple of positive real
numbers such that the value group of A{T1 /ρ1 , . . . , Tn /ρn } \ {0} has finite index over the
value group of A× . Then A{T1 /ρ1 , . . . , Tn /ρn } satisfies the Nullstellensatz condition with
respect to A.
Proof. We begin with two reductions. For each Ti , there is some positive integer ei such
that |Tiei | ∈ |A× |, so we can write A{T1 /ρ1 , . . . , Tn /ρn } as an integral extension of the ring
ArL,E {(T1 /ρ1 )e1 , . . . , (Tn /ρn )en }. This ring has the same value group as A× , and by going
up maximal ideals of A{T1 /ρ1 , . . . , Tn /ρn } restrict to maximal ideals of this ring. We can
therefore assume that the value groups of A{T1 /ρ1 , . . . , Tn /ρn } and A× are equal.
We also note that it is enough to show that m ∩ A 6= 0 whenever A is not a field. Given
x1 ∈ m ∩ A with x1 6= 0, the ring A{T1 /ρ1 , . . . , Tn /ρn }/(x1 ) = A/(x1 ){T1 /ρ1 , . . . , Tn /ρn }
is strongly noetherian and m/(x1 ) is a maximal ideal. We can therefore find x2 ∈ m/(x1 ) ∩
A/(x1 ) and iterate this process until A/(x1 , . . . , xn ) is a field. This must eventually happen
as A is noetherian, then m∩A = (x1 , . . . , xn ) is a maximal ideal of A as desired. We remark
that it isn’t clear that A/(x1 , . . . , xn ) must be a nonarchimedean field.
To show that m ∩ A 6= 0, we will be combining the proofs of [11, Theorem 3.2] (via
[11, Lemma 3.8]) and Lemma [12, 3.8]. The first proof deals specifically with the rings
A{T1 /ρ1 , . . . , Tn /ρn } while the second proves this version of the Nullstellensatz for a similar
ring.

PROPERTIES OF EXTENDED ROBBA RINGS

14

The proof will use ideas from the theory of Gröbner bases and an idea of Munshi [19].
We therefore begin by setting up the combinatorial construction.
Hypothesis 8.4. Let I = (i1 , . . . , in ) and J = (j1 , . . . , jn ) denote elements of the additive
monoid Zn≥0 of n-tuples of nonnegative integers.
Definition 8.5. We equip Zn≥0 with the componentwise partial order ≤ where I ≤ J if
and only if ik ≤ jk for i = 1, . . . , n. This is a well-quasi-ordering: any infinite sequence
contains an infinite nondecreasing sequence.
We also equip Zn≥0 with the graded lexicographic total order  for which I ≺ J if either
i1 + · · · + in < j1 + · · · jn , or i1 + · · · + in = j1 + · · · + jn and there exists k ∈ {1, . . . , n}
such that iℓ = jℓ for l < k and ik < jk . Since  is a refinement of ≤, it is a well-ordering.
The key properties for the proof is that  is a well-ordering refining ≤ and that for any
I, there are only finitely many J with J  I.
P
Definition 8.6. For x = I xI T I ∈ A{T1 /ρ1 , . . . , Tn /ρn }, define the leading index of x to
be the index of I which is maximal under  for the property that |xI T I | = |x|, and define
the leading coefficient of x to be the corresponding coefficient xI .
We proceed by contradiction: suppose that m is a maximal ideal of A{T1 /ρ1 , . . . , Tn /ρn }
with m ∩ A = 0. As we assumed that A is strongly noetherian, A{T1 /ρ1 , . . . , Tn /ρn } is
noetherian so m is closed by [1, Proposition 3.7.2/2].
Define the projection map ψ forgetting the constant term of x ∈ A{T1 /ρ1 , . . . , Tn /ρn }, it
is a bounded surjective morphism of Banach spaces with kernel A. Then m + A is a closed
subspace of A{T1 /ρ1 , . . . , Tn /ρn }, and V = ψ(m+A) is closed by the open mapping theorem
[1, § 2.8.1]. So ψ induces a bounded bijective map of Banach spaces m → V ; by the open
mapping theorem ψ −1 is also bounded. Define the nonconstant degree deg′ (x) = deg(ψ(x))
to be the leading index of ψ(x). Define the leading nonconstant coefficient of x to be xdeg′ (x) .
We now follow the proof of [11, 3.2] but using deg′ instead of deg in [11, 3.7]
Pand beyond
and using |ψ(·)| instead of | · |ρ . We obtain a finite set of generators mI =
mI,J T J for
m such that the leading index of ψ(mI ) is I. The key fact here is that for any x ∈ m with
leading index J, there is some mI with I  J.
Scale the mI by elements of A so that they all have norm 1. Let aI = mI,I be the
leading coefficient of mI , so now |aI | = 1. Define ǫ < 1 as in [11, 3.8] to be the largest
possible norm of some coefficient mI,J with I ≺ J. As the norm on A is multiplicative, the
ring oA /IA has no nonzero nilpotents so the nilradical
is {0}. We can therefore choose a
Q
nonzero prime ideal p of oA /IA not containing I∈S aI . Choose any ̟ ∈ oA reducing to a
nonzero element of p, so |̟| = 1. As m ∩ A = 0, we have ̟ 6∈ m, so by maximality we can
find x0 ∈ A{T1 /ρ1 , . . . , Tn /ρn } such that 1 + ̟x0 ∈ m.
Lemma 8.7. Let S be the multiplicative system generated by the aI . Given any c ∈
S and x ∈ A{T1 /ρ1 , . . . , Tn /ρn } with c + ̟x ∈ m, there exists some c′ ∈ S, x′ ∈
A{T1 /ρ1 , . . . , Tn /ρn } with c′ + ̟x′ ∈ m and |ψ(x′ )| ≤ ǫ|ψ(x)|.

