On G-(φ, ∇)-modules over the Robba ring
Dissertation
zur Erlangung des akademischen Grades doctor rerum naturalium
(Dr. rer. nat.) im Fach Mathematik
eingereicht an der
Mathematisch-Naturwissenschaftlichen Fakult¨at der Humboldt-Universit¨at zu Berlin
von
M.Sc. Shuyang Ye
Pr¨asidentin der Humboldt-Universit¨at zu Berlin Prof. Dr.-Ing. Dr. Sabine Kunst
Dekan der Mathematisch-Naturwissenschaftlichen Fakult¨at Prof. Dr. Elmar Kulke
Gutachter:
1. Prof. Dr. Elmar Große-Kl¨onne (Humboldt-Universit¨at zu Berlin) 2. Prof. Dr. Laurent Berger (´Ecole Normale Sup´erieure de Lyon) 3. Dr. habil. Adriano Marmora (Universit´e de Strasbourg) Tag der m¨undliche Pr¨ufung: 10. 07. 2019
Dedicated to my grandparents
Abstract
LetKbe a finite extension ofQpand letRbe the Robba ring with coefficients in K, equipped with an absolute Frobenius lift φ. Let F be the fixed field of K under φand let G be a connected reductive group over F. This thesis investigates G-(φ,∇)-modules over R, namely (φ,∇)-modules over R with an additional G-structure.
In Chapter 3, we construct a filtered fiber functor from the category of representations of G on finite-dimensional F-vector spaces to the category of Q-filtered modules over R, and prove that this functor is splittable. In Chapter 4, we prove a G-version of thep-adic local monodromy theorem. In Chapter 5, we prove a G-version of the logarithmicp-adic local monodromy theorem under certain assumptions. As an application, we attach to eachG- (φ,∇)-module a Weil-Deligne representation of the Weil group Wκ((t)) into G(Knr), whereκis the residue field ofK, andKnris the maximal unramified extension of K.
Zusammenfassung
Sei K eine endliche Erweiterung von Qp und seiR der Robba-Ring mit Ko- effizienten in K sein, die mit einem absoluten Frobenius-Liftφ ausgestattet sind. SeiF der Fixk¨oper vonK unterφund seiGeine verbundene reduktive Gruppe ¨uberF. Diese Arbeit untersucht G-(φ,∇)-Module ¨uber R, n¨amlich (φ,∇)-Module ¨uber R mit einer zus¨atzlicher G-Struktur.
In Kapitel 3 konstruieren wir einen gefilterten Faserfunktor aus der Darstel- lungskategorie vonGauf endlich-dimensionalenF-Vektorr¨aumenbis zur Kat- egorie von Q-gefilterten Modulen ¨uberR, und beweisen, dass dieser Funktor spaltbar ist. In Kapitel 4 beweisen wir eineG-Version des p-adischen lokalen Monodromie-Satzes. In Kapitel 5 beweisen wir eine G-Version des loga- rithmischen lokalen Monodromie-Satzes unter bestimmten Annahmen. Als Anwendung f¨ugen wir jedemG-(φ,∇)-Modul eine Weil-Deligne-Darstellung der Weil-Gruppe Wκ((t)) in G(Knr) an, wobei κ der Restklassenk¨orper von K, und Knr die maximal unverzweigte Erweiterung von K ist.
Acknowledgements
First and foremost, I would like to express my deep gratitude to my supervisor Prof. Dr. Elmar Große-Kl¨onne for his guidance and support. I feel blessed to have a supervisor who introduced such an interesting topic for me to work on, who was always ready to give constructive advises on my research, and who kept encouraging me for the past four years. His extremely well-organized lectures and seminars are also enjoyable and valuable to me.
I would also like to thank all members in the algebraic number theory group in Humboldt-Universit¨at, especially Dr. Claudius Heyer. I owe so much to his crash courses on various topics in mathematics during coffee breaks, to his interest in my topic along with some very enlightening sugges- tions.
I thank Prof. Dr. Laurent Berger and Dr. Adriano Marmora for gen- erously offering their time to co-examine my thesis. They gave me precious criticisms and suggestions in their reports.
I thank my master supervisor Prof. Kezheng Li from whom I learned a lot. I thank my master co-supervisor Prof. King-Fai Lai for his leading me to the world of p-adic differential equations, and his invaluable encouragement throughout the years.
Last but not least, I thank my family for their support and understanding.
In particular, I dedicate this thesis to my beloved grandparents.
Contents
0 Introduction 1
0.1 Background . . . 1
0.2 Outline of the thesis . . . 3
0.3 Notations and conventions . . . 8
1 The p-adic local monodromy theorem 9 1.1 φ-modules . . . 9
1.2 Mal’cev-Neumann series . . . 10
1.3 The Robba rings . . . 11
1.4 Pure φ-modules . . . 13
1.5 A theorem of Tsuzuki . . . 14
1.6 The slope filtration and the p-adic local monodromy theorem . 15 2 Tannakian categories 19 2.1 Tensor categories . . . 19
2.2 The tannakian duality . . . 22
2.3 The derivative of the tannakian duality . . . 24
2.4 Filtered and graded fiber functors . . . 27
3 G-isocrystals over the Robba ring 31 3.1 Definitions . . . 31
3.2 The Q-filtered fiber functor HNg . . . 33
3.3 Splittings of HNg . . . 37
3.4 The slope morphism . . . 40
4 G-(φ,∇)-modules over the Robba ring 45 4.1 Definitions and properties . . . 45
4.2 Pushforwards . . . 50
4.3 G-isocrystals attached to splittings . . . 52
4.4 G-(φ,∇)-modules attached to splittings . . . 55
4.5 A G-version of the p-adic local monodromy theorem . . . 61
5 Applications 67 5.1 Logarithmic G-(φ,∇)-modules . . . 67
5.2 Review of root data of split reductive groups . . . 69
5.3 A G-version of the logarithmicp-adic local monodromy theorem 72 5.4 (φ, N)-modules . . . 78 5.5 Weil-Deligne representations associated to G-(φ,∇)-modules . 81
Chapter 0 Introduction
0.1 Background
Let us fix a prime numberp. LetKbe a complete non-archimedean discretely valued field of characteristic 0 with residue field κ of characteristic p. For α∈(0,1), we put
Rα := {︂
∑︁
i∈Z
citi
⃓
⃓
⃓ ci ∈K, lim
i→±∞|ci|ρi = 0, ∀ρ∈[α,1) }︂
.
The Robba ring R is defined to be the union ⋃︁
α∈(0,1)
Rα. To put it briefly, R is the ring of bidirectional power series ∑︁
i∈Z
citi with one variable t and coefficients inK, which converge in an annulus with open outer radius 1 and closed inner radius 0< α < 1 (depending on the series). The subring E† of R consisting of series with bounded coefficients is called the bounded Robba ring. We define the 1-Gauss norm onE† by
⃓
⃓
∑︁
i∈Z
citi⃓
⃓1 := maxi{|ci|},
∑︁
i∈Z
citi ∈ E†.
