Ramification theory for D -modules in positive
characteristic
Lars Kindler
Freie Universität Berlin
Salt Lake City, July 2015
Fix a field k, X/k a smooth connected k-variety, and let DX/k be the sheaf of
differential operators on X relative to k. It is a sheaf of bi-OX -algebras acting
k-linearly on OX from the left. It is filtered by order
[ ≤n
DX/k =
DX/k .
n≥1
If char(k) = 0:
≤1
DX/k generated by DX/k
, i.e., by OX and k-linear derivations on OX .
If char(k) = p > 0:
for every n, there are operators of order pn which are not a product of
lower degree operators.
Corollary
In the situation above, let char(k) = 0. There is an equivalence
left-DX/k -modules ↔ OX -modules with flat connection
This is not true if char(k) > 0.
Lars Kindler (Freie Universität Berlin)
D -modules in positive characteristic
Salt Lake City, July 2015
2 / 11
Definition
Let k be a field, X a smooth k-variety. Define
Strat(X) := category of left-DX/k -modules which are coherent as OX -modules.
The objects are called stratified bundles (automatically locally free).
In characteristic 0, Strat(X) is equivalent to the category of vector bundles
with flat connection.
There is a good notion of regular singularity for stratified bundles.
Definition
Define Stratrs (X) ⊂ Strat(X) to be the full subcategory with objects the
regular singular stratified bundles.
Lars Kindler (Freie Universität Berlin)
D -modules in positive characteristic
Salt Lake City, July 2015
3 / 11
Both Strat(X) and Stratrs (X) are k-linear ⊗-categories; neutral
Tannakian if there exists x ∈ X(k). There is an associated quotient map
π1Strat (X, x) π1Strat,rs (X, x)
of affine k-group schemes.
Given f : Y → X finite étale, f∗ OY is canonically a stratified bundle,
obtain Cov(X) ,→ Strat(X), and
π1Strat (X, x) π1ét (X, x)
(considered as profinite k-group scheme).
Lars Kindler (Freie Universität Berlin)
D -modules in positive characteristic
Salt Lake City, July 2015
4 / 11
Theorem
Let k be algebraically closed.
(dos Santos) The profinite completion of π1Strat (X, x) is π1ét (X, x).
Strat,rs
(K.) The profinite completion of π1
commutative diagram
π1Strat (X, x)
(X, x) is π1tame (X, x) and there is a
π1Strat,rs (X, x)
profinite completion
π1ét (X, x)
profinite completion
π1tame (X, x)
Here, the notion of tameness is the one of Wiesend, Kerz-Schmidt: A finite
étale covering f : Y → X is tame if and only if f is tame with respect to all
geometric discrete valuations on k(X) (or, equivalently, if and only if it is
tamely ramified on all regular curves mapping to X ).
Lars Kindler (Freie Universität Berlin)
D -modules in positive characteristic
Salt Lake City, July 2015
5 / 11
What about higher singularities?
Let k = k̄, char(k) = p > 0 and C/k be a smooth curve, x ∈ C a closed point,
C := C \ {x}. Want to study “irregularity” of stratified bundles on C (formally)
locally around x.
d) ∼
Fix an identification Frac(O
C,x = k((t)).
If E ∈ Strat(C), then E ⊗OC k((t)) is a finite dimensional k((t))-vector
(n)
space, together with an action of operators ∂t
(n)
∂t “ = ”
, n ≥ 0. Heuristic:
1 n n
∂ /∂t .
n!
Such an object is sometimes called iterated differential module
(Matzat-van der Put).
Abusing notation slightly, write Strat(k((t))) for the category of such
objects.
For E ∈ Strat(k((t))), regular singularity means that there exists a
(n)
kJtK-lattice E ⊂ E , stable under t n ∂t
Lars Kindler (Freie Universität Berlin)
for all n.
