THE BAR CONSTRUCTION AND AFFINE STACKS
MARTIN OLSSON
1.
Introduction
The purpose of this note is to clarify a technical point which arose in our work on nonabelian
padic Hodge theory in [Ol]. While the main content of this paper is purely algebraic and
not concerned with
p
adic Hodge theory, let us first recall the motivation.
1.1.
Let
V
be a complete discrete valuation ring with perfect residue field of characteristic
p >
0 and fraction field
K
of characteristic 0. Fix an algebraic closure
K
⊂
K
. Let
X/V
be a smooth proper scheme, and let
D
⊂
X
be a divisor with normal crossings over
V
.
Assume that
X
K
is connected. Let
X
◦
denote
X

D
and assume given a point
x
∈
X
◦
(
V
).
We can then consider the prounipotent fundamental group of (
X
◦
, x
) in the various motivic
realizations (Betti, ´
etale, de Rham, and crystalline). The ´
etale realization
π
et
1
(
X
◦
K
, x
) is the
prounipotent group scheme over
Q
p
which is Tannaka dual to the category of unipotent
Q
p

local systems on the geometric generic fiber
X
◦
K
. There is a natural action of the Galois group
G
K
:= Gal(
K/K
) on
π
et
1
(
X
◦
K
, x
), and it is shown in [Ol] that the coordinate ring
O
π
et
1
(
X
◦
K
,x
)
is an indcrystalline representation (that is, a direct limit of crystalline representations).
Let us review what this means (see [Fo] for more details).
Let
W
denote the ring of
Witt vectors of the residue field of
V
, and let
K
0
⊂
K
denote the field of fractions of
W
.
Let
σ
:
W
→
W
be the canonical lifting of Frobenius, and write also
σ
for the induced
automorphism of
K
0
. Let B
cris
(
V
) denote Fontaine’s
K
0
algebra defined for example in [Fo].
Recall that B
cris
(
V
) comes equipped with an action of
G
K
, and a semilinear automorphism
ϕ
. The ring B
cris
(
V
) is contained in a
K
algebra B
dR
(
V
) which is a discrete valuation field.
In particular, the valuation on B
dR
(
V
) defines a filtration on B
dR
(
V
). For a continuous finite
dimensional
G
K
representation
M
over
Q
p
, define
D
(
M
) := (
M
⊗
Q
p
B
cris
(
V
))
G
K
,
where the action of
G
K
on
M
⊗
Q
p
B
cris
(
V
) is the diagonal action.
Then
D
(
M
) is a finite
dimensional
K
0
vector space with a semilinear automorphism induced by the automorphism
of B
cris
(
V
). There is a natural map
α
M
:
D
(
M
)
⊗
K
0
B
cris
(
V
)
→
M
⊗
Q
p
B
cris
(
V
)
which is always injective. The representation
M
is called
crystalline
if
α
M
is an isomorphism.
In this case we obtain an isomorphism
D
(
M
)
⊗
K
0
B
dR
(
V
)
M
⊗
Q
p
B
dR
(
V
)
,
which induces a filtration on
D
(
M
)
⊗
K
0
K
.
1
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2
MARTIN OLSSON
1.2.
It is shown in [Ol] that there is a canonical isomorphism of Hopfalgebras over
K
0
D
(
O
π
et
1
(
X
◦
K
,x
)
)
O
π
crys
1
(
X
◦
,x
)
,
where
π
crys
1
(
X
◦
, x
) denotes the Tannaka dual of the category of unipotent log isocrystals on
(
X, D
)
/W
. This isomorphism is compatible with the Frobenius automorphisms, where the
Frobenius automorphism on
π
crys
1
(
X
◦
, x
) is induced by Frobenius pullback on the category of
unipotent log isocrystals.
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 Spring '08
 Grunbaum,F
 Math, Algebra, Category theory, equivalences, Functor, MARTIN OLSSON, dgak, bar construction

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