PROPERTIES OF EXTENDED ROBBA RINGS

15

Proof. We will construct c′ and x′ iteratively. Given any cℓ ∈ S, xℓ ∈ A{T1 /ρ1 , . . . , Tn /ρn }
with cℓ + ̟xℓ ∈ m, we will construct cℓ+1 ∈ S, xℓ+1 ∈ A{T1 /ρ1 , . . . , Tn /ρn } with cℓ+1 +
̟xℓ+1 ∈ m and |ψ(xℓ+1 )| ≤ |ψ(xℓ )|. We will then use this construction to get c′ and x′ .
Choose λ ∈ A× so that |ψ(λ(cℓ + ̟xℓ ))| = 1, and let eIℓ T Iℓ be the leading term of
ψ(λxℓ ). Note that the leading index Iℓ is the same as the leading index of ψ(λ(cℓ + ̟xℓ ))
as ψ causes the contribution of cℓ to be forgotten and ̟ is a nonzero element of A so it
won’t affect the leading index. Then by the construction of the mI there is some mℓ with
leading index Jℓ ≤ Iℓ ; let aℓ be the leading coefficient of mℓ . Define
yℓ = aℓ λ(cℓ + ̟xℓ ) − ̟eIℓ mℓ T Iℓ −Jℓ ∈ m.
This has been chosen so that the coefficient of T Iℓ in yℓ is 0 and |ψ(yℓ )| ≤ 1. Let
xℓ+1 =

λ−1 yℓ − aℓ cℓ
= aℓ xℓ − λ−1 eIℓ mℓ T Iℓ −Jℓ ,
̟

cℓ+1 = aℓ cℓ .

