Slope filtrations in families

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by  Ruochuan Liu

Slope Filtrations in Families
Ruochuan Liu
8 Department of Mathematics, Room 2-229
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0 Massachusetts Institute of Technology
2
77 Massachusetts Avenue
p
e Cambridge, MA 02139
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[email protected]
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] September 1, 2008
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A
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t Abstract
a
m
In [1], Berger and Colmez introduced a theory of families of ´etale (ϕ,Γ)-modules
[ associatedtofamiliesofp-adicGaloisrepresentations. Inspiredbytheirworks,weconsider

1 families of φ-modules over a reduced affinoid algebra where the base is fixed by Frobenius
v actions. Weprovesomeresultsinthisframeworkincludingthesemicontinuity ofvariation
1
of HN-polygons. These results will be applied to families of (ϕ,Γ)-modules in subsequent
3
3 works.
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.
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0 Contents
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: 1 Slope theory of φ-modules over the Robba ring 3
v
1.1 The Robba ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
i
X
1.2 Slope theory of φ-modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
r
a
2 Generic and special slope filtrations 9
2.1 Dieudonn´e-Manin decompositions . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Generic slope filtrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Comparison of polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Variation of HN-polygons 13
3.1 Local constancy of generic HN-polygons of families of overconvergent φ-
modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Semicontinuity of HN-polygons of families of φ-modules . . . . . . . . . . 15
3.3 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1
Introduction
The slope filtration theorem gives a partial analogue, for a Frobenius module over the
Robba ring, of the eigenspace decomposition of a linear transformation. It was originally
introduced in the context of rigid cohomology by Kedlaya as the key ingredient in his
proof of the Crew’s conjecture. It also has important applications in p-adic Hodge theory
via Berger’s construction of the (ϕ,Γ)-modules associated to Galois representations. For
instance, it allows Berger to prove Fontaine’s conjecture that de Rham implies potentially
semistable and to give an alternate proof of the Colmez-Fontaine theorem that weakly
admissible implies admissible.
In [1], Berger and Colmez introduced a theory of families of ´etale (ϕ,Γ)-modules
associatedtofamiliesofp-adicGaloisrepresentations. Inspiredbytheirworks,weconsider
families of φ-modules over a reduced affinoid algebra A, i.e. φ-modules over the Robba
ring with coefficients in A. In fact, in rigid cohomology [5] one is also led to study such
families of φ-modules. The difference between these two cases is that in rigid cohomology
the Frobenius acts on A as a lift of a p-power map, but in families of (ϕ,Γ)-modules the
Frobenius does not move the base at all. The families we consider fit the latter setting.
Namely, we assume that φ(p) ⊆ p for any prime ideal p of A, and that φ induces an
isometric endomorphism on A/p.
In this paper we attempt to explore the variation of slope filtrations of families of φ-
modules. Recall thattheoriginalslopetheory requiresthecoefficient fieldstobecomplete
for a discrete valuation. In the course of the work, we initially get a partial extension of
the original slope theory to the spherically complete coefficient fields case. Unless stated
otherwise, K is a field of characteristic 0 and complete for a non-archimedean valuation
v which is not necessarily discrete. We make the assumption that the residue field is of
K
characteristic p.
Theorem 0.1. For K spherically complete, every φ-module over R admits a unique
K
Harder-Narasimhan filtration.
We expect that the semistability is preserved by tensor products in this case. This
would yield a further extension of the original slope theory,
Beforewestateotherresultsofthispaper,weintroducesomenotationsanddefinitions.
