Regular singular stratified bundles and tame ramification

Let X be a smooth variety over an algebraically closed field k of positive characteristic. We define and study a general notion of regular singularities for stratified bundles (i.e. O_X-coherent D_X-modules) on X without relying on resolution of singularities. The main result is that the category of regular singular stratified bundles with finite monodromy is equivalent to the category of continuous representations of the tame fundamental group on finite dimensional k-vector spaces. As a corollary we obtain that a stratified bundle with finite monodromy is regular singular if and only if it is regular singular along all curves mapping to X.


Introduction
If X is a smooth, connected, complex variety, then it is an elementary fact that a vector bundle with flat connection and finite monodromy at a closed point x ∈ X(C) is automatically regular singular. This implies, together with the Riemann-Hilbert correspondence as developed in [Del70], that the category of flat connections with finite monodromy is equivalent to the category Repf cont C πé t 1 (X, x) of continuous representations of theétale fundamental group of X on finite dimensional complex vector spaces equipped with the discrete topology. The goal of this article is to establish an analogous statement over an algebraically closed field k of positive characteristic p.
If X is a smooth, connected k-variety, then both the category of vector bundles with flat connection and the category of coherent O X -modules with flat connection lack many of the nice features that they have over the complex numbers. In particular they are not Tannakian categories over k.
To remedy this fact we pass to the perspective of modules over the ring of differential operators D X/k on X: Over C, the category of vector bundles with flat connection is equivalent to the category of left-D X/C -modules which are coherent as O X -modules. It turns out that if X is a smooth kvariety with k of positive characteristic, then the category of O X -coherent D X/k -modules is Tannakian over k. This was already worked out in [SR72]. Following Grothendieck and Saavedra we call an O X -coherent D X/k -module a stratified bundle (see Remark 2.2).
The content of this article is the definition and study of a sensible notion of regular singularity for stratified bundles in positive characteristic. The main result is the following: 2010 Mathematics Subject Classification. Primary: 14E20, 14E22. This work was supported by the Sonderforschungsbereich/Transregio 45 "Periods, moduli spaces and the arithmetic of algebraic varieties" of the DFG. Theorem 1.1. Let k be an algebraically closed field of positive characteristic. If X is a smooth, connected k-variety, and x ∈ X(k) a rational point, then a stratified bundle E on X is regular singular with finite monodromy at x if and only if E is trivialized on a finite tame covering.
The notion of tameness used here is due to Wiesend and extensively studied in [KS10]. From Theorem 1.1 we easily obtain the following, pleasantly conceptual statement: Corollary 1.2. If X is a smooth, connected k-variety, then after choice of a base point x ∈ X(k), the fiber E| x of a regular singular stratified bundle E carries a functorial π tame 1 (X, x)-action, and this functor induces an equivalence of categories between the category of regular singular stratified bundles with finite monodromy at x and the category Repf cont k π tame 1 (X, x) of finite dimensional continuous k-representations of π tame 1 (X, x).
From the main result of [KS10] we obtain our third main result: Corollary 1.3. If X is a smooth, connected k-variety, then a stratified bundle E on X with finite monodromy is regular singular, if and only if it is regular singular along all curves on X.
We make a few remarks about the definition of the notion of regular singularity and give a brief outline of the article: If the k-variety X has a good compactification, i.e. if there exists a smooth, proper k-variety X, such that X is a dense open subset of X with a strict normal crossings divisor as complement, then the notion of regular singularity is formally the same as in characteristic 0: A stratified bundle E is regular singular if and only if it extends to an O X -coherent, O X -torsion free D X/k (log X \ X)-module. This definition was explored in [Gie75]. Unfortunately, in characteristic p > 0, it is not known whether every X admits a good compactification. One first approach to a general definition of regular singularity could be to use [dJ96] to replace X by an alteration Y → X, such that Y admits a good compactification. Unfortunately, de Jong's theorem and its refinements (e.g. Gabber's prime-to-ℓ alterations) do not provide control over the wild part of the alteration Y → X, which implies that one cannot use Y to check stratified bundles on X for regular singularities 1 . For example, an Artin-Schreier covering f : is not. Instead, we use the fact that regular singularity should be a notion local in the codimension 1 points at "infinity", and we define a stratified bundle to be regular singular, if it is regular singular with respect to all good partial compactifications of X, see Section 4.
In Section 2 we establish the basic facts about stratified bundles and monodromy groups. Section 3 provides the technical basis for the discussion of regular singularities, which starts in Section 4. In Section 5 we prove an extension result for regular singular stratified bundles, which is exploited in Section 6. Here we prove Theorem 1.1 with respect to a fixed good 1 Purely inseparable alterations on the other hand could be very helpful, see [AO00, Question 2.9]. partial compactification. In Section 7 and 8 we finally complete the proofs of Theorem 1.1 and Corollary 1.3.
To close this introduction, we make a remark about the assumption that the base field k be algebraically closed. The argument at the end of the proof of [EM10,Prop. 2.4] shows that for X proper over k, a stratified bundle E on X is trivial if and only if E ⊗ kk is a trivial stratified bundle (with respect tok) on X × kk . This shows that Corollary 1.2 is false even for proper X, for example for X = Spec k, or X = P r k , if k is not algebraically closed (this was also noted in [dS07, 2.4]). Since establishing Corollary 1.2 as direct analogue to the classical situation over the complex numbers is the main motivation behind writing this article, we allow ourselves to always assume k to be algebraically closed.