Clearly cℓ+1 ∈ S, cℓ+1 + ̟xℓ+1 = λ−1 yℓ+1 ∈ m, and |ψ(xℓ+1 )| ≤ |ψ(xℓ )|.
As  is a well ordering, we have a bijection between indices of x and positive integers call the mth index Im . As ψ(xℓ ) is a convergent power series, there are only finitely many
terms of ψ(xℓ ) with coefficient norm greater than ǫ|ψ(x)|. We can therefore associate a
unique integer nℓ to each ψ(xℓ ) such that the mth term in the binary representation of
nℓ is 1 exactly when the coefficient of T Im in ψ(xℓ ) is greater than ǫ|ψ(x)|. We claim
that nℓ > nℓ+1 whenever nℓ > 0, so after finitely many steps we must have nk = 0. By
definition, this means that after finitely many steps every term of ψ(xk ) will have coefficient
with norm at most ǫ|ψ(x)|, so |ψ(xk )| ≤ ǫ|ψ(x)| as desired.
By the construction of ǫ, adding the multiple of mℓ required to go from xℓ to xℓ+1
won’t introduce any coefficients with norm greater than ǫ|ψ(x)| and index J ≻ Iℓ . By the
construction of xℓ+1 , the coefficient of T Iℓ in xℓ+1 is 0. So when we move from nℓ to nℓ+1 ,
the digit corresponding to Iℓ is changed from 1 to 0 and no higher digits are changed. So
nℓ > nℓ+1 as desired.

Starting with c0 = 1, we can iterate this process to get sequences {cℓ } ⊂ S, {xℓ } ⊂
A{T1 , . . . , Tn } so that for all ℓ, yℓ := cℓ + ̟xℓ ∈ m and |ψ(yℓ )| → 0. As the inverse of ψ is
bounded, we must have |yℓ | → 0. This implies that |cℓ + ̟xℓ,0 | → 0 as this is the constant
term of yℓ , which implies that for ℓ large cℓ − ̟xℓ,0 ∈ IA . This is a contradiction as we
chose ̟ so that cℓ is never divisible by ̟ in oA /IA .

Remark 8.8. We note that in our proof, it was essential that we could scale elements
of A{T1 /ρ1 , . . . , Tn /ρn } by elements of A× to get elements of norm 1. The result isn’t
generally true if we allow infinite extensions in the value group. For example, if we let
A = Qp {T1 /ρ} where ρ is irrational, the ring A{T2 /ρ−1 } has (T1 T2 − 1) as a maximal ideal,
but A ∩ (T1 T2 − 1) = {0}. This is analogous to the more standard example of the maximal
ideal (px − 1) of Zp [x].
I {T /ρ′ , . . . , T /ρ′ }
Remark 8.9. This result can be extended to some rings of the form BL,E
n m
1 1
by using Lemma [11, 4.9] to rewrite them as quotients of some ArL,E {T1 /ρ1 , . . . , Tn /ρn }.

PROPERTIES OF EXTENDED ROBBA RINGS

16

I , but it will
As mentioned in Remark 8.8, this argument will not hold for all of the BL,E
I , then it will also
work for intervals I = [s, r] with s ∈ Q. If the result does hold for BL,E
hold for any étale extension C.
I {T , . . . , T } are regular for I as in
Corollary 8.10. The rings ArL,E {T1 , . . . , Tn }, BL,E
1
n
Remark 8.9.
I {T , . . . , T }, the other case follows similarly. We must
Proof. We just show this for BL,E
1
n
I
show that for any maximal ideal m ⊂ BL,E {T1 , . . . , Tn }, the localization at m is a regular
I
local ring. By Theorem 8.3, m ∩ BL,E
= (m) for some maximal ideal (m) of the principal
I
I /(m) is a nonarchimedean field; we must check
ideal domain BL,E . We claim that BL,E
I /(m)) is a single point. By [11, Lemma 7.10], M(B I /(m)) is a finite
that M(BL,E
L,E
I /(m))
discrete topological space. By [13, Proposition 2.6.4], any disconnect of M(BL,E
I /(m). As B I /(m) is a field, this is impossible so
would induce a disconnect of BL,E
L,E
I
I /(m)){T , . . . , T } is therefore
M(BL,E /(m)) must be a point as desired. The ring (BL,E
1
n
a classical affinoid algebra so it is regular [15]. The result now follows from a general
commutative algebra statement: given a local ring (R, m) and an element m ∈ m \ m2 , if
R/(m) is regular then so is R. This is clear as in passing from R to R/(m), the dimension
of the local ring is reduced by 1 by Krull’s principal ideal theorem and the k-dimension of
m/m2 is reduced by 1 by our assumption on m.