Let M(A) denote the affinoid space associated to A. For M a family of φ-modules over
A
A and x ∈ M(A), we set the specialization of M at x as M = M ⊗ R ; this is a
A x A RA k(x)
φ-module over R . In general, we specialize M to any Berkovich point x by setting
k(x) A
M = M ⊗ R , where H(x) is the completion of the residue field of x. For K
x A RA H(x)
discrete and M a φ-module over R , by a model of M we mean a φ-submodule N over
K
Rbd such that N ⊗ R = M. We say it is a good model if the generic HN-polygon of
K Rbd K
K
N is the same as the HN-polygon of M. By a model of M we mean a φ-submodule N
A A
over Rbd such that N ⊗ R = M . We say it is a good model if N is a good model
A A Rbd A A x
A
of M for any x ∈ M(A). We say M (resp. N ) is a family of ´etale φ-modules (resp.
x A A
overconvergent φ-modules) if M (resp. N ) is ´etale at every x ∈ M(A). We say M
x x A
(resp. N ) is globally ´etale if M (resp. N ) has a φ-stable finite free Rint-submodule
A A A A
N′ such that φ∗N′ ∼= N′ and N′ ⊗ R = M (resp. N′ ⊗ Rbd = N ). More
A A A A Rint A A A Rint A A
A A
generally, we say M is globally pure of slope s, where s = c/d for coprime integers c,d
A
2
and d> 0, if ([d] M)⊗R (−c) is globally ´etale. It is obvious that if M is pure of slope
∗ A A
s, then M is pure of slope s for any x∈ M(A).
x
Our main results are the following theorems.
Theorem 0.2. (Semicontinuity of HN-polygons) Let M be a family of φ-modules over
A
A. Then for any x ∈ M(A), there is an affinoid neighborhood M(B) of x such that for
any y ∈ M(B) the HN-polygon of M lies above the HN-polygon of M and has the same
y x
endpoints.
Theorem 0.3. For M a family of φ-modules over M(A), if M is pure of slope s for
A x
some x ∈ M(A), then there exists an affinoid neighborhood M(B) of x such that M is
B
globally pure of slope s over M(B). Let NB be a good model of MB. If NB′ is a good
model for another affinoid neighborhood M(B′) of x, then x has an affinoid neighborhood
M(B′′) in M(B)∩M(B′) such that NB ⊗RbBd RbBd′′ = NB′ ⊗RbBd′ RbBd′′.
Theorem 0.2 is the strongest result we can expect in some sense. In fact, at the end
of this paper, we construct a family of rank 2 φ-modules M over the p-adic closed
B(O,1)
unit disk B(O,1) such that for any Berkovich point x with H(x) discrete, M is ´etale if
x
x is not O, and M has slopes (−1,1). Hence the HN-polygons are not continuous at O
O
over either classical points or Berkovich points.
In subsequent works we will apply these results to families of (ϕ,Γ)-modules.
Acknowledgement
I am grateful to my advisor, Kiran S. Kedlaya, for suggesting the topic of this paper, and
also for his useful comments and suggestions. In this last regard, thanks are due as well
to Liang Xiao. Thanks to Chris Davis for the help on English; any remaining mistakes
are my own. Thanks to the Institut de Math´ematiques de Jussieu, Paris for their kind
hospitality. Whenwritingthispaper,theauthorwasfoundedbyKedlaya’sNSFCAREER
grant DMS-0545904.
1 Slope theory of φ-modules over the Robba ring
This chapter contains some basic definitions and facts from the slope theory of φ-modules
over the Robba ring. The novel feature of our treatment is that we allow the coefficient
field to benon-discrete. We prove some new results in this case, particularly the existence
of HN filtrations for φ-modules over Robba rings with spherically complete coefficient
fields.
1.1 The Robba ring
As before, K is a field of characteristic 0 and complete for a non-archimedean valuation
v which is not necessarily discrete. Let O denote the valuation subring of K. Let m
K K K
denote the maximal ideal of O . Theresidue field k = O /m is of characteristic p. The
K K K
valuation v is normalized so that v (p) = 1 and the corresponding norm |·| is defined
K K
3
as |·| = p−vK(·). We extend v and |·| uniquely to a valuation and a norm of K which
K
are still denoted by v and |·|. Let A be a commutative Banach algebra over K and let
K b
M(A) be the spectrum of A in the sense of Berkovich [2].