Acknowledgements:
The results contained in this article are part of the author's dissertation [Kin12], written under the supervision of Hélène Esnault. The author is extremely grateful for the support, kindness, and generosity which Prof. Esnault offered during his graduate studies, and continues to offer. Additionally, the author wishes to express heartfelt thanks to Prof. Alexander Beilinson and the University of Chicago for the opportunity to visit Chicago in February and March 2012. Large parts of the arguments written down here were developed during this stay.
Notation: If k is a field and G an affine k-group scheme, then we write Vectf k (resp. Repf k G) for the category of finite dimensional k-vector spaces (resp. representations of G on finite dimensional k-vector spaces). Similarly, Vect k (resp. Rep k G) denotes the category of all k-vector spaces (resp. all representations of G on k-vector spaces). If T is a Tannakian category over k, and ω : T → Vectf k a fiber functor, then we write π 1 (T , ω) for the affine k-group scheme such that ω induces an equivalence T → Repf k π 1 (T , ω) ([DM82, Thm. 2.11]).
If k is a field and X an affine k-scheme, then we say that global sections x 1 , . . . , x n ∈ H 0 (X, O X ) are coordinates for X if the morphism X → A n k defined by them isétale.

Stratified Bundles and Monodromy
In this section we collect some facts about the category of stratified bundles and its Tannakian properties, and we recall the notion of the monodromy group of a stratified bundle. In all of this section, we fix an algebraically closed field k of positive characteristic p, and X will always denote a smooth, connected, separated k-scheme of finite type.
Definition 2.1. We write Strat(X) for the category of stratified bundles on k, i.e. for the category of left-D X/k -modules which are coherent as O Xmodules (when considering sections of O X as differential operators of order 0), together with D X/k -linear morphisms. Here D X/k is the sheaf of rings of differential operators of X relative to k, as developed in [EGA4,§16]. A stratified bundle is called trivial, if it is isomorphic to O n X together with its canonical diagonal left-D X/k -action.
If E is a stratified bundle, we write E ∇ for the sheaf of horizontal sections of E, defined by Note for example that (O n X ) ∇ (U ) = k n for all U ⊆ X open. Remark 2.2. Since X is smooth over k, a stratified bundle in our sense is precisely a coherent O X -module equipped with a "stratification" in the sense of [Gro68]. Moreover a stratified bundle is automatically a locally free O X -module, see e.g. [BO78,2.17], so the name "stratified bundle" is at least historically appropriate.
. The category Strat(X) is a k-linear Tannakian category, and a rational point x ∈ X(k) gives a neutral fiber functor Definition 2.4. Write E ⊗ for the smallest full sub-Tannakian subcategory of Strat(X) containing a stratified bundle E ∈ Strat(X). The objects of E ⊗ are subquotients of objects of the form P (E, E ∨ ), with P (r, s) ∈ N[r, s].
We analyze how the above categories behave when passing to an open subscheme of X: Lemma 2.5. Let U be an open dense subscheme of X. The following statements are true: (a) If E ∈ Strat(X), then the restriction functor ρ U,E : Proof. By our standing assumptions, X is connected. Clearly (b) follows directly from (a). We first prove (c). Assume codim X (X \ U ) ≥ 2. Denote by j : U ֒→ X the open immersion. Let E be a stratified bundle on U , in particular a locally free, finite rank O U -module. Then j * E is O X -coherent by the assumption on the codimension ([SGA2, Exp. VIII, Prop. 3.2]), and it carries a D X/k -action, since j * D U/k = D X/k . Thus it is also locally free. If E ′ is any other locally free extension of E to X, then we get a short exact the D X/k -structures. It follows that the functors j * and j * are quasi-inverse to each other, which proves that j * is an equivalence.
To prove (a), let codim X (X \ U ) be arbitrary. Let us first show that ρ U,E : E ⊗ → E| U ⊗ is essentially surjective. If F is an object of E ⊗ , and F ′ a subobject of ρ U (F ) = F | U , then j * F ′ ⊆ j * (F | U ). The quasi-coherent O X -modules j * F ′ and j * (F | U ) carry D X/k -actions, such that j * F ′ ⊆ j * (F | U ) and F ⊆ j * (F | U ) are sub-D X/k -modules. Then F ′ X := j * F ′ ∩ F is an O Xcoherent D X/k -submodule of F extending F ′ . Thus we have seen that the essential image of ρ U is closed with respect to taking subobjects.
But this also shows that F | U /F ′ can be extended to X: We just saw that F ′ extends to F ′ X ⊆ F , and since ρ U is exact, this means that (F/F ′ X )| U = F | U /F ′ . It follows that the essential image of ρ U is closed with respect to taking subquotients.