Using similar ideas, we show that ArL,E is strictly noetherian. We first recall the necessary
definitions.
Definition 8.11. Let (A, A+ ) be a Huber pair with A Tate. An A+ -module N is almost
finitely generated if for every topologically nilpotent unit u in A, there is a finitely generated
A+ -submodule N ′ of N such that uN is contained in N ′ .
Definition 8.12. A Huber pair (A, A+ ) is strictly noetherian if for every finite A+ -module
M , every A+ -submodule N of M is almost finitely generated.
Remark 8.13. We note that if (A, A+ ) is strictly noetherian, A must be noetherian.
Given an ideal H of A and topologically nilpotent unit u, u(H ∩ A+ ) is contained in some
finitely generated ideal hx1 , . . . , xn i of A+ . For any h ∈ H, multiplying
by a sufficiently
P
large power P
of u will give un h ∈ u(H ∩ A+ ), so we have un h =
ai xi with the ai in A+ ,
and so h =
u−n ai xi and so the xi generate H.
Remark 8.14. Kiehl was the first to consider the strict noetherian property, showing that
affinoid algebras are strictly noetherian in [16, Satz 5.1].
Proposition 8.15. For any nonnegative integer n and ρ1 , . . . , ρn ∈ R>0 , the pair (R, R◦ ) :=
(ArL,E {T1 /ρ1 , . . . , Tn /ρn }, ArL,E {T1 /ρ1 , . . . , Tn /ρn }◦ ) is strictly noetherian.
Proof. Quotients and direct sums preserve the almost finitely generated property, so it is
enough to check that every ideal H of R◦ is almost finitely generated. Fix any ideal H ⊂ R◦
and topologically nilpotent unit u, we will construct a finitely generated ideal H ′ ⊂ H such

PROPERTIES OF EXTENDED ROBBA RINGS

17

that uH ⊂ H ′ . Exactly following the construction of [11, Theorem 3.2] gives a finite subset
{xI } of HPsuch that for all y ∈ H, there exist aI ∈ R such that |aI |ρ |xI |ρ ≤ |y|ρ for all I
and y =
aI xI . Letting δ := min{|xI |ρ }, we see that if |y|ρ ≤ δ we have aI ∈ R◦ for all
I. The set {xI } therefore generates all of the elements of uH with norm at most δ.
Let c = |u|ρ , let m = ⌈logc δ⌉. As u is topologically nilpotent we have c < 1. For
each index I and k ∈ {1, . . . , m}, let dI,k be the smallest possible degree of the leading
coefficient of an element of H with leading index I and weighted Gauss norm ck . Following
[11, Definition 3.7], for each nonnegative integer d and k ∈ {1, . . . , n}, define Sd,k to be the
(finite) set of I which are minimal with respect to ≤ for the property that dI,k = d and let
Sk be the union of the Sd,k . For each I ∈ Sk , choose xI,k ∈ H \ {0} with leading index I,
weighted Gauss norm ck , and leading coefficient cI,k of degree dI .
We claim that the finite set {xI } ∪ {xI,1 : I ∈ S1 } ∪ · · · ∪ {xI,m : I ∈ Sm } generates every
element yP∈ uH. In the original proof, the key property of the chosen generators is that for
′
any y =
yJ T J in the ideal with leading term yJ ′ T J , there is some xI such that I ≤ J ′
and deg(cI ) ≤ deg(yJ ′ ). This allows for a series of approximations that can be shown to
converge using the fact that ArL,E is a Euclidean domain.
This argument continues to hold in our case, but we must also use generators with norm
at least that of y so that at each step of the approximation we are multiplying xI by an
′
element of R◦ . Let j = ⌈logc (|y|ρ )⌉ and let yJ ′ T J be the leading term of y. If j > m,
|y|ρ ≤ δ so y can be generated by elements of {xI }. Otherwise, we have u−1 y ∈ H and
|u−1 y|ρ > ck , so there is some unit v ∈ ArL,E ∩ R◦ with |vu−1 y|ρ = ck . Multiplication in R
by units will not change leading indices or leading degrees, so by construction, we can find
some xI,j with I ≤ J ′ and deg(cI,j ) ≤ deg(yJ ′ ). As |xI,j |ρ = ck ≥ |y|ρ , this is the desired
element, the rest of the proof is identical to [11].

