Towards non-abelian p-adic Hodge theory in the good reduction case

170 Pages·2011·1.59 MB·English

by  Martin C. Olsson

MEMOIRS
of the
American Mathematical Society
Number 990
Towards Non-Abelian
p-adic Hodge Theory
in the Good Reduction Case
Martin C. Olsson
March 2011 • Volume 210 • Number 990 (end of volume) • ISSN 0065-9266
American Mathematical Society
Number 990
Towards Non-Abelian
p-adic Hodge Theory
in the Good Reduction Case
Martin C. Olsson
March2011 • Volume210 • Number990(endofvolume) • ISSN0065-9266
Library of Congress Cataloging-in-Publication Data
Towardsnon-abelianp-adicHodgetheoryinthegoodreductioncase/MartinC.Olsson.
p.cm. —(MemoirsoftheAmericanMathematicalSociety,ISSN0065-9266;no. 990)
“March2011,Volume210,number990(endofvolume).”
Includesbibliographicalreferences.
ISBN978-0-8218-5240-8(alk. paper)
1.Hodgetheory. 2.p-adicanalysis. 3.Variants. I.Title.
QA564.O58 2011
516.3(cid:2)5—dc22 2010046756
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Contents
Chapter 1. Introduction 1
Chapter 2. Review of some homotopical algebra 9
Chapter 3. Review of the convergent topos 19
Chapter 4. Simplicial presheaves associated to isocrystals 29
Chapter 5. Simplicial presheaves associated to smooth sheaves 47
Chapter 6. The comparison theorem 63
Chapter 7. Proofs of 1.7–1.13 73
Chapter 8. A base point free version 85
Chapter 9. Tangential base points 97
Chapter 10. A generalization 123
Appendix A. Exactification 125
Appendix B. Remarks on localization in model categories 135
Appendix C. The coherator for algebraic stacks 139
(cid:2)
Appendix D. B (V)-admissible implies crystalline. 147
cris
Bibliography 155
iii

Abstract
We develop a non–abelian version of p–adic Hodge Theory for varieties (pos-
sibly open with “nice compactification”) with good reduction. This theory yields
in particular a comparison between smooth p–adic sheaves and F–isocrystals on
the level of certain Tannakian categories, p–adic Hodge theory for relative Malcev
completions of fundamental groups and their Lie algebras, and gives information
about the action of Galois on fundamental groups.
ReceivedbytheeditorMay8,2006.
ArticleelectronicallypublishedonSeptember15,2010;S0065-9266(2010)00625-2.
2010 MathematicsSubjectClassification. Primary14-XX;Secondary11-XX.
(cid:2)c2010 American Mathematical Society
v