Then E† is a discretely valued henselian field w.r.t. the 1-Gauss norm with residue field κ((t)).
Fix a powerq of p, a relative q-power Frobenius lift on Ris an endomor- phism φ: R → R given by
∑︁
i∈Z
citi ↦−→
∑︁
i∈Z
φ(ci)ui,
for some isometry φ on K and u = φ(t) ∈ R, such that |u−tq|1 < 1. If φ induces the q-power map on κ, then φ is said to be an absolute q-power Frobenius lift on R; we will fix one. We also equip R with the derivation
∂ = dtd. Let Ω1R be the freeR-module generated by the symbol dt, we define dφ: Ω1R−→Ω1R, f dt↦−→µφ(f)dt,
whereµ=∂(φ(t)).
A (φ,∇)-module overRis a triple (M,Φ,∇), whereM is a finite freeR- module, Φ is aφ-linear endomorphism ofM (i.e.an additive map satisfying Φ(f m) = φ(f)Φ(m) for f ∈ R and m ∈ M) which induces an isomorphism M⨂︁R,φR → M of R-modules where R is viewed as an R-module via φ, and ∇: M → M⨂︁Ω1R is a connection (i.e. an additive map satisfying the Leibniz rule: ∇(f m) = f∇(m) +m⊗∂(f)dt for f ∈ R and m ∈ M), such that the diagram
M M⨂︁RΩ1R
M M⨂︁RΩ1R
Φ
∇
Φ⊗dφ
∇
(1)
commutes. For any finite separable extension L of κ((t)), we define RL :=
R⊗E†EL† whereEL† is the unique finite unramified extension ofE†with residue field L. A (φ,∇)-module over R is said to be quasi-unipotent if there is a finite separable extension Lof κ((t)) such thatM⊗RRL admits a filtration by (φ,∇)-submodules such that each quotient admits a basis of elements in the kernel of ∇. We are now ready to state:
Theorem 0.1.1(p-adic local monodromy theorem).Any (φ,∇)-module over R is quasi-unipotent.
Theorem 0.1.1 was originally conjectured by Crew [Cre98, 10.1], then reformulated by Tsuzuki [Tsu98b, Theorem 5.2.1]. The case of absolute Frobenius lifts was proved independently by Andr´e [And02], Kedlaya [Ked04]
and Mebkhout [Meb02]. The proofs of Andr´e and Mebkhout both rely on a Hasse-Arf decomposition theorem for differential modules over the Robba ring due to Christol and Mebkhout [CM00; CM01], while the proof of Kedlaya relies on a slope filtration theorem forφ-modules over the Robba ring. Later, Kedlaya gave a proof of Theorem 0.1.1 for relative Frobenius lifts in [Ked10].
0.2. Outline of the thesis
We remark that a (φ,∇)-module over R described above is the same as a finite free differential module over R with a Frobenius structure in loc. cit..
Accordingly, Theorem 0.1.1 may be rephrased as Theorem 1.6.10 in the main text of this thesis.
Theorem 0.1.1 has various applications in arithmetic algebraic geometry.
For a first example, in [Ked06], Kedlaya used this theorem to prove that the rigid cohomology spacesHrigi (X/K,E) are finite-dimensionalK-vector spaces for alli≥0, whereE is an overconvergentF-isocrystal on a separated scheme X of finite type over κ.
For a second example, in [Ber02], Berger constructed a faithful and es- sentially surjective exact tensor functor from the category of de Rham rep- resentations to that of (φ,∇)-modules over R. Moreover, employing Theo- rem 0.1.1, he gave a positive answer to Fontaine’sp-adic monodromy conjec- ture that everyp-adic de Rham representation is potentially semistable.
For a third example, in [Mar08], Marmora used Theorem 0.1.1 to con- struct a functor from the category of (φ,∇)-modules over R to that of Knr-valued Weil-Deligne representations of the Weil group Wκ((t)), where Knr is the maximal unramified extension of K in a fixed algebraic closure K¯ of K. Using this functor, Lazda and P´al proved some p-adic weight- monodromy theorems (e.g. [LP16, Theorem 5.33 and Theorem 5.58]). The aforementioned functor is also fundamental in the work of Chiarellotto and Lazda [CL18], where they proved several cases of theℓ-independence conjec- ture.
0.2 Outline of the thesis
Chapters 1 and 2 are preliminaries. There is nothing new except that in Section 2.3, we prove a tannakian duality result concerning the Lie algebra of an affine algebraic group (Proposition 2.3.2).
Chapters 3, 4 and 5 comprise the core of this thesis. As in the main body of the thesis, we fix in this section local fields F and K as in Hypoth- esis 3.0.1. In short, K is a complete non-archimedean discretely valued field of characteristic 0, equipped with a Frobenius automorphism φ with fixed field F. We assume the cardinality of the residue field of F is a p-power q.
By R, we mean the Robba ring with coefficients in K, which is equipped
with an absoluteq-power Frobenius liftφ and a derivation ∂ = dtd. Let κbe the residue field of K. Let G be an affine algebraic F-group and let g be a fixed element inG(R).
Let Γ be a finitely generated abelian group, we denote by Γ-GradR (resp.
Γ-FilR) the category of Γ-graded modules (resp. Γ-filtered modules) overR.
A Γ-filtered (resp. Γ-graded) fiber functor is an an exactF-linear tensor func- tor from the categoryRepF(G) of representations ofGon finite-dimensional F-vector spaces to the category Γ-GradR (resp. Γ-GradR). Note that the notion of exactness in Q-FilR is a bit subtle, which we shall discuss in Sec- tion 2.4.
In Chapter 3, we use Kedlaya’s slope filtration theorem to construct a Z- filtered fiber functor HNg(see Theorem 3.2.2). We then reduce HNg to a Q- filtered fiber functor HNZg : RepF(G)→Z-FilR (see Lemma 3.3.4). Then a result of Ziegler (Theorem 2.4.4) immediately implies that HNZg is splittable, i.e., it factors through a Z-graded fiber functor (see Proposition 3.3.5). In particular, for any splitting of HNZg, we construct a morphism λg: Gm,R → GR of R-groups in Section 3.4, which is called the Z-slope morphism ofg.
In Chapter 4, we first introduce the notion of G-(φ,∇)-modules over R.
Given (V, ρ) ∈ RepF(G), we write VR = V ⊗F R. For any g ∈ G(R), we define
gφ: VR−→VR, v⊗f ↦−→ρ(g)(v⊗1)φ(f),
then gφ is a φ-linear endomorphism of VR which induces an isomorphism VR⨂︁
R,φR →VR of R-modules.
Let gR be the Lie algebra of G overR. For any X ∈gR, we define
∇X: VR−→VR⨂︁Ω1R, v⊗f ↦−→(v⊗1)⊗∂(f)dt+X(v⊗f)⊗dt, then ∇X is a connection (i.e. an additive map satisfying the Leibniz rule).