I , C are strictly noetherian.
Corollary 8.16. The rings BL,E

Proof. By [11, Lemma 4.9] and Hypothesis 7.1, these rings are quotients of some ring of
the form ArL,E {T1 /ρ1 , . . . , Tn /ρn }.

9. Excellence
I , and C are excellent when
Finally, we show that the rings ArL,E {T1 /ρ1 , . . . , Tn /ρn }, BL,E
the ρi are chosen as in Theorem 8.3, adapting the argument of [20, Theorem 101]. The idea
∂
of ArL,E {T1 /ρ1 , . . . , Tn /ρn } give us derivations which
is that the n partial derivatives ∂T
i
satisfy a Jacobian criterion that implies excellence.

Remark 9.1. The methods of this section only work for rings of characteristic 0, but if E is
characteristic p our rings will also be characteristic p. In this case, ArL,E {T1 /ρ1 , . . . , Tn /ρn }
is a ring of convergent power series over L. As L is perfect of characteristic p, ArL,E {T1 /ρ1 , . . . , Tn /ρn }
has a finite p-basis given by ̟, T1 , . . . , Tn . It is therefore excellent by a theorem of Kunz
[17, Theorem 2.5]. Excellence of the other rings of interest follow as in the characteristic 0
case, see Corollary 9.15.

PROPERTIES OF EXTENDED ROBBA RINGS

18

We begin the mixed characteristic case by recalling some results from Matsumura [20,
Sections 32, 40].
Definition 9.2. A noetherian ring R is J-0 if Reg(Spec(R)) contains a non-empty open
subset of Spec(R), and J-1 if Reg(Spec(R)) is open in Spec(R).
Lemma 9.3. For a noetherian ring R, the following conditions are equivalent:
(1) Any finitely generated R-algebra S is J-1;
(2) Any finite R-algebra S is J-1;
(3) For any p ∈ Spec(R), and for any finite radical extension K ′ of the residue field
κ(p), there exists a finite R-algebra R′ satisfying R/p ⊆ R′ ⊆ K ′ which is J-0 and
whose quotient field is K ′ .
If these conditions are satisfied, we say that R is J-2.
Proof. [20, Theorem 73]



Corollary 9.4. Given a noetherian ring R containing Q, if R/p is J-1 for all p ∈ Spec(R)
then R is J-2.
Proof. By [20, Chapter 32, Lemma 1], the first condition in Lemma 9.3 is equivalent to
the following condition: Let S be a domain which is finitely generated over R/p for some
p ∈ Spec(R), then S is J-0. Let k and k′ be the quotient fields of R and S respectively. As
Q ⊂ R, k′ is a separable extension of k by [20, Paragraph 23.E]. This is now Case 1 of the
proof of [20, Theorem 73], so the result follows.

Our goal is to understand when quotients of a regular local ring are again regular. The
following lemmas explain how to use derivations to do this. We first set some notation,
following [20, Section 40].
Given a ring A, elements x1 , . . . , xr ∈ A, and derivations D1 , . . . , Ds ∈ Der(A), we write
J(x1 , . . . , xr ; D1 , . . . , Ds ) for the Jacobian matrix (Di xj ). Given a prime ideal p ⊂ A, we
write J(x1 , . . . , xr ; D1 , . . . , Ds )(p) for the reduction of the Jacobian mod p. If p contains
x1 , . . . , xr , the rank of the Jacobian mod p depends only on the ideal I generated by the
xi , so we denote it rank J(I; D1 , . . . , Ds )(p). Given a set ∆ of derivations of A, we define
rank J(I; ∆)(p) to be the supremum of rank J(I; D1 , . . . , Ds )(p) over all finite subsets
{D1 . . . , Ds } ⊂ ∆.
Lemma 9.5. Let (R, m) be a regular local ring, let p be a prime ideal of height r and ∆
be a subset of Der(R). Then:
(1) rank J(p; ∆)(m) ≤ rank J(p; ∆)(p) ≤ r,
(2) if rank J(f1 , . . . , fr ; D1 , . . . , Dr )(m) = r and f1 , . . . , fr ∈ p, then p = (f1 , . . . , fr )
and R/p is regular.
Proof. [20, Theorem 94]