Definition 1.1. For any subinterval I ⊂ (0,∞], define
+∞
RI = {f = a Ti, where a ∈ K and f(T) is convergent on v (T)∈ I}.
K i i K
i=X−∞
Equivalently, RI is the ring of rigid analytic functions on the annulus v (T) ∈ I over
K K
K. For any s ∈ I, we define a valuation w on RI as w (f) = v (a )+si. We set the
s,i K s,i K k
valuationws onRIK asws(f)= infi∈Z{ws,i(f)}. Thecorrespondingnormis|f|s = p−ws(f).
Note that |·| is multiplicative for any s ∈ I. Geometrically, |f| is the maximal value of
s s
f on the circle |x| = p−s. It is clear that RI is Fr´echet complete with respect to w for
K s
s ∈ I. For I = (0,r], we also use Rr instead of R(0,r]. Let R be the union of Rr for
K K K K
all r > 0. The ring R is called the Robba ring over K.
K
Definition 1.2. Let Rint be the subring of R consisting of series with coefficients in
K K
O . Let Rbd be the subring of R consisting of series with bounded coefficients. We
K K K
equip Rbd with a valuation w by setting w(f) = inf {v (a )}; the corresponding norm
K i∈Z K i
is |f| = sup |a |. By the following proposition, w is additive. If v is discrete, then
i∈Z i K
Rint is a Henselian local ring with residue field k((T)), and Rbd is the fraction field of
K K
Rint [4].
K
Proposition 1.3. For u,v ∈ Rbd, we have w(uv) = w(u)+w(v).
K
Proof. Wefirstshowthatforanyf = +∞ a Ti ∈ Rbd,wehavelim w (f)= w(f).
i=−∞ i K r→0+ r
Foranyǫ > 0,picki0 suchthatw(f) ≤PvK(ai0)< w(f)+2ǫ;henceforr ≤ r0 = 2ǫi0,wehave
w (f)≤ ri +v (a ) < w(f)+ǫ. ChooseN ∈ Nsufficientlylargesuchthatr i+v (a )≥
r 0 K i0 0 K i
w(f)foranyi≤ −N. Foranyr ≤ r = min{r , ǫ },ifi≤ −N,thenwehaveri+v (a )≥
1 0 N K i
r i+v(a ) ≥ w(f); if i≥ −N, then we have ri+v (a )≥ w(f)−rN ≥ w(f)−ǫ. So we
0 i K i
get that |w (f)−w(f)| ≤ ǫ for any 0< r ≤ r . This shows that lim w (f)= w(f). It
r 0 r→0+ r
followsthatw(uv) = lim w (uv) = lim w (u)+lim w (v) = w(u)+w(v).
r→0+ r r→0+ r r→0+ r
Remark 1.4. If v is discrete, we have that the units in R are precisely the nonzero
K K
elements of Rbd by Lazard’s work [8]. If v is not discrete, Rbd is no longer a field, let
K K K
alone the fraction field of Rint. But we still have that the units of R are contained in
K K
Rbd by the following proposition.
K
Proposition 1.5. The units of R are contained in Rbd.
K K
Proof. Let f = a Ti be a unit of R . Consider the Newton polygon P of f, i.e. the
i∈Z i K
boundary of thePlower convex hull of the set of points (−i,vK(ai)) ∈ R2. Then the slopes
of P are exactly the valuations of the roots of f [8]. Since f is a unit in R , it has only
K
finitely many roots in the open unit disk. Hence P has only finitely many positive slopes.
Let r be the smallest one. Suppose the two vertices of the segment having slope r are
(−i,vK(ai)),(−i′,vK(ai′)) with i > i′. Then it is clear that vK(aj) > vK(ai) if j < i and
v (a ) ≤ v (a ) if j > i. Hence f lies in Rbd.
K j K i K
4
Similarly, we define the Robba ring over the Banach algebra A as follows.
Definition 1.6. For any subinterval I ⊂ (0,∞], we set
+∞
RI = {f = a Ti, where a ∈ A and f(T) is convergent on v (T) ∈ I}.