Definition 0.2.1 (Definition 4.1.2). Let V ∈ RepF(G). Let g ∈G(R) and letX ∈gR such that the diagram
VR VR⨂︁RΩ1R
VR VR⨂︁
RΩ1R
gφ
∇X
gφ⊗dφ
∇X
commutes. We say that (VR, gφ,∇X) is a G-(φ,∇)-module overR.
0.2. Outline of the thesis
Note that a G-(φ,∇)-module over R is a (φ,∇)-modules over R in the classical sense, if we forget the additionalG-structure. (Compare the diagram above with diagram (1).) For any X ∈gR, we define
Γg(X) := Ad(g)(X)−dlog(g)∈gR,
where Ad denotes the adjoint representation of G, and the map dlog : G(R) → gR is defined in Construction 4.1.3. Suppose that X = Γg(︁
µφ(X))︁
, then (VR, gφ,∇X) is a G-(φ,∇)-module over R for all V ∈ RepF(G) (see Lemma 4.1.9). The main result of this chapter is aG-version of thep-adic local monodromy theorem (w.r.t. an absolute Frobenius liftφ):
Theorem 0.2.2 (Theorem 4.5.1). LetGbe a connected reductiveF-group.
If g ∈ G(R) and X ∈ gR satisfy X = Γg(︁
µφ(X))︁
, then there exists a finite separable extension L of κ((t)) and an element b ∈ G(RL) such that Γb(X)∈Lie(︁
UGR(−λg))︁
RL.
Here, UGR(−λg) is the unipotent radical of the parabolic subgroup PGR(−λg) of GR associated to the cocharacter −λg (cf. Remark 4.4.2 for a general construction). In particular, Theorem 0.2.2 implies that the (φ,∇)- module (VR, gφ,∇X) over R is quasi-unipotent (cf. Remark 4.5.3). Thus, Theorem 0.2.2 recovers Theorem 0.1.1 for absolute Frobenius lifts.
The proof of Theorem 0.2.2 follows the strategy of Kedlaya’s proof of The- orem 0.1.1 in [Ked04]. More specifically, we fix a splitting of HNg, then attach to anyG-(φ,∇)-module (VR, gφ,∇X) a newG-(φ,∇)-module (VR, zφ,∇X0), wherez ∈G(R) andX0 ∈gRare constructed out of the splitting (see Propo- sition 4.4.4). We next reduce the situation to the unit-root case by showing Corollary 0.2.3 (Corollary 4.4.5). (︁
VR, λg(π−1)[dg]∗(z)φdg,∇X0)︁
is a unit- root G-(φdg,∇)-module over R for all (V, ρ)∈RepF(G).
Here, dg ∈ N is the least positive integer such that dgx ∈ Z for all non-zero slopes of the φ-module (VR, gφ) and all V ∈ RepF(G) (cf. Con- struction 3.3.2), and [dg]∗(z) = zφ(z)· · ·φdg−1(z) ∈ G(R). Briefly put, the φdg-linear action λg(π−1)[dg]∗(z)φdg on VR is obtained by first replacing zφ by its dg-power and then multiplying it with λg(π−1) (the two operations correspond to applying the pushforward functor and twisting, respectively, in the sense of [Ked08]). Theorem 0.2.2 then follows from a theorem of
Tsuzuki (Theorem 1.5.5), a theorem of Steinberg (Lemma 4.5.2), together with a tannakian argument.
Chapter 5 is devoted to a generalization of Marmora’s construction of the functor from the category of (φ,∇)-modules over R to that of Knr-valued Weil-Deligne representations of Wκ((t)) (cf. [Mar08, Proposition 3.4.3]). We fix a split connected reductive F-group Gand an element g ∈G(R). A key step in our approach is the decomposition of UGR(−λg) into a product of root subgroups (see Lemma 5.3.7). To achieve this, we need to assume that the element g in question satisfies
Hypothesis 0.2.4(Hypothesis 5.3.1). λg(Gm,R) is contained in a split max- imal R-torus T inGR.
In Example 5.3.2, we show that the hypothesis holds true for allg ∈G(R) where G is the special linear group. We expect the hypothesis to hold for all g ∈ G(R) where G is a split connected reductive F-group (cf. Con- jecture 5.3.3). Moreover, we explain in Remark 5.3.4 that if the answer to Question 5.3.5 is positive, namely the first Zariski cohomology setHZar1 (R,L) is trivial for all split reductiveR-groupsL), then Hypothesis 5.3.1 holds true for all split reductiveF-groupsG and all g ∈G(R).
The main result of the chapter is:
Theorem 0.2.5 (Theorem 5.3.8). Assume that g ∈ G(R) satisfies Hy- pothesis 0.2.4. Let L be a finite separable extension of κ((t)). For any Y ∈ Lie(︁
UGR(−λg))︁
RL, there exists u ∈ UGR(−λg)(︁
RL[logt])︁
such that Γu(Y) = 0.
Here, logt denotes a free variable over R, with which we define in Sec- tion 5.1 the notion of logarithmic G-(φ,∇)-modules. Combining Theo- rem 0.2.5 and Theorem 0.2.2, we deduce the following corollary, which gener- alizes the classical logarithmicp-adic monodromy theorem (Theorem 5.3.10).
Corollary 0.2.6 (Corollary 5.3.9). Assume thatg ∈G(R) satisfies Hypoth- esis 0.2.4. Let X ∈ gR satisfy X = Γg
(︁µφ(X))︁
, we find a finite separable extensionL of κ((t)) and c∈G(RL[logt]) such that Γc(X) = 0.
The idea of the proof of Theorem 0.2.5 could be better understood through going over the proof of Theorem 5.3.10 given in [Ked04]. Let M be
0.2. Outline of the thesis
a (φ,∇)-module over R, then Theorem 0.1.1 provides a basis for M⨂︁RRL over RL for a finite separable extension L of κ((t)), such that the matrix N of action of∇is upper-triangular with 0 in the diagonal. The proof of Theo- rem 5.3.10 then proceeds by induction on the rankn of M: in the i-th step, we eliminate the (i+ 1)-th column of N overRL[logt] for 1 ≤i≤n−1 (the first column is already zero). In effect, each step above, say the i-th step, may be decomposed into i−1 sub-steps in a na¨ıve way: we eliminate first the (1, i)-entry, and then the (2, i)-entry, and so on, until the (i−1, i)-entry.
The latter method is actually more general, because if we view N as an el- ement in Lie(GLn)(RL), the Lie algebra of GLn over RL, then each entry above the diagonal corresponds to the root space of a positive root (w.r.t.
the split maximal torus of all diagonal matrices). It indicates that, for a general connected reductive group G, we should eliminate root-by-root an element Y ∈Lie(︁
UGR(−λg))︁
RL as in Theorem 0.2.5, and such an approach will need the aforementioned decomposition of UGR(−λg) into a product of root subgroups.