Lemma 9.6. Let R, p, and ∆ be as in the preceding lemma. Then the following two
conditions are equivalent:
(1) rank J(p; ∆)(p) = ht p,

PROPERTIES OF EXTENDED ROBBA RINGS

19

(2) let q be a prime ideal contained in p, then Rp /qRp is regular if and only if rank J(q; ∆)(p) =
ht q.
Proof. [20, Theorem 95]



Definition 9.7. The weak Jacobian condition holds in a regular ring R if for every p in
Spec(R), rank J(p; Der(R))(p) = ht p. In this case we say that (WJ) holds in R.
Mizutani and Nomura showed that rings satisfying (WJ) and containing Q are excellent:
their proof is [20, Theorem 101]. A key step in the proof is the following proposition, we
go through it in detail as we will adapt it when showing that A{T1 , . . . , Tn } is excellent.
Proposition 9.8. Every regular ring R satisfying (WJ) is J-2.
Proof. This is roughly the argument of [20, Paragraph 40.D]. By Corollary 9.4, it is enough
to show that for every q ∈ Spec(R), the set Reg(R/q) is open in Spec(R/q). Fix any prime
p ⊇ q such that the image P of p in Spec(R/q) is regular. We will construct an open set
around P contained in Reg(R/q).
As (WJ) holds in R, we have rank J(p; Der(R))(p) = rank J(p; Der(Rp )) = ht(p). By
Lemma 9.6, this implies that rank J(q; Der(Rp ))(p) = ht(q) as Rp /qRp is regular. Let
r = ht(q), then we have f1 , . . . , fr ∈ q and D1 , . . . , Dr ∈ Der(R) such that det(Di fj ) 6∈ (p).
By Lemma 9.5, this implies that qRp = (f1 , . . . , fr )Rp . We therefore have some g ∈ R − p
such that qRg = (f1 , . . . , fr )Rg . Let h = det(Di fj ). By definition g and h are not in p or
q, so the reduction gh is nonzero in R/q. For any prime p′ reducing to some P ′ ∈ D(gh) ⊂
Spec(R/q), we have rank J(f1 , . . . , fr ; D1 , . . . , Dr )(p′ ) = r as h 6∈ P ′ , so by Lemma 9.5
ht(qRp′ ) ≥ r. As g 6∈ P ′ , f1 , . . . , fr generate qRp′ so ht(qRp′ ) ≤ r. So ht(qRp′ ) = r and we
can apply Lemma 9.5 to see that Rp′ /qRp′ is regular. So Reg(R/qR) contains the open set
D(gh) containing P , so it is open in Spec(R/qR) as desired.

We can now state the hypotheses we need to prove excellence.
Hypothesis 9.9. Let A be a regular integral domain containing Q, let R be a ring such that
∂
. Assume
A[T1 , . . . , Tn ] ⊂ R ⊂ A[[T1 , . . . , Tn ]] and that is stable under the n derivatives ∂T
i
that (R, A) satisfies the Nullstellensatz condition of Definition 8.1 and that R ⊗A Frac(A)
is weakly Jacobian as in Definition 9.7.
It is clear that for (WJ) to hold in R we must have dimR (Der(R)) ≥ dim(R). We
have n natural derivations to work with in our setup, so we’d prefer to work with rings
of dimension at most n. We therefore tensor with Frac(A) to reduce the dimension to
something that (WJ) can apply to. We now check that the rings of interest satisfy our
hypothesis.
Proposition 9.10. Let A = ArL,E and R = ArL,E {T1 /ρ1 , . . . , Tn /ρn }, then Hypothesis 9.9
is satisfied.
Proof. Everything but the weak Jacobian condition has already been checked or is clear.
Choose p ∈ Spec(R ⊗A Frac(A)), let the height of p be h. Let p′ ∈ Spec(R) be the