A i i K
i=X−∞
For any s ∈ I, we define a valuation w on RI as w (f) = v (a ) + is. We set
s,i A s,i A k
the valuation w on RI as w (f) = inf {v (a ) + is}. The corresponding norm is
s A s i∈Z A i
|f| = p−ws(f). It is clear that RI is Fr´echet complete with respect to w for s ∈ I. For
s A s
general A, |·| is no longer multiplicative; instead we have that w (fg) ≥w (f)+w (g),
s s s s
|fg| ≤ |f| |g| for any f, g ∈ RI . Similarly, for I = (0,r], we use Rr instead of R(0,r].
s s s A A A
We let R be the union of Rr for all r > 0. It is called the Robba ring over A.
A A
Definition 1.7. Let Rint be the subring of R consisting of series with coefficients hav-
A A
ing norm at most 1. Let Rbd be the subring of R consisting of series with bounded
A A
coefficients. For f ∈ Rbd, define w(f) = inf {v (a )} and |f| = sup {|a |}. Then
A i∈Z A i i∈Z i
Rint = {f ∈Rbd,w(f) ≥ 0}.
A A
Let Rint,r denote the intersection of Rint and Rr .
A A A
Definition 1.8. Fix an integer q > 1. A relative (q-power) Frobenius lift on the Robba
ring R (resp. R ) is a homomorphism φ :R → R (resp. φ: R → R ) of the form
K A K K A A
+∞ a Ti 7→ +∞ φ (a )Si
i=−∞ i i=−∞ K i
P P
(resp. +∞ a Ti 7→ +∞ φ (a )Si), where φ (resp. φ ) is an isometric endomor-
i=−∞ i i=−∞ A i K A
phism oPf K (resp. A) aPnd S ∈ Rint (resp. S ∈ Rint) is such that w(S −Tq)> 0. If k has
K A
characteristic p > 0 and q is a power of p, we define an absolute (q-power) Frobenius lift
as a relative Frobenius lift for which φ or φ is a q-power Frobenius lift.
K A
Convention 1.9. For any valuation v (resp. norm | · |) and a matrix A = (A ), we
ij
use v(A) (resp. |A|) to denote the minimal valuation (resp. maximal norm) among the
entries.
In the rest of the paper, we always equip K and A with isometries φ and φ re-
K A
spectively. They then become a φ-field and a φ-ring respectively in the sense of following
definition.
Definition 1.10. Define a φ-ring/field to be a ring/field R equipped with an endomor-
phism φ; we say R is inversive if φ is bijective. Define a (strict) φ-module over a φ-ring
R to be a finite free R-module M equipped with an isomorphism φ∗M → M, which we
also think of as a semilinear φ-action on M; the semilinearity means that for r ∈ R and
m ∈ M, φ(rm) = φ(r)φ(m). Note that thecategory of φ-modules admits tensor products,
symmetric and exterior powers, and duals.
Fromnowon,ifRisaφ-ring/field,thenwealwaysuseφ todenotetheendomorphism
R
φ from the definition. For a φ-subring of a φ-ring R, we mean a subring of R stable under
φ with the restricted φ-action.
5
Definition 1.11. We can view a φ-module M over a φ-ring R as a left module over
the twisted polynomial ring R{X}. For a positive integer a, define the a-pushforward
functor [a] from φ-modules to φa-modules to be the restriction along the inclusion
∗
R{Xa} ֒→ R{X}. Define the a-pullback functor [a]∗ from φa-modules to φ-modules to be
the extension of scalars functor M → M ⊗ R{X}.
R{Xa}
In therest of the paper,a φ-field is always complete with anon-archimedean valuation
such that the φ-action is an isometric endomorphism. For a φ-subring, we mean a subring
with the induced φ-action and valuation. We always equip the Robba ring R (resp.
K
R ) with a Frobenius lift φ, and our main objects will be φ-modules over R (resp.
A K
R ). Note that by the definition of Frobenius lift, we have that φ maps Rbd or Rbd to
A K A
itself. Therefore we can also talk about φ-modules over Rbd (resp. Rbd). They are called
K A
overconvergent φ-modules.