We now assume that the residue field κ of K is finite, and denote by G- WDKnr(Wκ((t))) the set of equivalence classes of Weil-Deligne representations of the Weil group Wκ((t)) into G(Knr) (cf.Definition 5.5.4). Put
B∇(G,R) := {︁
(g, X)∈G(R)×gR |X = Γg(︁
µφ(X))︁}︁/︂
∼,
where (g, X) ∼ (g′, X′) if and only if g′ = xgφ(x−1) and X′ = Γx(X) for some x∈G(R) (cf. Definition 5.5.5).
Suppose that g ∈ G(R) satisfies Hypothesis 5.3.1 and X ∈ gR satisfies X = Γg(︁
µφ(X))︁
, we show that, for any (V, ρ) ∈ RepF(G), there exists an element dlogN(u) in gKnr, the Lie algebra of G over Knr, such that NV = Lie(ρV)(−dlogN(u)) is the nilpotent operator attached to the Weil-Deligne representation associated to the (φ,∇)-module (VR, gφ,∇X) in Marmora’s construction. (See Corollary 5.4.5 and its proof.) Applying this result and a tannakian argument to Marmora’s construction, we prove:
Theorem 0.2.7 (Theorem 5.5.6). Let G be a split connected reductive F- group such that eachg ∈G(R) satisfies Hypothesis 5.3.1. There is an injec- tion
Ψ : B∇(G,R)−→G-WDKnr(Wκ((t))).
In particular, for any pair (g, X)∈G(R)×gR with g satisfying Hypoth- esis 5.3.1 (for example, when G is the general linear group or the special linear group) andX = Γg(︁
µφ(X))︁
, Ψ gives a Weil-Deligne representation of Wκ((t)) into G(Knr). Theorem 0.2.7 may be thus thought as a G-version of Marmora’s construction.
0.3 Notations and conventions
Let k be a field. By a k-algebra, we always mean a commutative k-algebra R. Let G be an affine algebraic k-group and let R be ak-algebra.
Throughout this thesis, we adopt the following notations and conventions.
• Veck := the category of finite-dimensional k-vector spaces.
• ModR:= the category of finitely generated modules over R.
• Algk:= the category of (commutative) k-algebras.
• Repk(G) := the category of representations of G on finite-dimensional k-vector spaces.
• k[G] := the Hopf algebra of G.
• GR :=G×SpeckSpecR, the base change of Gfrom Speck to SpecR.
• VR := V ⊗kR for all k-vector spaces V; αR := α ⊗R, the R-linear extension ofα, for allk-linear maps α between k-vector spaces.
• H1(k, G) :=H1(︁
Gal(ksep/k), G(ksep))︁
.
• By a reductivek-group, we mean a (not necessarily connected) affine al- gebraick-groupGsuch that every smooth connected unipotent normal subgroup of Gk¯ is trivial, where k¯ is an algebraic closure of k.
Chapter 1
The p-adic local monodromy theorem
1.1 φ-modules
This section is a recollection of [Ked10, 14], where the author uses the notion difference-modules, instead of φ-modules.
A φ-ring (R, φ) is a commutative ring R with 1, equipped with a ring endomorphismφ: R→R. Ifφis an automorphism onR, then (R, φ) is said to be inversive.
A φ-module (M,Φ) over (R, φ) is an R-module M equipped with a map Φ : M → M which is additive and φ-linear in the sense that Φ(rm) = φ(r)Φ(m) for all r ∈ R and m ∈ M. Φ and φ are sometimes omitted if they are clear in the context.
We define
MΦ=1 :={m ∈M |Φ(m) = m}.
A φ-module M is said to be trivial if it admits a basis in MΦ=1.
A finite free φ-module M over R is said to be standard if it admits a basis v1,· · · ,vd such that Φ(vi) = vi+1 for 1 ≤ i ≤ d−1 and Φ(ed) = re1 for some r∈R×. We call v1,· · · ,vd a standard basis for M.
A morphism of φ-modules (M,Φ) and (M′,Φ′) over (R, φ) is anR-linear mapf: M →M′ such thatf◦Φ = Φ′◦f. Finite free φ-modules over (R, φ) form a category, which is denoted by φ-ModR.
A φ-ring (R, φ) may be viewed as an R-module via φ: R → R, we thus
have an φ-module M ⊗R,φR. (Note that rm⊗R,φ 1 = m⊗R,φ φ(r) in this module.)
LetM be a finite free R-module and Φ : M →M an additive map. Then Φ is φ-linear if and only if the linearization map
Φ∗: M ⊗R,φR−→M, m⊗R,φ r↦−→rΦ(m)
isR-linear. A finite freeφ-module (M,Φ) over (R, φ) is said to bedualizable if the linearization map is an isomorphism ofR-modules. If (R, φ) is inversive and (M,Φ) is dualizable, then Φ is invertible on M.
Let (M,Φ) be dualizable and letM∨ = HomR(M, R) be the dual module.
Then there is a unique way to equipM∨ with aφ-module structure (M∨,Φ∨) such that the diagram
M M
R R
Φ
f Φ∨(f)
φ
commutes.
A φ-field k is said to be weakly difference-closed if every finite dualizable φ-module overkis trivial. We saykisstrongly difference-closed if it is weakly difference-closed and inversive. (Cf. [Ked10, 14.3].)
1.2 Mal’cev-Neumann series
Definition 1.2.1. Let R be a ring. The ring of Mal’cev-Neumann series R((tQ)) overR consists of formal sums x=
∑︁
i∈Q
citi with ci ∈R such that for eachx=
∑︁
citi the support
Supp(x) = {i∈Q|ci ̸= 0}
is well-ordered (i.e. contains no infinite decreasing subsequence).
Remark 1.2.2. Suppose that k is a field, we have the following properties.
• k((tQ)) is a field (cf.[Poo93, Corollary 2]).
• We have thet-adic valuation v onk((tQ)), where v(x) := min Supp(x).
The value group isQ and the residue field is k.
1.3. The Robba rings
1.3 The Robba rings
Hypothesis 1.3.1. For the remainder of this chapter, we fix an inversive φ-field (K, φ), where K is a complete discretely valued field of characteristic 0 and of residue characteristic p > 0. We denote by OK its ring of integers, mK the maximal ideal ofOK,π a uniformiser ofK, and κK the residue field.
For α∈(0,1), we put Rα := {︂
∑︁
i∈Z
citi
⃓
⃓
⃓ ci ∈K, lim
i→±∞|ci|ρi = 0, ∀ρ∈[α,1)}︂
.
For any ρ∈[α,1), we define the ρ-Gauss norm on R˜α by setting
⃓
⃓
∑︁
i
citi⃓
⃓ρ:= supi{|ci|ρi}.
The Robba ring is defined to be the union R := R(K, t) := ⋃︁
α∈(0,1)
Rα. For any
∑︁
i
citi ∈ R, we define ⃓
⃓
∑︁
i
citi⃓
⃓1 := supi{|ci|} ∈R≥0∪ {∞}.