D -modules in positive characteristic
Salt Lake City, July 2015
6 / 11
Example
For α ∈ Z p , let O(α) = k((t)) with
(n)
∂t (1)
α −n
:=
t .
n
O(α) is regular singular, and every rank 1 object of Strat(k((t))) is
isomorphic to O(α) for some α (Gieseker).
Consider the Artin-Schreier extension L := k((t))[u]/(u p − u − t −1 ) and
define
E := 1 · k((t)) ⊕ u · k((t)) ⊂ L.
(n)
This is a rank 2 stratified bundle, i.e., stable under ∂t
(n)
extension of k((t)) by itself (with its canonical ∂t
regular singular! Proof:
n (pn )
t p ∂t (u)
Lars Kindler (Freie Universität Berlin)
n (pn )
= t p ∂t
for all n. It is an
-operation), and it is not
1 1
1
+ p + . . . + pn
t t
t
D -modules in positive characteristic
.
Salt Lake City, July 2015
7 / 11
In fact, these two examples are characteristic:
Proposition (Matzat-van der Put, Gieseker)
Let E ∈ Strat(k((t))).
E is a successive extension of 1-dimensional stratified bundles.
∼ Ln O(αi ), for
E ∈ Strat(k((t))) is regular singular, if and only if E =
i=1
some α1 , . . . , αn ∈ Z p , i.e., if and only if E is semi-simple.
Unlike in characteristic 0, the irregularity of a stratified bundle on k((t)) is not
encoded in its irreducible constituents, but in the extension data.
Question
Is there an invariant Irr(E) ∈ Z≥0 or Q≥0 “measuring” the irregularity of E ?
Philosophically analogous to irregularity number of a differential module on
C((t)) and the Swan conductor of an `-adic sheaf on k((t)). But note that both
are additive in short exact sequences!
Lars Kindler (Freie Universität Berlin)
D -modules in positive characteristic
Salt Lake City, July 2015
8 / 11
Both the Swan conductor of an `-adic sheaf, and the irregularity of a complex
differential module come from a filtration of the absolute (differential) Galois
group.
Theorem (K.)
Let char(k) = p > 0. The category Strat(k((t))) admits a neutral fiber functor.
There is a section to the restriction functor
Strat(P1k \ {0, ∞})
[
⊗ Frac(O
)
P1 ,0
k
Strat(k((t))).
“canonical extension”
∼
[
Here we use the identification Frac(O
P1 ,0 ) = k((t)).
Compare to to canonical extension of `-adic sheaves (Katz-Gabber) and complex
diff. modules (Katz).
Lars Kindler (Freie Universität Berlin)
D -modules in positive characteristic
Salt Lake City, July 2015
9 / 11
Given a fiber functor ω , we write Gdiff (ω) for the associated affine k-group
scheme, the absolute differential Galois group of k((t)).
Corollary
Let k be algebraically closed and fix a separable closure of k((t)).
There is a short exact sequence
0
Gdiff (ω)
P(ω)
Grs
diff (ω)
0
where Grs
diff (ω) is the affine group scheme attached to the category
rs
Strat (k((t))) and ω .
The profinite completion of this sequence is (up to inner automorphism)
0
I 0+
Gal(k((t))sep /k((t)))
b (p0 )
Z
0
where I 0+ is the wild ramification group.
P(ω) is smooth, unipotent.
Lars Kindler (Freie Universität Berlin)
D -modules in positive characteristic
Salt Lake City, July 2015
10 / 11
Definition
For λ ∈ R>0 define P(ω)λ as the preimage of I λ ⊂ I 0+ .
This defines a filtration on Gdiff (ω) lifting the ramification filtration of
Gal(k((t))sep /k((t))).
Question
How to describe P(ω)λ in D -module theoretic terms? In other words, how to
describe the subcategory
Repk Gdiff (ω)/P(ω)λ ,→ Strat(k((t)))?
Lars Kindler (Freie Universität Berlin)
D -modules in positive characteristic
Salt Lake City, July 2015
11 / 11