1.2 Slope theory of φ-modules
Definition 1.12. Let M be a φ-module over R (resp. Rbd) of rank n. Then its top
K K
exterior power nM has rank 1 over R (resp. Rbd). Let v bea generator of nM, and
K K
suppose φ(v) =Vλv for some λ ∈ Rbd. Define the degree of M by setting deg(MV) = w(λ).
K
Note that this is independentof the choice of v because w is additive and φ is an isometry
on Rbd. If M is nonzero, define the slope of M by setting µ(M)= deg(M)/rank(M).
K
For c ∈ Z, we set R (c) as the rank 1 φ-module over R with a generator v such
K K
that φ(v) = pcv. For a φ-module M over R , we set M(c) as M ⊗R (c). The following
K K
formal properties are easily verified; see [6] for the proof.
(1) If 0 → M → M → M → 0 is exact, then deg(M) = deg(M )+deg(M ).
1 2 1 2
(2) We have µ(M ⊗M ) = µ(M )+µ(M ).
1 2 1 2
(3) We have deg(M∨) =−deg(M) and µ(M∨) = −µ(M).
(4) We have µ(M(c)) =µ(M)+c.
(5) If M is a φ-module, then µ([a] M)= aµ(M).
∗
(6) If M is a φa-module, then µ([a]∗M) = a−1µ(M).
Definition1.13. Wesayaφ-moduleM overR is(module-)semistable ifforanynonzero
K
φ-submodule N, we have µ(N) ≥ µ(M). We say M is (module-)stable if for any proper
nonzero φ-submodule N, we have µ(N) > µ(M). Note that both properties are preserved
under twisting (tensoring with a rank 1 module).
Lemma 1.14. For a∈ R , if there exists a λ ∈ R× such that w(λ) ≤ 0 and φ(a) = λa,
K K
then we have a∈ R×.
K
Proof. Recall that a is a unit in R if and only if there exists an r > 0 such that a
K
has no roots in the annulus 0 < v (T) < r. By [6, Proposition 1.2.6] (although this is
K
under the hypothesis that v is discrete, the proof works for the general v ), we get that
K K
a∈ Rbd. Note that there exists an r > 0 such that for any r ∈ (0,r ), φ induces a finite
K 0 0
´etale map of degree q from the annulus 0 < v (T) < r/q to the annulus 0 < v (T) < r
K K
6
and v (φ(x)) = qv (x) for 0 < v (x) < r/q. Now if a is not a unit in R , then it has
K K K K
roots c ,c ,··· in the open unit disk such that limv (c ) = 0. Moreover, by the theory
1 2 K i
of Newton polygons, we see that v (c ) < ∞ for some α since a∈ Rbd. Now
0<vK(ci)≤α K i K
we choose an r < min{α,r0} suchPthat λ has no roots in the annulus 0 < |T| < r. Pick a
root c of a in this annulus. For any c′ such that φ(c′) = φ (c), we have that it is a root
K
of φ(a), hence a root of a because φ(a) = λa. Now we can find q different roots c′ in the
annulus 0 < v (x) < r/q such that φ(c′) = φ (c); each of them has valuation v (c)/q.
K K K
Iterating the above process, we get q2 roots of a having valuations v (c)/q2, and so on.
K
But qi(v (c)/qi) = ∞; this yields a contradiction.
i∈N K
P
Remark 1.15. In the rest of the paper, we fix an r as in the above proof.
0
Proposition 1.16. Any rank 1 φ-module over R is stable.
K
Proof. Bytwisting, weonly needtoprovethisforthetrivialφ-moduleM ∼= R . Suppose
K
that N is a nonzero φ-submodule of M; we write N = R a for some a ∈ R . Then we
K K
havethatλ = φ(a)/a ∈ R×,andµ(N) = w(λ)bydefinition. Ifw(λ) = µ(N) ≤ µ(M)= 0,
K
then φ(a) = λa implies that a ∈R× by the above lemma. Hence N = M. In other words,
K
µ(N) > µ(M) unless N = M, as desired.