We define the bounded Robba ring E† = E†(K, t) to be the subring of R consisting of the elements with bounded coefficients. We also define
E :=E(K, t) :={︂
∑︁
i∈Z
citi
⃓
⃓
⃓ ci ∈K, lim
i→−∞|ci|= 0,supi{|ci|}<∞}︂
.
Remark 1.3.2. Let R be E† or E, we define the 1-Gauss norm by |x|1 = supi{|ci|} for x=
∑︁
i
citi ∈R. Then E† is a henselian discretely valued field andE is its completion w.r.t. the 1-Gauss norm. The residue fields ofE†and E are both isomorphic to κK((t)). For more details, we refer to [Ked10, 15.1 and 16.2].
Definition 1.3.3. Fix a power q of p and let R ∈ {R,E†,E}. A relative q-power Frobenius lift onR is an endomorphism φ: R→R given by
∑︁
i∈Z
citi ↦−→
∑︁
i∈Z
φ(ci)ui
for some isometry φ on K and u = φ(t) ∈ R, such that |u−tq|1 < 1 (if R =R, this forces u ∈ E†). If φ induces the q-power map on κK, then φ is said to be anabsolute q-power Frobenius lift onR. A Frobenius lift (relative or absolute) φis said to be standard if φ(t) =tq.
Definition 1.3.4. [Ked08, Definition 2.2.4] Forα ∈(0,1), letR˜α be the set of formal sums
∑︁
i∈Q
citi with ci ∈K, subject to the following properties.
• For anyc >0, the set {i∈Q| |ci| ≥c} is well-ordered.
• For anyρ∈[α,1), we have
i→±∞lim |ci|ρi = 0.
R˜
α is a ring under the formal power series addition and multiplication. For any ρ∈[α,1), we define the ρ-Gauss norm on R˜
α by setting
⃓
⃓
∑︁
i
citi⃓
⃓ρ= supi{|ci|ρi}.
We define the extended Robba ring
R˜ :=R˜ (K, t) = ⋃︂
α∈(0,1)
R˜
α.
Definition 1.3.5. Let E˜† be the subring of R˜ consisting of elements with bounded coefficients. We define the 1-Gauss norm on E˜† by setting
⃓
⃓
∑︁
i
citi⃓
⃓1 = supi{|ci|} for all
∑︁
i
citi ∈ E˜†. We call E˜† the extended bounded Robba ring.
Definition 1.3.6. Let E˜ be the set of formal sums x =
∑︁
i∈Q
citi with ci ∈K subject to the following properties.
• For anyc >0, the set {i∈Q| |ci| ≥c} is well-ordered.
• supi{|ci|}<∞.
• lim
i→−∞|ci|= 0.
Remark 1.3.7. The following properties are discussed in [Liu13, 1.4].
(i) E˜ is a complete discretely valued field w.r.t. the 1-Gauss norm.
(ii) The natural inclusion E˜†↪→ E˜ is an isometry, and identifies E˜ with the completion ofE˜† w.r.t. the 1-Gauss norm.
1.4. Pure φ-modules
(iii) The residue fields of E˜† and E˜ are isomorphic to κK((tQ)) the field of Mal’cev-Neumann series overκK.
Definition 1.3.8. Fix a power qof p. Let φbe an isometry onK, we define the Frobenius lift onR˜ by
φ: R˜ −→ R˜,
∑︁
i∈Q
citi ↦−→
∑︁
i∈Q
φ(ci)tiq. .
Remark 1.3.9. Given a relative q-power Frobenius lift φ on R, we have a φ-equivariant embedding ψ : R → R˜ (i.e. an embedding of φ-rings) such that |ψ(x)|ρ = |x|ρ for ρ sufficiently close to 1. (Cf. [Ked08, Proposition 2.2.6].)
1.4 Pure φ-modules
Let R ∈ {E†,R,E˜†,R˜} equipped with a Frobenius lift φ. Let (M,Φ) be a finite free φ-module over a φ-ring (R, φ). Let d be a positive integer, then (M,Φd) is a φd-module over (R, φd). We define thed-pushforward functor:
[d]∗: φ-ModR−→φd-ModR, (M,Φ)↦−→(M,Φd).
Let s ∈ Z, we define the twist M(s) of (M,Φ) by s to be the φ-module (M, πsΦ), where π is a uniformizer of K.
Definition 1.4.1. Let M be a finite free dualizable φ-module over R of dimension d.
(i) We say that M is unit-root (or ´etale) φ-module if there exists a basis v1,· · · ,vd of M over R in which Φ acts via an invertible matrix in GLd(OE†) if R ∈ {E†,R}, or GLd(OE˜†) if R∈ {E˜†,R˜}.
(ii) Let µ=s/r ∈Q with r >0 and (s, r) = 1. We say that M is pure of slope µif ([r]∗M)(−s) is unit-root.
1.5 A theorem of Tsuzuki
Adifferential ring (R, d) is a commutative ringRequipped with a derivation d: R−→R,i.e., an additive map satisfying the Leibniz rule:
d(xy) = xd(y) +yd(x), ∀x, y ∈R.
Adifferential module(M, D) over a differential ring (R, d) is anR-module M equipped with a differential operator relative to (R, d), i.e., an additive map D: M−→M satisfying
D(rx) = rD(x) +xd(r), ∀r∈R, x ∈M.
We define
MD=0 :={m∈M |D(m) = 0},
its elements are calledhorizontal sections of M. A differential module M is said to betrivial if it admits a basis of horizontal sections.
Definition 1.5.1. Let R ∈ {R,E†,E}, equipped with the derivation ∂ =
∂t = dtd and a Frobenius lift φ. A Frobenius structure on a finite free dif- ferential module (M, D) over (R, ∂) w.r.t. the Frobenius lift φ is a φ-linear map Φ :M →M satisfying
(i) (M,Φ) is a dualisable φ-module over (R, φ);
(ii) D(Φ(m)) = ∂(φ(t))Φ(D(m)), m∈M.
Let φ be an absolute q-power Frobenius lift on E†. Recall that E† is a henselian discretely valued field. We then have the following definition.
Definition 1.5.2. LetLbe a finite separable extension ofκK((t)), we denote byEL† the unique unramified extension of E† of residue field L.
Lemma 1.5.3. Suppose E† is equipped with an absoluteq-power Frobenius lift φ. Let L be a finite separable extension of κK((t)) and let EL† be the unique unramified extension of E† with residue field L. Then there exists a unique q-power absolute Frobenius lift φ˜ onEL† which extends φ.
1.6. The slope filtration and the p-adic local monodromy theorem
Proof. Let F = κK((t)), we may write L =F(α¯) for some α ∈ OE†
L
by the primitive element theorem, where OE†
L is the ring of integers of EL†. We have thatEL† =E†(α) by Hensel’s lemma. LetP(T) = Tm+am−1Tm−1+· · ·+a0 ∈ OE†[T] be the minimal polynomial of α overE†. If such a liftφ˜ exists, then φ˜(α) is a root of
Q(T) :=Tm+φ(am−1)Tm−1+· · ·+φ(a0)∈ OE†[T].