CHAPTER 1
Introduction
1.1. The aim of this paper is to study p–adic Hodge theory for non–abelian
invariants. Let us begin, however, by reviewing some of the abelian theory.
Let k be a perfect field of characteristic p>0, V the ring of Witt vectors of k,
K the field of fractions of V, and K (cid:2)→K an algebraic closure of K. Write G for
K
the Galois group Gal(K/K). Let X/V be a smooth proper scheme, and denote by
X the generic fiber of K. Let D ⊂X be a divisor with normal crossings relative
K
toV, andset Xo :=X−D. Denote by H∗(Xo)the´etalecohomology of Xo with
et K K
coefficients Q and by H∗ (Xo) the algebraic de Rham cohomology of Xo. The
p dR K K
visecatofirltsepraecdeFH–e∗its(oXcrKoy)sthaal.s aThnaattuirsa,ltahcetiospnaρceXKoHo∗f(GXKo,)acnodmtehseesqpuaicpepeHdd∗Rw(iXthKoa)
dR K
·
filtration Fil and a semi–linear (with respect to the canonical lift of Frobenius
Xo
to V) FrobeniKus automorphism ϕ : H∗ (Xo) → H∗ (Xo). The theory of p–
adic Hodge theory implies that tXheKo twodcRollecKtions ofddRataK(H∗(Xo),ρ ) and
(Hd∗R(XKo),Fil·XKo,ϕXKo) determine each other [Fa1, Fa2, Fa3, Tets1]K. XKo
1.2. Moreprecisely,letB (V)denotetheringdefinedbyFontaine[Fo1,Fo2,
cris
Fo3], MF the category of K–vector spaces M with a separated and exhaustive
K
filtration Fil· and a semi–linear automorphism ϕ :M →M, and let Repcts(G )
M Q K
bethecategoryofcontinuousrepresentationsofG onQ –vectorspaces. Thpering
K p
B (V) comes equipped with an action of G , a semi–linear Frobenius automor-
cris K
phism, and a filtration. There is a functor
(1.2.1) D:Repcts(G )−→MF
Q K K
p
sending a representation L to (L⊗Qp Bcris(V))GK with the semi–linear automor-
phism and filtration induced by that on B (V). For any L∈Rep(G ) there is a
cris K
natural transformation
(1.2.2) αL :D(L)⊗K Bcris(V)−→L⊗Qp Bcris(V)
which by [Fo1, 5.1.2 (ii)] is always injective. The representation L is called crys-
talline if the map α is an isomorphism, in which case L and D(L) are said to
L
be associated. The precise statement of the comparison between between ´etale
and de Rham cohomology in the above situation is then that (H∗(Xo),ρ ) and
(H∗ (Xo),Fil· ,ϕ ) are associated. In particular there is aetnaKturalXiKosomor-
dR K XKo XKo
phism
(1.2.3) Hd∗R(XKo)⊗K Bcris(V)(cid:6)He∗t(XKo)⊗Qp Bcris(V)
compatible with the actions of G , the filtrations, and the Frobenius automor-
K
phisms.
There is also a version of the comparison 1.2.3 with coefficients. In [Fa1,
Chapter V (f)] Faltings defines a notion of a crystalline sheaf on the scheme Xo,
K
1
2 MARTINC.OLSSON
which is a smooth Q –sheaf L on Xo which is associated in a suitable sense to a
p K,et
filtered log F–isocrystal (E,Fil ,ϕ ) on X /K (see 6.13 for a precise definition).
E E k
For such a sheaf L, the´etale cohomology H∗(Xo ,L ) is a Galois representation
K,et K
(here L denotes the restriction of L to Xo ) and the log de Rham cohomology
K K,et
H∗ (X ,E) is naturally viewed as an object of MF . In [Fa1, 5.6], Faltings
dR K K
shows that the representation H∗(Xo ,L ) is crystalline and is associated to
K,et K
H∗ (X ,E).
dR K
The main goal of this paper is to generalize to the level of certain homotopy
types the above comparisons between cohomology.
1.3. Before explaining the main results of the paper, let us discuss an example
whichprovidedmotivationforthisworkandhopefullyhelpsputthemoretechnical
results below in context. This example is not discussed in the body of the paper
and the reader who so desires can skip to 1.5. Let g ≥ 3 be an integer and Mo
g
the moduli space of smooth curves of genus g. If we fix a point p ∈ Mo(C)
g
corresponding to a curve C, the fundamental group π (Mo(C),p) (where Mo(C)
1 g g
is viewed as an orbifold) is naturally identified with the so–called mapping class
group Γ which has a long and rich history (see [Ha1, Ha3, H-L, Na] for further
g
discussion). The first homology of the universal curve over Mo defines a local
g,C
system on Mo which by our choice of base point defines a representation
g,C
(1.3.1) Γ −→Aut(H (C,Z)).
g 1
The kernel is the so–called Torelli group denoted T . Associated to the represen-
g
tation 1.3.1 is a pro–algebraic group G called the relative Malcev completion and a
factorization
(1.3.2) Γ →G →Aut(H (C,C)).
g 1
ThekernelofG →Aut(H (C,C))isapro–unipotentgroupwhichwedenotebyU .
1 g
If T denotes the Malcev completion of T , then there is a natural map T → U
g g g g
whose kernel is isomorphic to Q [Ha3] (here the assumption g ≥ 3 is used). Let
u denote the Lie algebra of U . The interest in the group u derives from the fact
g g g