PROPERTIES OF EXTENDED ROBBA RINGS

20

contraction of p, it also has height h. Let q ∈ Spec(R) be a maximal ideal containing p′ ,
let Q = Rq /p′ , this is a local ring of dimension n + 1 − q.
Let ∆ be the derivations of R induced by derivations of Der(R ⊗A K)), then ∆ is
∂
. The derivations in ∆ are exactly the A-derivations of R.
generated by the n elements ∂T
i
We have rank J(p; Der(R ⊗A K)) = rank J(p′ ; ∆) ≤ h by Lemma 9.5; we must show that
we have equality. Following the proof of [20, Theorem 100], we see that rank DerA (Q) =
n − rank J(p′ ; ∆), so it is enough to show that rank DerA (Q) = n − h.
To do this, we largely follow [20, Theorem 98]. By Theorem 8.3, we have q∩ A = (x0 ) for
a principal maximal ideal (x0 ). We extend x0 to a system of parameters x0 , . . . , xr of Q,
b is an integral extension
here r = n − h. By the Cohen structure theorem, we have that Q
of F [[x0 , . . . , xr ]] where F = R/q. Writing F = (R/(x0 ))/(q/(x0 )), we see that it is the
quotient of an affinoid algebra over the field A/(x0 ), so it is an integral extension.
b Then D
Take any A-linear derivation D of Q vanishing on x1 , . . . , xr and extend it to Q.
vanishes on x0 as x0 ∈ A, and it vanishes on F as it vanishes on A/(x0 ) and F is an integral
b
extension of A/(x0 ). So D vanishes on all of F [[x0 , . . . , xr ]], so it must also vanish on Q
and therefore Q. Indeed, given any y ∈ Q, there is some integral relation f (T ) for y over
F [[x0 , . . . , xn ]] of minimal degree. Then 0 = D(f (y)) = f ′ (y)D(y) and f ′ (y) is nonzero as
the characteristic is 0, so D(y) = 0. So D is determined by the tuple (D(x1 ), . . . , D(xm )),
and so rank DerA (Q) ≤ n − h. As rank J(p′ ; ∆) ≤ h, we must have equalities in both
equations as desired.

Remark 9.11. Keeping notation as in Proposition 9.10, we note we could apply [20,
Theorem 100] to get (WJ) for R ⊗A Frac(A) if we knew that every maximal ideal m ⊂
R ⊗A Frac(A) has residue field algebraic over Frac(A). While this seems plausible, we
were unable to prove it directly. It is tempting to try to prove a version of Weierstrass
preparation and division for R ⊗A Frac(A), but following the standard proof for affinoid
algebras over a field doesn’t quite work.
Proposition 9.12. Any ring R satisfying Hypothesis 9.9 is J-2.
Proof. We proceed by induction on the dimension of A. As A is an integral domain, the
base case is when A is a field. By Proposition 9.10, R is (WJ) in this case, so by Proposition
9.8 it is J-2. For the inductive step, by Corollary 9.4 it is again enough to show that for
every q ∈ Spec(R), the set Reg(R/q) is open in Spec(R/q).
Let q ∩ A = Q, then we can reduce to the case where Q = {0} by noting that R/q ∼
=
(R/Q)/(q/Q) and that (A/Q)[T1 , . . . , Tn ] ⊂ R/Q ⊂ (A/Q)[[T1 , . . . , Tn ]]. In this case, q is
in the image of the injection Spec(R ⊗A Frac(A)) ֒→ Spec(R), so we can work with the
preimage q′ . By Hypothesis 9.9, (WJ) holds in R ⊗A Frac(A), so Proposition 9.8 implies
that this ring is J-2.
In particular, letting p = q′ in Proposition 9.8 we get a nonempty open set D(gh)
containing only regular primes. The construction of gh gives the required generators
f1 , . . . , fr ∈ p ⊗A Frac(A) and derivations D1 , . . . , Dr ∈ Der(R ⊗A Frac(A)) for Lemma
9.5 to apply. Finding a common denominator in Frac(A) gives an element d ∈ A such that
the entire argument works in Rd , so the set Reg((R/q)d ) is open.