The objects of E| U ⊗ are subquotients of stratified bundles bundles of the form P (E| U , (E| U ) ∨ ), with P (r, s) ∈ N[r, s]. We can lift all of the objects It remains to check that ρ U,E : , and similarly over U , we may replace F 1 by O X . Moreover, by (c), we may remove closed subsets of codimension ≥ 2 from X, so without loss of generality we may assume that X \ U is the support of a smooth irreducible divisor D with generic point η. To finish the proof, we may shrink X around η. Thus we may assume that X = Spec A for some finite type k-algebra A, that F 2 corresponds to a free A-module, say with basis e 1 , . . . , e n , and that we have global coordinates x 1 , . . . , x n , such that D = (x 1 ).
and in particular that . By induction, we assume ∂ (a) x 1 (φ i ) ∈ φ i A for every a ≤ m. But then we compute to see that ∂ (m) x 1 (φ i ) ∈ φ i A, so this holds for all m ≥ 1. In particular we see that the pole order of ∂ (m) x 1 (φ i ) along D is at most the pole order of φ i along D, for all m ≥ 0. This shows that φ i ∈ A for every i, so φ is defined over all of X.
Corollary 2.6. If E is a stratified bundle, then E ∇ is the constant sheaf associated with a finite dimensional k-vector space. We will usually identify E ∇ with this vector space.
Proof. A horizontal section of E over some open U spans a trivial substratified bundle of E| U , which lifts to X by the lemma. Now we proceed to the notion of monodromy, which is completely analogous to the classical notion in characteristic 0.
Definition 2.7. If ω : E ⊗ → Vectf k is a fiber functor, we denote by G(E, ω) := π 1 ( E ⊗ , ω) the monodromy group of E with respect to ω, i.e. the affine, finite type k-group scheme associated via Tannaka duality with the Tannakian category E ⊗ ⊆ Strat(X) and the fiber functor ω. The following theorem (relying on the assumption that k is algebraically closed) is essential for what is to follow: Theorem 2.9 ([dS07, Cor. 12]). If E ∈ Strat(X) and ω : Strat(X) → Vectf k a fiber functor, then the k-group scheme G(E, ω) is smooth.
Remark 2.10. In [dS07, Cor. 12], the above theorem is only formulated in the case that G(E, ω) is finite over k, but the given proof holds more generally. Indeed, by [dS07, Thm. 11], the affine group scheme π 1 (Strat(X), ω) is perfect and thus reduced. This means that any quotient of π 1 (Strat(X), ω) is also reduced.
Definition 2.11. An object E ∈ Strat(X) is said to be finite or to have finite monodromy if there is a fiber functor ω : E ⊗ → Vectf k , such that G(E, ω) is finite over k. Since k is algebraically closed, Theorem 2.9 implies that this is equivalent to G(E, ω) being a constant k-group scheme associated with a finite group.
Remark 2.12. Note that in our situation E ⊗ always has k-linear fiber functors by [Del90,Cor. 6.20]. Thus E is finite if and only if every object of E ⊗ is isomorphic to a subquotient of E ⊕n for some n, see [DM82,Prop. 2.20]. In particular, E has finite monodromy if and only if G(E, ω) is finite for every fiber functor ω : E ⊗ → Vectf k .
Proposition 2.13. If E is a stratified bundle on X, then the following statements are equivalent: Proof. This follows directly from Lemma 2.8.
Remark 2.14. Before we can state the next proposition, we have to recall some facts about Tannakian categories over k; general references are [SR72], [DM82], and [Del90].
This representation is an algebra over the trivial representation, which also is its maximal trivial object (because G acting on itself does not have any invariants), and it has the property that there exists a functorial isomorphism T is a Tannakian category over k, and ω : T → Vectf k a fiber functor, then ω induces a ⊗-equivalence T ∼ = − → Repf k π 1 (T , ω) with G := π 1 (T , ω) an affine k-group scheme, and such that ω is naturally Thm. 2.11]). The right-regular representation (O G , ∆) corresponds via ω to an object A ω ∈ Ind(T ), which is an algebra over the unit object of T , and such that the functor ω : T → Vectf k can be written as T → H 0 (Ind(T ), T ⊗ A ω ).
Proposition 2.15. Let E ∈ Strat(X) be a stratified bundle, and ω : E ⊗ → Vectf k a fiber functor. Write G := G(E, ω). Then there exists a smooth Gtorsor h E,ω : X E,ω → X with the following properties: (a) Every object of E ⊗ has finite monodromy if and only if h E,ω is finiteétale.
From now on assume that G is a finite (thus constant by Theorem 2.9) group scheme, and hence h E,ω finiteétale. Then h E,ω has the following properties.
i.e. a finiteétale Galois covering with group G(k), and Y the sub-Tannakian subcategory of Strat(X) generated by those objects which become trivial after pullback along f , then: Proof. This proposition is fairly well-known, but the author does not know of a complete reference. Certainly all the ideas from the theory of Tannakian categories are contained in [Del90] (particularly [Del90, §9.]) and [DM82]. We use the the facts recalled in Remark 2.14. The main ingredient not intrinsic to the theory of Tannakian categories is the fact that if E ∈ Strat(X) is a stratified bundle then G( E ⊗ , ω) is a smooth k-group scheme by dos Santos' theorem 2.9. In particular, if G( E ⊗ , ω) is finite, then it is finité etale and hence constant if k is algebraically closed.
Back to the notations of the proposition: Let ρ : Strat(X) → Coh(X) denote the forgetful functor. With the fiber functor ω : E ⊗ → Vectf k we associate in Remark 2.14 an object A E,ω of Ind(Strat(X)). Via ρ we can consider A E,ω as a quasi-coherent O X -algebra with D X/k -action, and such that A E,ω corresponds to the right-regular representation of G in Rep k G.