Corollary 1.17. Suppose that N ⊂ M are two φ-modules over R of the same rank;
K
then µ(N) ≥ µ(M), with equality if and only if N = M.
Proof. Suppose that rankM = n. Apply the above lemma to the inclusion nN ⊆
nM. V
V
Definition 1.18. Let M be a φ-module over R . A semistable filtration of M is a
K
filtration 0 = M ⊂ M ··· ⊂ M = M of M by saturated φ-submodules, such that
0 1 l
each successive quotient M /M is a semistable φ-module of some slope s . A Harder-
i i−1 i
Narasimhan (HN) filtration of M is a semistable filtration such that s < ··· <s . These
1 l
s are called the slopes of M.
i
Definition 1.19. LetM beaφ-moduleoverR . Definetheslope multiset ofasemistable
K
filtration of M as the multiset in which each slope of a successive quotient occurs with
multiplicity equal to the rank of that quotient. Define the slope polygon of this filtration
as the Newton-polygon of the slope multiset. For the definition of Newton polygon of a
multiset, we refer to [4, Definition 3.5.1].
Proposition 1.20. For K spherically complete, every φ-module over R has a unique
K
maximal φ-submodule of minimal slope, which is semistable.
Proof. LetM beaφ-moduleover R . From [8],R isaB´ezout domainforK spherically
K K
complete. It follows that the saturation of a φ-submodule of M and the sum of two φ-
submodules of M are still φ-submodules of M. So we can talk about the saturation of a
φ-submodule and the sum of two φ-submodules.
We will prove this proposition by induction on the rank of the φ-modules. For d = 1,
we have µ(N) ≥ µ(M) for any φ-submodule N of M, with equality if and only if N = M
by Corollary 1.17. Hence M itself is the desired submodule. Now suppose the theorem is
trueforφ-modulesoverR ofrank≤ d−1forsomed≥ 2. LetM bearankdφ-moduleof
K
7
slope s. If M is semistable, then we are done. Otherwise, let P be a φ-submoduleof slope
less than s and of maximal rank. Following Corollary 1.17, the saturation P˜ of P satisfies
µ(P˜) ≤ µ(P). Hence by passing to the saturation P˜, we suppose that P is saturated. If
rankP = d, then we have µ(M) = µ(P)< s. That is a contradiction. So P is a φ-module
of rank ≤ d−1. By the inductive hypothesis, P has a uniquemaximal φ-submoduleP of
1
minimalslope. WeclaimthatP isalsotheuniquemaximalφ-submoduleofM ofminimal
1
slope. In fact, suppose that Q is a φ-submodule of M such that µ(Q) ≤ µ(P ) < s. If
1
Q * P, consider the following exact sequence
0 −→ P ∩Q −→ P ⊕Q −→ P +Q −→ 0.
Sinceµ(P∩Q)≥ µ(P ) ≥ µ(Q), we concludethatµ(P+Q)≤ max{µ(P),µ(Q)} < s. But
1
rank(Q+P) > rankP, which contradicts the definition of P. We conclude that Q ⊂ P.
Therefore we must have µ(Q) = µ(P ) and Q ⊆ P by the definition of P . The induction
1 1 1
step is finished.
Theorem 1.21. For K spherically complete, every φ-module over R admits a unique
K
HN filtration.
Proof. This is a formal consequence of the above proposition. In fact, by the definition of
HN filtration, for any i≥ 1, M can be characterized as the unique maximal φ-submodule
i
of M/M of minimal slope.
i−1
Definition 1.22. Let M be a φ-module over R . If K is spherically complete, then
K
M admits a unique HN filtration by the above corollary. The slope polygon of the HN
filtration of M is called the HN-polygon of M.
Proposition 1.23. The HN-polygon lies above the slope polygon of any semistable filtra-
tion and has the same endpoint.
Proof. This is a formal consequence of the definition of HN filtration. We refer to [4,
Proposition 3.5.4] for a proof.