We observe that the existence and uniqueness of φ˜ is equivalent to the ex- istence and uniqueness of a root β ∈ OE†
L of Q(T) such that β¯ = α¯q. Now letα =α1,· · · , αm ∈ E† be the distinct roots of P(T), then α1,· · · , αm ∈F¯ are the distinct roots of P¯ (T) = Tm +am−1Tm−1 +· · ·+a0 because P¯ is separable. ThereforeQ¯ (T) =Tm+am−1qTm−1+· · ·+a0q has distinct roots αq,· · · , αmq. Thus, Q(T) has a unique root β ∈ OE†
L
which is the lift of αq
by Hensel’s lemma, as desired. ■
Remark 1.5.4. LetLbe a finite separable extension of κK((t)). By [Mat95, Proposition 3.4], there exists a finite unramified extension E of K such that for any uniformiser u of L, EL† is isomorphic to E†(E, u), where E†(E, u) is the bounded Robba ring with series parameter uand with coefficients in E.
Moreover, to E†(E, u) extends the 1-Gauss norm on E†. Then the derivation
∂ onE† extends uniquely toEL†, which is still denoted by ∂.
Theorem 1.5.5. Let (M, D) be a finite differential module overE†admitting a unit-root Frobenius structure Φ w.r.t. some absolute q-power Frobenius lift φ. Then there exists a finite separable extension L of κK((t)) such that M ⊗E† EL† is a trivial differential module.
This is [Ked04, Proposition 6.11]. The case where κK is algebraically closed and q=p is due to Tsuzuki, [Tsu98a, Theorem 5.11].
1.6 The slope filtration and the p-adic local monodromy theorem
Theorem 1.6.1 (The slope filtration theorem). [Ked10, Theorem 17.4.3]
LetM be a finite free differential module overR equipped with a Frobenius
structure w.r.t. some Frobenius lift φ on R. Then there exists a unique filtration
0 = M0 ⊆M1 ⊆ · · · ⊆Ml=M
of differential submodules preserved by the Frobenius structure such that (i) each successive quotient Mi/Mi−1 is finite free and descends uniquely
to a differential module over E† with an induced Frobenius structure that is pure of some slope µi;
(ii) µ1 < µ2 <· · ·< µl.
Definition 1.6.2. Let M be a finite free dualizable φ-module over R. We call the filtration given by Theorem 1.6.1 the slope filtration of M. We call µ1,· · · , µl the jumps of the slope filtration. The (uniquely determined, not neccesarily strictly) increasing sequence (µ1,· · · , µ1,· · · , µl,· · · , µl), with each µi appearing rkR(Mi/Mi−1) times, is said to be the Newton slope se- quence for M. We call rkR(Mi/Mi−1) the multiplicity of µi for all 1≤i≤l.
Definition 1.6.3. (Recall that Hypothesis 1.3.1 is conserved.) By anexten- sion of (K, φ), we mean a φ-field (E, φE), where E is a complete discretely valued field containingK and φE extends φ. We say that E is admissible if it has the same value group asK.
Lemma 1.6.4 ([Liu13, Lemma 1.5.3]). The field K admits an admissible extensionE such that the residue field κE of E is strongly difference-closed.
Lemma 1.6.5. Let (M,Φ) be a finite free φ-module over R. Let E be an admissible extension ofK such that κE is strongly difference-closed.
(i) If M is pure of some slope µ, thenM ⊗RR˜ (E, t) is pure of slope µ.
(ii) Tensoring the slope filtration of M with R˜ (E, t) gives the slope filtra- tion of M ⊗RR˜ (E, t).
We will give a sketch of proof. Indeed, both assertions are involved in the proof of the slope filtration theorem of φ-modules over R ([Ked08, Theorem 1.7.1]), which essentially shows that the HN-filtration of a φ-module over R has successive pure quotients. For the notions of semistability and HN- filtration of a φ-module overR, we refer to Section 1.4 inloc. cit..
1.6. The slope filtration and the p-adic local monodromy theorem
Proof. Assertion (i) is deduced from the proof of [Ked08, Theorem 3.1.3]. For assertion (ii), we let M be a semistable dualizable finite freeφ-module over R. Then M⊗RR˜ (E, t) is also semistable by [Ked08, Theorem 3.1.2]. Since κE is strongly difference-closed by assumption, we have that M ⊗RR˜ (E, t) is pure of some slope by [Ked08, Theorem 2.1.8]. It follows from assertion (i) thatM is pure of the same slope, assertion (ii) then follows. ■ Lemma 1.6.6. Suppose that κK is strongly difference-closed.
(i) Let 0−→M1−→M−→M2−→0 be a short exact sequence of φ- modules overR˜ , such that the slopes ofM1 are all less than the smallest slope ofM2, then the sequence splits.
(ii) Everyφ-module overR˜ admits aDieudonn´e-Manin decomposition,i.e., it is a direct sum of standard φ-submodules.
Proof. The first assertion is [Liu13, Proposition 1.5.11], and the section as-
sertion is Proposition 1.5.12 in loc. cit.. ■
Lemma 1.6.7. [Ked10, Lemma 16.4.3] Let M (resp. N) be a finite free φ-module over R with Newton slope sequence (µM,1,· · ·, µM,1,· · · , µM,m,· · · , µM,m) with multiplicities dM,1,· · ·, dM,m (resp. (µN,1,· · ·, µN,1,· · · , µN,n,· · · , µN,n) with multiplicities dN,1,· · · , dN,n). Then the multiset of the Newton slopes for M ⊗R N are µM,i+µN,j with multiplicities dM,idN,j for all 1≤i≤m,1≤j ≤n.
Lemma 1.6.8. [Ked08, Proposition 1.4.18] Let R ∈ {R,R˜} and let M and N be finite free dualizable φ-modules over R. If the slopes of M are all less than the smallest slope ofN, then no non-zero morphism from M to N exists.
Definition 1.6.9. Let M be a differential module over R. We put RL :=
R ⊗E†EL† for all finite separable extension L of κK((t)).
(i) We say thatM is quasi-constant if M ⊗RRL is trivial for some finite separable extension Lof κK((t)).
(ii) We say that M is quasi-unipotent if there exists a filtration 0 = M0 ⊆ M1 ⊆ · · · ⊆Ml=M of differential submodules such that eachMi+1/Mi is quasi-constant.
Theorem 1.6.10 (p-adic local monodormy theorem). [Ked10, Theorem 20.1.4] Let M be a finite free differential module over R equipped with a Frobenius structure w.r.t. some Frobenius lift φ on R. Then M is quasi- unipotent.
For the remainder of the thesis, we only consider absolute Frobenius lifts.