Write h E,ω : X E,ω := Spec A E,ω → X for this G-torsor. The property from Remark 2.14 that Now assume that G is a finiteétale group scheme on k, hence constant. Then h E,ω is a finiteétale morphism; in particular, A E,ω is an O X -algebra in the category Strat(X). Moreover, the . Now everything follows fairly directly from general theory: (d) follows from the fact that Isom ⊗ k (ω, ω ′ ) is an fpqc-torsor on k, but k is algebraically closed, so the torsor is trivial, hence ω ∼ = ω ′ .
Corollary 2.16. If f : Y → X is a finiteétale morphism, and f ′ : Y ′ → X its Galois closure with Galois group G, then Proof so the claim follows. Definition 2.18. Let E ∈ Strat(X) be a stratified bundle and ω : E ⊗ → Vectf k a fiber functor. Write h E,ω : X E,ω → X for the smooth G( E ⊗ , ω)torsor associated with E ⊗ and ω in Proposition 2.15. We call h E,ω the Picard-Vessiot torsor associated with E and ω.
Corollary 2.19. A stratified bundle E on X has finite monodromy if and only if there exists a finiteétale covering f : Proof. If E has finite monodromy then an associated Picard-Vessiot torsor is finiteétale and trivializes E.
Remark 2.20. A caution is in order: If E is a stratified bundle with infinite monodromy, then it is true that the D is not trivial, because it also carries an action of the differential operators relative to X E,ω → X, which were trivial in theétale case.

Logarithmic Differential Operators
We continue to denote by k an algebraically closed field of positive characteristic p.
(a) Let X be a smooth, separated, finite type k-scheme, and X ⊆ X an open subscheme such that the boundary divisor D X := X \ X has strict normal crossings. We denote such a datum by (X, X) and call it a good partial compactification. (b) If (X, X) is a good partial compactification, then D X/k (log D X ) denotes the sheaf of subalgebras of D X/k generated by those differential operators which locally fix all powers of the ideal of the boundary divisor.
(a) A good partial compactification (X, X) gives rise to a log-scheme over Spec k with its trivial log-structure, in the sense of [Kat89]. The sheaf D X/k (log D X ) is an invariant of this relative log-scheme, which can be constructed using an appropriate notion of thickenings in the category of fine saturated log-schemes. Moreover, if the strict normal crossings divisor U \ X is defined by x 1 · . . . · x r = 0, r ≤ n, then D U /k (log D X ∩ U ) is generated by operators of the form To proceed, we need to recall a few elementary facts about congruences for binomial coefficients: Lemma 3.3. Let p be a prime number.
(a) Lucas' Theorem: For a 0 , . . . , a n , b 0 , . . . , b n integers in [0, p − 1], a := a 0 + a 1 p + . . . + a n p n , b : (1 + x p k ) a k mod p. We have the similar functoriality results for D X/k (log D X ) as for D X/k : Proposition 3.4. Let (X, X) and (Y, Y ) be good partial compactifications with boundary divisors D X and D Y , andf : Y → X a morphism such that f (Y ) ⊆ X, i.e. such thatf induces a morphism of the associated log-schemes. Write f :=f | Y . Then the following statements are true: (a)f There exists a canonical morphism fitting in the commutative diagram where the lower horizontal morphism is the classical one arising from the functoriality of the diagonal morphism and its thickenings. Now assume thatf is finite, and fétale. Thenf is faithfully flat. Moreover, (c)f ♯ is an isomorphism iff is tamely ramified with respect to the strict normal crossings divisor D X .
Proof. Everything follows fairly easily from the general point of view of logarithmic structures of [Kat89], sincef being tamely ramified implies that the induced morphism of log-schemes is log-étale. For the sake of self-containedness, we give an explicit proof for the case thatf is finite and fétale, which is the only case needed in the sequel. In this case the proof is essentially a question about finite extensions of discrete valuation rings, by localizing at the generic points of the components of the boundary divisor. Let A ֒→ B be such an extension, x ∈ A and y ∈ B uniformizers. Then x = uy e for some e ≥ 1 and u ∈ B × . We know that K(B) ⊗ D B/k (log (y)) ∼ = − → K(B) ⊗ A D A/k (log (x)) is an isomorphism. Statement (a) is clear, and for (b) we need to show that the above isomorphism maps D B/k (log (y)) to B ⊗ A D A/k (log (x)). We claim that where δ (p m ) y = y p m ∂ (p m ) y as usual. Clearly (1) implies (b). Let us now prove that (1) holds. We compute: Note that er p m x r = c+d=m e p c δ (p d ) x (x r ) by Lemma 3.3, so (2) shows that δ For (c) assume that A ֒→ B is tamely ramified. It suffices to show that δ (p m ) x is in the image off ♯ : D B/k (log (y)) →f * D A/k (log (x)) for every m ≥ 0, becausef ♯ is injective by the separability of the residue extensions. Consider the completions of A and B: A ֒→ B. Replacing B by anétale extension does not change differential operators, so we may assume that u = v e in B. Indeed, by Hensel's Lemma, u has an e-th root in B, if and only if it has an e-th root modulo y, and since e is prime to p by assumption, the extension of the residue fields obtained by adjoining an e-th root is separable. Replacing y by vy, we may assume that x = y e . Then (2) shows x is in the image off ♯ . We proceed inductively: x ∈imf ♯ which completes the proof.