In case v is discrete, we can say more about semistable φ-modules over R .
K K
Definition 1.24. Let M be a φ-module over R (resp. Rbd). We say M is ´etale if M
K K
has a φ-stable Rint-submodule M′ such that φ∗M′ ∼= M′ and M′ ⊗ R = M (resp.
K Rint K
K
M′ ⊗ Rbd = M). More generally, suppose µ(M) = s = c/d, where c,d are coprime
Rint K
K
integers with d > 0. We say M is pure if for some φ-module N of rank 1 and degree −c,
([d] M)⊗N is ´etale.
∗
Itisnotdifficulttoprovethatapureφ-moduleisalwayssemistable[6,Theorem1.6.10]
(although the proof is under the hypothesis that v is discrete, it works for the general
K
case). But the converse is much more difficult. We know this for discrete v thanks to
K
the following theorem of Kedlaya [6].
Theorem 1.25. (Slope filtration theorem) In case v is discrete, every semistable φ-
K
module over R is pure. As a consequence, every φ-module M over R admits a unique
K K
filtration 0 = M ⊂ M ⊂ ··· ⊂ M = M by saturated φ-submodules whose successive
0 1 l
quotients are pure with µ(M /M ) <··· < µ(M /M ).
1 0 l l−1
8
It follows that for v discrete, the tensor product of two semistable φ-modules is
K
semistable.
Question 1.26. If K is merely spherically complete, do we have that semistable implies
pure? If this is not true, do we still have that the tensor product of two semistable
φ-modules is semistable?
2 Generic and special slope filtrations
In this chapter, we will freely use the language and results of the theory of difference
algebras, for which the readers may consult [7, Chapter 14]. In this setting, for a φ-ring
R the category of φ-modules over R is exactly the category of dualizable finite difference
modules over the difference algebra R.
2.1 Dieudonn´e-Manin decompositions
In this section, F is φ-field. We are concerned with the existence of Dieudonn´e-Manin
decompositions of φ-modules over F.
Definition 2.1. For λ ∈ F and d a positive integer, let V be the φ-module over F with
λ,d
basis e , ..., e such that φ(e ) = e , ..., φ(e ) = e , φ(e ) = e , φ(e ) = λe ;
1 d 1 2 d−2 d−1 d−1 d d 1
any such a basis is called a standard basis. For a φ-module V over F, a Dieudonn´e-
Manin decomposition of V is a direct sum decomposition V = ⊕n V into standard
i=1 λi,di
φ-submodules. The slope multiset of such a decomposition is the union of the multiple
sets consisting of v (λ )/d with multiplicity d for i= 1,...,n.
F i i i
Lemma 2.2. Suppose that F is spherically complete and that the residue field k is
F
strongly difference-closed. Then we have the following
(1) For any a ∈ F, there exists x∈ F with |x|≤ |a| and φ(x)−x= a;
(2) For any a ∈ F, there exists x∈ F with φ(x) = a.
Proof. Give F a partial order as follows. For x,y ∈ F, we say x > y if |φ(x)−x−a| <
|x−y| ≤ |φ(y)−y−a|. Suppose x < x < ... is a chain in the set {x | |x| ≤ |a|}. Put
1 2
r = |φ(x )−x −a|; then B(x ,r ) ⊇ B(x ,r ) ⊇ ···. By spherical completeness, we
i i i 1 1 2 2
have a point x inside all the discs; then |x| ≤ |a| and x > y for all i. Applying Zorn’s
i
lemma, we have a maximal x′ in the set {x | |x| ≤ |a|}; then it satisfies φ(x′)−x′ = a
because k is strongly difference-closed. We prove (2) similarly.
F
The following theorem generalizes [7, Theorem 14.6.3] to spherically complete fields.
Theorem 2.3. For spherically complete F, if the residue field k is strongly difference-
F
closed, then every φ-module over F admits a Dieudonn´e-Manin decomposition.