In this case, the proof in [Ked04] substantially builds on Theorem 1.6.1, by which the situation is reduced to the unit-root case and the theorem follows from Theorem 1.5.5. We will follow this strategy in our generalization of this theorem.
Chapter 2
Tannakian categories
In this chapter, k is an arbitrary field.
2.1 Tensor categories
Let C be a category and let ⊗: C × C → C be a functor. An associativity constraint for (C,⊗) is a functorial isomorphism φX,Y,Z: X ⊗(Y ⊗Z) → (X⊗Y)⊗Z, such that for all objectsX, Y, Z, T, the diagram
X⊗(Y ⊗(Z⊗T))
X⊗((Y ⊗Z)⊗T))
(X⊗(Y ⊗Z))⊗T ((X⊗Y)⊗Z)⊗T (X⊗Y)⊗(Z ⊗T) Id⊗φY,Z,T
φX,Y⊗Z,T
φX,Y,Z ⊗Id
φX,Y,Z⊗T
φX⊗Y,Z,T
commutes. Acommutativity constraint for (C,⊗) is a functorial isomorphism ψX,Y: X⊗Y →Y ⊗X, such thatψX,Y◦ψY,X: X⊗Y →X⊗Y is the identity morphism onX⊗Y for all objects X, Y. An associativity constraint φand a commutativity constraintψ are calledcompatible if, for all objects X, Y, Z,
the diagram
X⊗(Y ⊗Z)
X⊗(Z ⊗Y)
(X⊗Z)⊗Y (Z⊗X)⊗Y Z⊗(X⊗Y) (X⊗Y)⊗Z Id⊗ψY,Z
φX,Z,Y
ψX,Z⊗Id
φX,Y,Z ψX⊗Y,Z
φZ,X,Y
commutes.
A pair (1, e) comprising an object 1 of C and an isomorphism e: 1 → 1⊗1is called anidentity object ofC if X↦→1⊗X: C → C is an equivalence of categories.
Definition 2.1.1. A system (C,⊗, φ, ψ), in which φ and ψ are compatible associativity and commutativity constraints, respectively, is atensor category if there exists an identity object.
Proposition 2.1.2. [DM82, Proposition 1.3] Let (1, e) be an identity object of the tensor category (C,⊗).
(i) For eachX ∈Ob(C), there exists a unique isomorphismℓ: Id→1⊗( ) of functors such thatℓ(1) = e and the diagrams
X⊗Y 1⊗(X⊗Y)
X⊗Y (1⊗X)⊗Y
ℓ
φ ℓ⊗Id
X⊗Y (1⊗X)⊗Y
X⊗(1⊗Y) (X⊗1)⊗Y
ℓ⊗Id
Id⊗ℓ ψ⊗Id
φ
commute.
(ii) If (1′, e′) is a second identity object, then there exists a unique isomor- phisma: 1→1′ making the diagram
1 1⊗1
1′ 1′⊗1′
e
a a⊗a
e′
commute.
2.1. Tensor categories
Definition 2.1.3. Let (C,⊗) be a tensor category. For X, Y ∈ Ob(C), we define
φX,Y: Copp−→Set, T ↦−→HomC(T ⊗X, Y).
ForT, S ∈Ob(C) andf ∈HomCopp(T, S), we define
φX,Y(f) : HomC(T ⊗X, Y)−→HomC(S⊗X, Y) φ↦−→φ◦(f⊗IdX).
ThenφX,Y is a functor fromCopp toSet. If the functorφX,Y is representable, we call the representing object Hom(X, Y) the internal Hom. Assume that φX,Y is representable for all X, Y ∈Ob(C).
(i) Let (1, e) be an identity object, we define the dual object X∨ of X to be Hom(X,1).
(ii) Let evX,Y ∈ HomC(Hom(X, Y)⊗ X, Y) be the evaluation morphism corresponding to IdHom(X,Y).
Definition 2.1.4. [Del07, 2.5] A tensor category (C,⊗) is said to be rigid, if any object X admits a (unique) dual X∨, together with morphisms evX: X∨⊗X →1 and coevX: 1→X⊗X∨ such that the compositions
X ℓ(X) 1⊗X coevX⊗IdX X⊗X∨⊗X IdX⊗evX X⊗1ℓ(X) X
−1◦ψ
and
X∨ψ◦ℓ(X XX∨ ⊗1 X∨⊗X⊗X∨ 1⊗X∨ X∨
∨) coevX∨⊗IdX∨ evX∨⊗IdX∨ ℓ(X∨)−1
are identities.
Example 2.1.5. The categoriesVeck andModR are rigid tensor categories for all k-algebras R.
Definition 2.1.6. Let (C,⊗) and (C′,⊗′) be two tensor categories. A tensor functor from C to C′ is a pair (F, c) consisting of a functor F:C → C′ and a functorial isomorphism c: ⊗ ◦(F, F)−→F ◦ ⊗ of functors from C × C toC. We require that F(1, e) = (F(1), F(e)) is an identity object in C′ and (F, c) to become compatible with the unit object, the commutativity constraints and the associativity constraints of C resp. C′. For more details, we refer to [DM82, Definition 1.8].
Definition 2.1.7. Let (F, c),(G, d) : (C,⊗) → (C′,⊗′) be tensor functors between tensor categories. A morphism of tensor functors is a morphism λ: F →Gof functors such that, for all finite families of objects (Xi)i∈I inC, the diagram
⨂︁′
i∈IF(Xi) F(⨂︁i∈IXi)
⨂︁′
i∈IG(Xi) G(⨂︁i∈IXi)
c
⊗i∈Iλ(Xi) λ(⊗i∈IXi) d
commutes.
Proposition 2.1.8. [DM82, Proposition 1.13] Let (F, c),(G, d) : (C,⊗) → (C′,⊗′) be tensor functors between tensor categories. If C and C′ are rigid, then every morphism of tensor functors λ: F →Gis an isomorphism.
Definition 2.1.9. Let C → C′ be tensor categories.
(i) We define Hom⊗(F, G) (resp. Isom⊗(F, G)) to be the class of mor- phisms (resp. isomorphisms) (F, c) → (G, d) of tensor functors. We define End⊗(F) := Hom⊗(F, F), and Aut⊗(F) := Isom⊗(F, F).
(ii) Let (F, c) and (G, d) be tensor functors C → Veck. For any R ∈ Algk, we define Hom⊗(F, G)(R) := Hom⊗(FR, GR), Isom⊗(F, G)(R) := Isom⊗(FR, GR), End⊗(F) := End⊗(FR), and Aut⊗(F)(R) := Aut⊗(FR). Then Hom⊗, Isom⊗, End⊗, and Aut⊗ are functors from Algk toSet.
2.2 The tannakian duality
Definition 2.2.1. Let (C,⊗) be an abelian k-linear rigid tensor category.
(i) Let R ∈ Algk, an R-valued fiber functor for C is an exact faithful k-linear tensor functor ω: (C,⊗)→(ModR,⊗).