Corollary 3.5. We continue to use the notations from Proposition 3.4. If Iff is finiteétale and tamely ramified with respect to X \ X, and F a D Y /k (log D Y )-module, thenf * F is a D X/k (log D X )-module.

(X, X)-Regular Singular Stratified Bundles
Definition 4.1. If (X, X) is a good partial compactification, then a stratified bundle E is called (X, X)-regular singular if it extends to an O X -torsionfree, O X -coherent D X/k (log D X )-module E on X.
We define Strat rs ((X, X)) to be the full subcategory of Strat(X) consisting of (X, X)-regular singular bundles.
Remark 4.2. The notion of an (X, X)-regular singular stratified bundle with (X, X) an actual good compactification (i.e. X proper) was studied in [Gie75]. Many of his arguments do not use the properness of X and carry over to our setup.
The following proposition shows that to check that a stratified bundle is (X, X)-regular singular, we may always assume that X \ X is a smooth divisor, and we may shrink X around the generic points of X \ X.
Proposition 4.3. Let (X, X) be a good partial compactification, η 1 , . . . , η r the generic points of X \ X, and E a stratified bundle on X. Then E is (X, X)-regular singular, if and only if there are open neighborhoods U i of η i , i = 1, . . . , r, such that E| U i ∩X is (U i ∩ X, U i )-regular singular.
Proof. Only the "if" direction is interesting. Given open neighborhoods U i of η i , i = 1, . . . , r, as in the proposition, we may assume, by shrinking the U i if necessary, that η j ∈ U i if i = j and then that X ⊆ We also have the notion of a pullback of (X, X)-regular singular stratified bundles: Proposition 4.4. Let (X, X), (Y, Y ) be good partial compactifications and f : X → Y a k-morphism, such thatf (X) ⊆ Y . Then for every (X, X)regular singular stratified bundle on X, Proof. This is a direct consequence of Corollary 3.5.
Proposition 4.5. Let (X, X) be a good partial compactification and E, E ′ be (X, X)-regular singular bundles. Then the following stratified bundles are also (X, X)-regular singular: (a) Every substratified bundle F ⊆ E.
. It follows that Strat rs ((X, X)) is a sub-Tannakian subcategory of Strat(X), and, if ι : Strat rs ((X, X)) ֒→ Strat(X) denotes the inclusion functor, then for a (X, X)-regular singular bundle E, restriction of ι gives an equivalence Proof. Again we write D X := X \ X, j : X ֒→ X. For (a), let E be an O Xtorsion-free O X -coherent D X/k (log D X )-module extending E. Then j * F and E are both D X/k (log D X )-submodules of j * E; let F be their intersection.  . Let (X, X) be a good partial compactification such that D X := X \ X is a smooth divisor, and i : D X ֒→ X the closed immersion (D X reduced). Write If E is a locally free O X -coherent D X/k (log D X )-module, then D acts O D Xlinearly on E| D X , and there exists a decomposition such that θ ∈ D acts on F α via θ(s) =ᾱs, whereᾱ is defined as follows: If y is a local defining equation for D X , p N > ord θ and β ∈ N, such that β ≡ α mod p N , then θ(y β ) =ᾱy β mod y β+1 . (4) Remark 4.7.
(a) Let us unravel the definition of the F α after the choice of local coordinates x 1 , . . . , x n as in Remark 3.2 (b), such that D X is locally cut out by, say, x 1 : A section s of E has the property that s + x 1 E ∈ F α if and only if δ Definition 4.8. Let E be an O X -locally free D X/k (log D X )-module with D X := X \ X smooth. The elements α ∈ Z p such that F α = 0 in the decomposition (3), are called exponents of E along D X . If D X is not smooth, but D X = r i=1 D i , D i smooth divisors, then the exponents of E along D i are defined by restricting E to an open set U ⊆ X intersecting D i , but not D j , for j = i. Finally, if E is O X -torsion-free, but not locally free, then the exponents are defined by restricting to an open subset U ⊆ X such that codim X X \ U ≥ 2 and such that E| U is locally free.
Propostion 4.6 shows that this definition does not depend on the choices made.
We state another analogy to the characteristic 0 situation (see [GL76]): Theorem 4.9. Let (X, X) be a good partial compactification, and E an O Xtorsion-free, O X -coherent D X/k (log D X )-module. If the exponents of E do not differ by integers, then E is locally free if and only if it is reflexive.
Remark 4.10. We will see later on (Theorem 5.2) that an (X, X)-regular singular stratified bundle always has an extension to an O X -locally free D X/k (log D X )-module.
Proposition 4.11. Let (Y, Y ) and (X, X) be good partial compactifications with boundary divisors D Y and D X , andf : Y → X a finite morphism, such thatf (Y ) ⊆ X, and such that f :=f | Y isétale. Let D ′ Y be a component of D Y mapping to the component D ′ X of D X . If E an (X, X)-regular singular stratified bundle, then the exponents of the (Y, Y )-regular singular bundle f * E along D ′ Y are the exponents of E along D ′ X multiplied by the ramification index of D ′ Y over D ′ X . Proof. This is again a question about discrete valuation rings. Let A ֒→ B be a finite extension of discrete valuation rings, such that the extension of fraction fields K(A) ֒→ K(B) is separable. Let x be a uniformizer for A and y a uniformizer for B, x = uy e with u ∈ B × . We use the computation from Proposition 3.4. Let E be an A-module with D A/k (log (x))-action. If a ∈ E is such that δ (p m ) x (a) = α p m a + xE for some α ∈ Z p , then (1) shows that which proves the proposition.