PROPERTIES OF EXTENDED ROBBA RINGS

21

To complete the proof, we just need to show that
Reg(R ∩ V (d) ⊂ Spec(R}/q) ∩ V (d)
is open. This is equivalent to checking that Reg((R/d)/(q, d)) is open in Spec(R/d)/(q, d)).
We just need to check this for each of the finitely many minimal primes. As R/(d) satisfies
Hypothesis 9.9 for A = A/(d) and dim(A/d) < dim(A) these follow from the inductive
hypothesis.

Proposition 9.13. Any ring R satisfying Hypothesis 9.9 is a G-ring.
Proof. Here we are adapting [20, Theorem 101]. By [20, Theorem 75], it is enough to show
that for every maximal ideal m of R, the local ring Rm has geometrically regular formal
fibers. As Q ⊂ R, it is enough to check that the formal fibers are regular; the argument
is the same as in Corollary 9.4. Concretely, it is enough to show that for every prime
cm ), the local ring (R
cm )p /p is regular. As in Proposition 9.12, we can reduce to
p ∈ Spec(R
the case where p ∩ A = {0} by replacing A with A/(p ∩ A).
When p ∩ A = (0), we look at the image p′ of p in R ⊗A Frac(A). Here (WJ) holds
by Proposition 9.10, so we get derivations D1′ , . . . , Dr′ and f1′ , . . . , fr′ ∈ p ⊗A Frac(A) such
that rank J(f1′ , . . . , fr′ ; D1′ , . . . , Dr′ )(p′ ) = r. We can multiply by an element of A to clear
the denominators of the matrix and restrict the derivations to R. This gives f1 , . . . , fr ∈ p
with rank J(f1 , . . . , fr ; D1 , . . . , Dr )(p) = r.
cm and view the fi as elements of p(R
cm )P to get
We can extend the derivations to R
cm = ht p =
c
rank J(f1 , . . . , fr ; D1 , . . . , Dr )(p(Rm )P ) = r. By [20, Theorem 19], we have ht pR
cm )P /p is regular as desired.

r so we can again apply 9.5 to see that (R
Combining these gives the desired theorem.

Theorem 9.14. Any ring R satisfying Hypothesis 9.9 is excellent.
Proof. This follows from 8.10, 9.12, and 9.13.



I {T , . . . , T } are excellent.
Corollary 9.15. The rings ArL,E {T1 /ρ1 , . . . , Tn /ρn } and BL,E
1
n
As excellence is stable under passage to finitely generated algebras, this implies that the
rings C of Hypothesis 7.1 arising from étale morphisms are also excellent.

Corollary 9.16. The stalks of the adic Fargues-Fontaine curve are noetherian.
Proof. Temkin proved this for rigid analytic spaces, making essential use of the fact that
affinoid algebras are excellent. His proof works just as well for the Fargues-Fontaine curve
now that we have proven that the extended Robba rings are excellent. We give a brief
sketch of the proof, a more detailed version is given in Proposition [4, 15.1.1].
The local ring Ox at a point x of the Fargues-Fontaine curve can be written the direct
limit of a directed system (Ai ) of rational domains Spa(Ai , A+
i ) containing x where the Ai
are all extended Robba rings. Let m denote the maximal ideal of the local ring Ox , and let
mi ∈ Spec(Ai ) be the image of m in the map Spec(A) → Spec(Ai ) coming from the direct
limit. Letting Bi = (Ai )mi , the directed system of the Bi also has limit Ox .

PROPERTIES OF EXTENDED ROBBA RINGS

22

Huber showed that the transition maps Ai → Aj are flat [8, II.1.iv]. The directed system
(Bi ) therefore consists of local noetherian rings with flat local transition maps, it is shown
in EGA that the limit is noetherian if for sufficiently large i, we have mi Bj = mj for all
j ≥ i. As the transition maps are flat, we have
dim(Bj ) = dim(Bi ) + dim(Bj /mi Bj ) ≥ dim(Bi )
for j ≥ i. As the dimension of the Bj is bounded above, we must have some i0 such that
dim(Bi ) = dim(Bi0 ) for all i ≥ i0 and so dim(Bi /mi0 Bi ) = 0 for i ≥ i0 .
So it is enough to show that Bi /mi0 Bi must be reduced. This is the localization of a

fiber algebra of a map of excellent rings, so it is reduced by the argument in [4].
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