(ii) (C,⊗) is atannakian category overk if End(1)∼=k and there exists an R-valued fibre functor ω for some R ∈Algk .
(iii) A tannakian categoryC is calledneutral if it has ak-valued fibre func- tor.
2.2. The tannakian duality
Example 2.2.2. Let G be an affine algebraic k-group and let Repk(G) be the category of G-representations on finite-dimensional k-vector spaces.
ThenRepk(G) is a neutral tannakian category equipped with a fiber functor ωG: = forg : Repk(G) → Veck. In particular, it is essentially small (that is, HomG(V, W) is a set for allV, W ∈Repk(G)).
The following tannakian duality will be repeatedly used in this thesis, whose proof can be found, e.g., in [Mil17, Theorem 9.2].
Theorem 2.2.3. Let G be an affine algebraic k-group and let R ∈ Algk. Suppose that for any (V, ρV) ∈ Repk(G) we are given an R-linear map λV : VR →VR. If the family {λV |(V, ρV)∈Repk(G)} satisfies
(i) λV⊗W =λV ⊗λW for all V, W ∈Repk(G);
(ii) λ1 is the identity map where 1 is the trivial representation on k;
(iii) for all G-equivariant maps α: V →W, we have λW ◦αR=αR◦λV. Then there exists a unique g ∈G(R) such that λV =ρV(g) for all V.
Corollary 2.2.4. [Mil17, Note 9.8] LetGbe an affine algebraick-group, we then have an isomorphismG∼= Aut⊗(ωG) of affine algebraick-groups.
Proposition 2.2.5. Let G be a smooth affine algebraic k-group. Let K/k be a field extension and letη: Repk(G)→VecK be a fibre functor over K, then Hom⊗(ωG, η) is a G-torsor over K. In particular, if H1(K, G) = {1}
and G(K)̸=∅, then ωG is isomorphic to η over K.
Proof. Notice that we have an action
Hom⊗(ωG, η)×Aut⊗(ωG)−→Hom⊗(ωG, η)
by pre-composition. By [DM82, Theorem 3.2 (i)], Hom⊗(ωG, η) is an Aut⊗(ωG)-torsor. In particular, it is a G-torsor over K by Corollary 2.2.4.
Because G is a K-group variety, G-torsors over K are K-varieties by [Mil17, Proposition 2.69], whose isomorphism classes are classified by H1(K, G). It follows from the triviality ofH1(K, G) that Hom⊗(ωG, η)(K)∼= G(K), hence Hom⊗(ωG, η)(K) ̸= ∅. Proposition 2.1.8 then implies the sec- ond assertion.
■
2.3 The derivative of the tannakian duality
We first recall the notion of the Lie algebra of ak-group functor. For a more detailed discussion, we refer to [DG70a, II, §4].
For any R ∈ Algk, we have the R-algebra of dual numbers R[ε] :=
R[X]/(X2) where ε := X+ (X2), and the canonical projection πR: R[ε] → R, ε↦→0. Let G be ak-group functor, we define
Lie(G)(R) := KerG(πR).
Letf:G→H be a morphism ofk-group functors, the commutative diagram Lie(G)(R) = Ker(G(πR)) Lie(H)(R) = Ker(H(πR))
G(R[ϵ]) H(R[ϵ])
G(R) H(R)
ιG ιH
f(R[ϵ])
G(πR) H(πR)
f(R)
(2.1)
implies that f(R[ϵ])◦ιG(X)∈ Lie(H)(R) for all X ∈ Lie(G)(R). We define Lie(f) :=f(R[ϵ])◦ιG: Lie(G)(R)→Lie(H)(R). Hence, Lie( )(R) is functor from the category ofk-group functors to that of abelian groups.
For an affine algebraic k-groupG, we write I for the kernel of the counit ϵG: k[G]→k. We have the following familiar group isomorphisms
g:= Lie(G)(k)∼= Homk(I/I2, k)∼= Derk(A, k).
Moreover, we have Lie(G)(R)∼=gR. The Lie bracket on Derk(A, k) then gives a Lie bracket ongRand hence on Lie(G)(R). We will identify Lie(G)(R) and gR, and call it theLie algebra of GoverR, wheneverGis affine algebraic. In this case, Lie( )(R) is a functor from the category of affine algebraick-groups to that of Lie algebras over R.
Remark 2.3.1. For any d-dimensional G-representation (V, ρV), we write glV := Lie(GLV)(k). We then have glV,R = {Id+εB | B ∈ Matd,d(R)}, after choosing a k-basis for V. Then Id +εB ↦→ B gives a group isomor- phism from glV,R to EndR(VR). Henceforth, we will identify Lie(ρV)(X) as an endomorphism ofVR, for all X ∈gR.
2.3. The derivative of the tannakian duality
Replacing H with GLV and f with ρV in diagram (2.1), we obtain a morphism Lie(ρV) = ρV(R[ϵ]) ◦ ιG: gR → glV,R of Lie algebras over R. Let (W, ρW) ∈ Repk(G), and let α ∈ HomG(V, W). We then have αR◦Lie(ρV)(X) = Lie(ρW)(X)◦αR for all X ∈gR.
By the tannakian duality (Theorem 2.2.3), we have an isomorphism G ∼= Aut⊗(ωG) of affine algebraic k-groups (Corollary 2.2.4). Applying the functor Lie( )(R) on both sides of the isomorphism then gives us an isomor- phism gR∼= Lie(Aut⊗(ωG))(R) of Lie algebras overR. The following lemma indicates that the elements in Lie(Aut⊗(ωG))(R) are exactly the derivatives (in the sense of taking derivations of conditions (i,ii,iii) in Theorem 2.2.3) of elements in Aut⊗(ωG)(R); it might thus be thought as the derivative of the tannakian duality.
Proposition 2.3.2. Let G be an affine algebraic k-group and let R be a k-algebra. Suppose that for any (V, ρV)∈Repk(G) we are given anR-linear endomorphism λV of VR subject to the conditions
(i) λV⊗W =λV ⊗IdWR+ IdVR⊗λW for all V, W ∈Repk(G);
(ii) λ1 = 0 where 1=k is the trivial G-representation;
(iii) λW ◦αR=αR◦λV for all α ∈HomG(V, W).
Then there exists a unique element X ∈ gR such that λV = Lie(ρV)(X) for all (V, ρV)∈Repk(G).
Proof. For any (V, ρV)∈Repk(G) andλV : VR →VR, we define the following R[ε]-linear map
ελV : VR[ε]−→VR[ε], v⊗(x+yε)↦−→λV(v⊗x)ε.
We then define the following R[ε]-linear endomorphism λ˜V := IdVR[ε]+ελV : VR[ε]−→VR[ε].
Then λ˜V ∈Lie(GLV)(R)⊆GLV(R[ε]), because πR(λ˜V) = IdVR. We claim that the family
{︁λ˜
V : VR[ε]→VR[ε] |(V, ρV)∈Repk(G)}︁
(2.2)