Proposition 4.12. Let (X, X) be a good partial compactification such that D X := X \ X is smooth. Let E be an (X, X)-regular singular stratified bundle, and let E and E ′ be two locally free O X -coherent D X/k (log D X )- as in Proposition 4.6, and define Exp( Proof. Let η be the generic point of X \X. To prove the proposition, we may shrink X around η, so that we can assume that X = Spec A is affine, E, E ′ are free, and that there are coordinates x 1 , . . . , x n ∈ H 0 (X, O X ) such that D X = (x 1 ). Write j : X ֒→ X. Then E ∩ E ′ , E, E ′ ⊆ j * E are D X/k (log D X )submodules, and replacing E by E ∩ E ′ , we may assume that E ⊆ E ′ is a We may now consider the situation over O X,η , which is a discrete valuation ring with uniformizer x 1 . For some n ∈ N we have Thus it suffices to show that if α ∈ Z p is an exponent of E η , then α + N α is an exponent of E ′ η for some N α ∈ Z. Let α 1 , . . . , α r be the exponents of E η . If e ∈ E η \ x 1 E ′ η is an element such that δ (m) x 1 (e) = α 1 m e + x 1 E η , then α 1 also is an exponent of E ′ η . If there is no such e, let e 1 , . . . , e r be a lift of a basis of E η /x 1 , such that δ (m) x 1 (e i ) = α i m e i + x 1 E η for all m ≥ 0, and define E −1 as the submodule of E ′ η spanned by x −1 1 e 1 , e 2 . . . , e r . Note that E η E −1 . It is readily checked that E −1 is stable under the δ (m) x 1 , and we show that α 1 − 1 is an exponent of E −1 . Since O X,η /(x 1 ) is a field, and since e i ∈ x 1 E −1 for i > 1, one finds t ∈ E −1 , such that t, e 2 , . . . , e r is a lift of a basis of E −1 /x 1 E −1 , and such that δ (m) x 1 (t) = β m t + x 1 E −1 , for some β ∈ Z p . We compute: for some f ∈ e 2 , . . . , e r . On the other hand, writing x −1 1 e 1 = λ 1 t + j>1 λ j e j with λ i ∈ O X,η , we get Since x −1 1 e 1 ∈ e 2 , . . . , e r , we have λ 1 ∈ (x 1 ). Hence, comparing coefficients shows that β = α 1 − 1, which shows that α 1 − 1 is an exponent of E −1 . If η as above, with exponent α 1 − 2. Since E ′ η /E η has finite length, this process has to terminate, so there is some n such that α 1 − n is an exponent of E ′ η .
Definition 4.13. Let (X, X) be a good partial compactification, such that Remark 4.14. We emphasize that by definition the exponents of an (X, X)regular singular bundle are elements of Z p /Z, while the exponents of an O X -locally free D X/k (log D X )-module are elements of Z p .

τ -Extensions of (X, X)-Regular Singular Stratified Bundles
In this section, we study in which ways a given (X, X)-regular singular bundle E can extend to a O X -locally free D X/k (log D X )-module.
Definition 5.1. Let (X, X) be a good partial compactification, D X the boundary divisor, and τ : Z p /Z → Z p a set-theoretical section of the projection Z p → Z p /Z. If E is an (X, X)-regular singular bundle, then a τextension of E is a finite rank, O X -locally free D X/k (log D X )-module E τ such that the exponents of E τ lie in the image of τ .
Theorem 5.2. If (X, X) is a good partial compactification, D X := X \X, E an (X, X)-regular singular bundle on X, and τ : Z p /Z → Z p a section of the projection, then a τ -extension of E exists and is unique up to isomorphisms which restrict to the identity E → E on X.
Proof. This proof is an extension of the method of [Gie75, Lemma 3.10]. From Proposition 4.3 and Theorem 4.9 it follows easily that we may assume without loss of generality that D X is a smooth divisor with generic point η, and to prove the proposition we may shrink X around η. Hence, we assume that X is affine with coordinates x 1 , . . . , x n , such that D X = (x 1 ), that E is free of rank r, and that there exists an O X -free D X/k (log D X )extension E of E.
Let Exp(E) = {α 1 , . . . , α r } be the set of exponents of E along D X . To prove the existence of a τ -extension, we proceed in two steps: Applying step (a) for an appropriate a ∈ Z + , and then step (b) repeatedly for various i, we obtain a τ -extension.