Proof. By the lemma above, we have that F is inversive. Let V be a d-dimensional φ-
moduleover F. We firstprove that if V is pureof norm 1, then V is trivial. This amounts
to showing that for any A ∈ GL (O ), there exists U ∈ GL (O ) such that U−1Aφ(U) =
d F d F
I . Since k is strongly difference-closed, we pick U such that |U−1Aφ(U )−I | = c< 1.
d F 1 1 1 d
Choosing π ∈ F such that |π| = c, we construct a sequence U ,U ,··· ∈ GL (O )
1 2 d F
inductively such that
9
U ≡ U and U−1Aφ(U )≡ I (mod πi)
i i+1 i i d
for every i. Given U , since U−1Aφ(U ) − I ≡ 0 (mod πi), by Lemma 2.2 we take
i i i d
X ∈M (O ) satisfying
i d F
φ(πiX )−πiX +(U−1Aφ(U )−I ) ≡ 0 (mod πi+1).
i i i i d
PutU = U (I +πiX );ashortcomputationshowsthatU−1Aφ(U ) ≡ I (mod πi+1).
i+1 i d i i+1 i+1 d
It follows from Lemma 2.2 that, if V is trivial, then H1(V) = V/(φ−1) = 0. By [7,
Theorem 14.4.13], we reduce the desired result to the case V is pure of norm s > 0. Let
m be the smallest positive integer such that sm ∈ |F×|, and choose a π ∈ F such that
|π| = sm. Then the above argument implies that π−1φm fixes some nonzero element of V;
this induces a nonzero map from V to V. This map is injective since V is irreducible
π,m π,m
by [7, Lemma 14.6.2]. Repeating this argument we have that V is a successive extension
of of copies of V . Finally observe that Ext1(V ,V ) = H1(V∨ ⊗V ) = 0 since
π,m π,m π,m π,m π,m
V∨ ⊗V is pure of norm 1, yielding the desired result.
π,m π,m
2.2 Generic slope filtrations
Throughout this section, K is discrete.
Definition2.4. LetE bethecompletionofRbd withrespecttow; theinducedvaluation
K K
onE isstilldenotedbyw. ThenE isjustthesetofLaurentseriesf = +∞ a Ti with
K K i=−∞ i
ai ∈ K, satisfyingtheconditions thatthe|ai|’s areboundedandthat |ai|P→ 0as i→ −∞.
It follows that E is a field. The valuation w can be written as w(f) = min {v (a )}.
K i∈Z K i
The φ-action on Rbd induces a φ-action on E . In this way, E is equipped with a φ-field
K K K
structure.
Definition2.5. ForS acommutativering,letS((uQ))denotetheHahn-Malcev-Neumann
algebra of generalized power series c ui, where each c ∈ S and the set of i with
i∈Q i i
ci 6= 0 is well-ordered (has no infiniPte decreasing subsequence); these series form a ring
under formal series multiplication, with a natural valuation v given by v( c ui) =
i∈Q i
min{i | ci 6= 0} P
Let R˜ , R˜bd be the extended Robba ring and the extended bounded Robba ring
K K
defined in [6].
Definition 2.6. Let E˜ be the ring of formal sums f = a ui with bounded coeffi-
K i∈Q i
cients in K satisfying the conditions that for each c> 0, thPeset of i ∈Q such that |ai| ≥ c
is well-ordered and that |a | → 0 as i → −∞. It is clear that E˜ is a field. Define the
i K
valuation w on E˜ by setting w(f) = min v (a ). Then E˜ is the completion of R˜bd
K i∈Q K i K K
with respect to w. Let O denote the ring of integers of E˜ . The residue field of O is
E˜K K E˜K
k((uQ)). We set the φ-action on E˜ as φ( a ui) = φ(a )uqi. In this way, E˜ is
K i∈Q i i∈Q i K
equipped with a φ-field structure. P P
Proposition 2.7. There exists a φ-equivariant isometric embedding ψ : E ֒→ E˜ . We
K K
thus view E as a φ-subfield of E˜ via ψ.
K K
10