that it carries a natural mixed Hodge structure which Hain and others have used
to obtain information about the group T .
g
The construction of the mixed Hodge structure on u suggests that the Lie
g
algebra u should in a suitable sense be a motive over Z. In particular, there
g
should be an ´etale realization, de Rham realization, and a p–adic Hodge theory
relating the two similar to the cohomological theory. A consequence of the work in
this paper is that this is indeed the case.
To explain this, choose the curve C to be defined over Q and assume C
p
has good reduction. Let π : Co → Mo be the universal curve and let E =
g,Qp
R1π∗(Ω•Co/Mog,Qp) be the relative de Rham cohomology of Co which is a module
with connection on Mo, and let L = R1πoQ be the relative p–adic ´etale coho-
g ∗ p
mology. The module with connection E has a natural structure of a filtered log
F–isocrystal (E,Fil ,ϕ ) and is associated to the smooth Q –sheaf L [Fa1, 6.3].
E E p
(cid:3)
Let (cid:8)E(cid:9) be the smallest Tannakian subcategory of the category of modules with
⊗
connection on Mo which is closed under extensions and contains E, and let
g,Qp
1. INTRODUCTION 3
(cid:8)(cid:2)L (cid:9) denote the smallest Tannakian subcategory of the category of smooth Q –
Q p
p ⊗
sheavesonMo whichisclosedunderextensionsandcontainsL (therestriction
g,Qp Qp
of L to Mo ). The point p ∈ Mo (Q ) defined by C defines fiber functors for
g,Qp g,Qp p
these Tannakian categories, and using Tannaka duality we obtain pro–algebraic
groups π ((cid:8)(cid:3)E(cid:9) ) and π ((cid:8)(cid:2)L (cid:9) ) over Q . The group π ((cid:8)(cid:3)E(cid:9) ) comes equipped
1 ⊗ 1 Q p 1 ⊗
p ⊗
with a Frobenius automorphism ϕ (really a semi–linear automorphism but we are
working over Q ) and the group π ((cid:8)(cid:2)L (cid:9) ) comes equipped with an action of the
p 1 Q
p ⊗
Galois group Gal(Q /Q ). Let udR (resp. uet) denote the Lie algebra of the pro–
p p g g
unipotent radical of π ((cid:8)(cid:3)E(cid:9) ) (resp. π ((cid:8)(cid:2)L (cid:9) )). The Frobenius automorphism
1 ⊗ 1 Q
p ⊗
ϕ induces an automorphism ϕ of udR and the Galois action on π ((cid:8)(cid:2)L (cid:9) ) in-
udgR g 1 Qp ⊗
duces a Galois action ρ on uet. By [Ha2, 3.1] any embedding Q (cid:2)→ C induces
uegt g p
a natural isomorphisms of Lie algebras ug (cid:6) udgR⊗Qp C and ug (cid:6) uegt⊗Qp C. It is
therefore natural to call udR (resp. uet) the de Rham (resp. ´etale) realization of
g g
u . The p–adic Hodge theory studied in this paper (in particular 1.10 below) now
g
yields the following result:
Theorem 1.4. The Gal(Q /Q )–representation uet is pro–crystalline, and
p p g
there is a natural isomorphism D(uet) (cid:6) udR compatible with the Frobenius au-
g g
tomorphisms.
Similar results also hold for the moduli spaces Mo of n–pointed genus g
g,n
curves. Note that Mo (cid:6) P1−{0,1,∞} is the case studied by Deligne in [De2].
0,4
Theorem 1.4 for P1 −{0,1,∞} had previously been obtained by Hain and Mat-
sumoto [H-M, 9.8], as well as Shiho [Sh3] and Tsuji. Pridham has also obtained
some of the results of this paper using a different method [Pr].
1.5. Asin[Ol1]weworkinthispapersystematicallywithsimplicialpresheaves
and stacks [Bl, H-S, Ja, To1]. For any ring R, let SPr(R) denote the category of
simplicial presheaves on the category of affine R–schemes, and let SPr∗(R) denote
the category of pointed objects in SPr(R). By [To1, 1.1.1], there is a natural
model category structure on SPr(R) and SPr∗(R), and we write Ho(SPr(R)) and
Ho(SPr∗(R)) for the resulting homotopy categories.
Let X/V be as in 1.1, and assume given a section x:Spec(V)→Xo. Let L be
acrystallinesheafonXo associatedtosomefilteredlogF–isocrystal(E,Fil ,ϕ ).
K E E
Let C denote the smallest full Tannakian subcategory of the category of smooth
et
Q –sheaves on Xo closed under extensions and containing L . Similarly let C
p K K dR
denotethesmallest full Tannakiansubcategoryofthecategoryofmoduleswithin-
tegrable connection on Xo/K closed under extensions and containing E. Also, let
K
(cid:8)LK(cid:9)⊗ (resp. (cid:8)E(cid:9)⊗) denote the Tannakian subcategory of the category of smooth
sheaves on Xo generated by L (resp. the Tannakian subcategory of the cate-
K K
gory of modules with integrable connection on Xo/K generated by E), and let
K
π1((cid:8)LK(cid:9)⊗,x¯) (resp. π1((cid:8)E(cid:9)⊗,x)) denote the Tannaka dual of (cid:8)LK(cid:9)⊗ (resp. (cid:8)E(cid:9)⊗)
with respect to the fiber functor defined by x.
Assumption 1.6. Assume that the groups π1((cid:8)LK(cid:9)⊗,x¯) and π1((cid:8)E(cid:9)⊗,x) are
reductiveandthatE hasunipotent localmonodromy(see 4.16for whatthis means).