We construct E i as in step (b) for i = 1. Assume that α 1 = . . . = α ℓ and α j = α 1 for j > ℓ. After perhaps shrinking X around η there exists a lift e 1 , . . . , e r ∈ E of a basis of E/x 1 E, such that δ (m) x 1 (e i ) = α i m e i + x 1 E for all m ≥ 0. If j : X ֒→ X denotes the inclusion, define E 1 to be the O Xsubmodule of j * E spanned by x 1 e 1 , . . . , x 1 e ℓ , e ℓ+1 , . . . , e r . Then x 1 (x 1 e i ) ∈ E 1 , and hence E 1 is stable under the δ (m) x 1 . Moreover, since x 1 e i ∈ x 1 E 1 , it follows that that δ (m) x 1 (x 1 e i ) = α 1 +1 m x 1 e i ≡ 0 mod x 1 E 1 , so α 1 + 1 is an exponent of E 1 . To compute the other exponents, note that for every i > ℓ there exists f i ∈ x 1 e 1 , . . . , x 1 e ℓ , such that If g 1 , . . . , g r is a lift of a basis of E 1 /x 1 such that δ (m) x 1 (g i ) = β i m g i + x 1 E 1 , for some β i ∈ Z p , and β 1 = . . . = β h = α 1 + 1 for ℓ ≤ h ≤ r, then for i > ℓ we write e i = r j=1 λ j g j , λ j ∈ O X . There are two cases: If λ j ∈ (x 1 ) for all j > h, then and compare coefficients with (5). It follows that β j 0 = α i , since f i ∈ g 1 , . . . , g h . This finishes the proof of the existence of a τ -extension. Its unicity follows from the following lemma: Lemma 5.3. Let X = Spec A be an affine k-variety and x 1 , . . . , x n ∈ A coordinates such that D X = (x 1 ). Let X be X \ D X = Spec A[x −1 1 ] and τ : Z p /Z → Z p a section of the canonical projection. If E is a free O Xmodule of rank r with D X/k -action, and if E 1 , E 2 are free τ -extensions of E, via φ : Proof. This argument is an adaption of [AB01,Prop 4.7]. Let M be the free A-module corresponding to E 1 and E 2 . Denote by the two D X/k (log D X )-actions on M , coming from the actions on E 1 , E 2 . By Proposition 4.12 we know that ∇ 1 and ∇ 2 have the same exponents α 1 , . . . , α r along D X . Let s 1 , . . . , s r be a basis of M such that and let s ′ 1 , . . . , s ′ r be a basis of M , such that Assume that there is an integer ℓ i > 0, such that the maximal pole order of the f ij along x 1 is ℓ i . Then . Tensoring the equality with k(x 1 ), we obtain In the completion of the discrete valuation ring A (x 1 ) , we can write f ij = d ij s=−ℓ i a ijs x s 1 , with x 1 not dividing in the a ijs , and a ijs = 0 for s = −ℓ i and some j. But then we can compute Then (6) gives in k(x 1 ) the equation both sides of which are 0, except when s = −ℓ i . But this implies α i k = α j −ℓ i k for all k, and hence α i = α j − ℓ i by Lucas' Theorem 3.3 (a), which is is impossible, as α 1 , . . . , α r lie in the image of τ , and thus map injectively to Z p /Z. Thus ℓ i = 0 and hence φ(s i ) ∈ M .
Corollary 5.4. The essential image of the fully faithful restriction functor is the full subcategory of (X, X)-regular singular bundles with exponents equal to 0 ∈ Z p /Z.
Proof. By the τ -extension theorem 5.2, we have to show that a finite rank O X -locally free D X/k (log D X )-module E with exponents 0 has a canonical D X/k -action. For this we may assume that X is affine with coordinates x 1 , . . . , x n such that D X = (x 1 ), and that E is free. Then having exponents 0 means that for every e ∈ E, and for all m ≥ 1, x 1 (e) = 0 · e + x 1 E.

Thus we can define ∂
(1) x 1 (e) := δ (1) x 1 (e) x 1 . In particular, the D X/k (log D X )-action defines an honest flat connection with p-curvature 0 on E. Then, by Cartier's Theorem ([Kat70, Thm. 5.1]), if (−) (1) denotes Frobenius twist, then X )-module obtained as the sheaf of sections s of E such that ∂ (1) x i (s) = 0 for all i. Moreover, E 1 also has exponents 0, and δ (p) x 1 acts as δ (1) x (1) 1 on E 1 . We reapply the argument to give meaning to the action of ∂ (1) x 1 on E. Then we apply Cartier's Theorem again, etc.
Remark 5.5. Corollary 5.4 reveals a big difference to the classical situation over the complex numbers: If (X, X) is a good partial compactification over C, then a flat connection on X can be (X, X)-regular singular with all exponents 0 (i.e. with nilpotent residues), but still not extend to a flat connection on X.

(X, X)-Regular Singular Stratified Bundles with Finite Monodromy
We are ready to prove Main Theorem 1.1 with respect to a fixed good partial compactification. As before, let k denote an algebraically closed field of characteristic p > 0.
Theorem 6.1 ((X, X)-Main Theorem). Let (X, X) be a good partial compactification (Definition 3.1). Then for a stratified bundle E ∈ Strat(X), the following statements are equivalent: (a) E is (X, X)-regular singular and has finite monodromy.
(b) There exists a finiteétale covering f : Y → X, tamely ramified with respect to X \ X, such that f * E ∈ Strat(Y ) is trivial.
Remark 6.2. Recall that in the situation of Theorem 6.1 the morphism f is tamely ramified with respect to X \ X, if the discrete rank 1 valuations of k(X) associated with the codimension 1 points of X \ X are tamely ramified in k(Y ).
The theorem will be deduced from the following lemma, which is the technical heart of the proof: Lemma 6.3 (Main Lemma). Let (X, X) be a good partial compactification and f : Y → X a finite Galoisétale morphism. Then the stratified bundle f * O Y is (X, X)-regular singular, if and only if f is tamely ramified with respect to X \ X.
Proof of Theorem 6.1 (assuming Lemma 6.3). Let E be an (X, X)-regular singular stratified bundle with finite monodromy. Let ω : E ⊗ → Vectf k be a fiber functor, and h E,ω : X E,ω = Spec A E,ω → X the Picard-Vessiot torsor associated with E and ω (Definition 2.18). Then A E,ω ∈ E ⊗ is (X, X)-regular singular, so the Main Lemma 6.3 implies that h E,ω is tamely ramified with respect to X \ X. By construction h * E,ω E is trivial. Conversely, if f : Y → X is finiteétale and tamely ramified with respect to X \ X, as stratified bundles, and f * O Y is (X, X)-regular singular by the Main Lemma 6.3. Now to the proof of the Main Lemma 6.3: Proof. Corollary 3.5 implies that f * O Y is (X, X)-regular singular if f is tamely ramified with respect to X \ X. Indeed, we may assume that D X := X \ X is a smooth divisor and that X = Spec A is affine, such that X \ X is cut out by a regular element t ∈ A. Then Y = Spec B is affine, and we can write Y for the normalization of X in k(B). After shrinking X around the generic point of D X if necessary, and after replacing Y by the preimage of the smaller X, D Y := Y \ Y is a strict normal crossings divisor, and we get a commutative diagram Then Corollary 3.5 applies and shows that f * O Y is (X, X)-regular singular.
The converse is more involved: Again we may assume without loss of generality that D X is a smooth irreducible divisor with generic point η, and in the construction we may shrink X around η. We proceed in five steps: (a) Note that the exponents of f * O Y are torsion in Z p /Z, because by Proposition 4.11, pulling back an D X/k (log X \X)-extension of f * O Y alongf multiplies the exponents by the ramification indices off along D X , and clearly 1 ], and leth : Z 1 → X be the associated covering. Let Z 1 :=h −1 (X) and h =h| Z 1 . Then h isétale,h finite and tamely ramified with respect to X \ X, and h * f * O Y has exponents equal to 0 in Z p /Z, which means that by Corollary 5.4 there exists a stratified bundle E 1 ∈ Strat(Z 1 ) extending h * f * O Y . (d) Now we claim that there exists a finiteétale coveringḡ : Z → Z 1 such thatḡ * E 1 is trivial. Indeed, this is true for E 1 | Z 1 = h * f * O Y because it is true for f * O Y , and by Proposition 2.13 restriction functor E 1 ⊗ → h * f * O Y ⊗ is an equivalence. (e) We can finish up: Write Z :=ḡ −1 (Z 1 ), g =ḡ| Z and h = h 1 g. Then we have the following diagram of finiteétale maps and h is tamely ramified with respect to X \ X by construction. But also by construction h * f * O Y is trivial, and since h * f * O Y = f Z, * O Y × X Z , Corollary 2.17 shows that f Z is the trivial covering. But this means that the covering h : Z → X dominates f : Y → X, so f is tamely ramified with respect to X \ X.
Now write D X := X \ X and denote by π D X 1 (X, x) the profinite group associated with the Galois category of finiteétale coverings of X which are tamely ramified with respect to D X .
Corollary 6.4. Let (X, X) be a good partial compactification, and D X := X \ X. Let x ∈ X(k) be a rational point. Then the fiber functor ω x : Strat rs ((X, X)) → Vectf k induces an equivalence of the category of (X, X)regular singular stratified bundles with finite monodromy with the category Repf cont k π D X 1 (X, x). In other words: If π D X 1 (X, x) = lim ← −i G i with G i finite, then the maximal pro-étale quotient of π 1 (Strat rs ((X, X)), ω x ) is π D X 1 (X, x) k := lim ← −i (G i ) k , where ω x is the neutral fiber functor associated with x.

Regular Singular Stratified Bundles in General
Since resolution of singularities is not available in positive characteristic, we unfortunately cannot use good compactifications to define regular singularity of stratified bundles in positive characteristic. In this section we present a definition which works in general, and we generalize the results from the previous sections to this new notion of regular singularity.
We continue to denote by k an algebraically closed field of characteristic p > 0, and by X a smooth, connected, separated k-scheme of finite type.
Definition 7.1. A stratified bundle E on X is called regular singular if it is (X, X)-regular singular for all good partial compactifications (X, X) of X. The category Strat rs (X) is defined to be the full subcategory of Strat(X) with objects the regular singular stratified bundles.
where i is one of the natural inclusions C ′ ֒→ C ′ . Define ω φ : E| C ⊗ → Vect k by F → H 0 (Strat(C ′ ), f * F ), see Remark 2.14. This is a k-linear fiber functor since f * (E| C ) is trivial, and we obtain a commutative diagram Indeed, again by Proposition 2.15, we know that (with notations from (7)) for every object N ∈ E ⊗ . By [DM82, Prop. 2.21b] this implies that G( E| C ⊗ , ω φ ) ֒→ G( E , ω) is a closed immersion, and that f : C ′ → C is a G( E| C , ω φ )-torsor; in fact it is the Picard-Vessiot torsor associated with E| C and ω| C , according to the following elementary lemma: From Lemma 8.1 and the results of [KS10] it follows that regular singularity (at least for stratified bundles with finite monodromy) is a property determined on the "2-skeleton" of X: Theorem 8.3. Let X be a smooth, finite type k-scheme, Then a stratified bundle E on X with finite monodromy is regular singular, if and only if E| C := φ * E is regular singular for every k-morphism φ : C → X, with C a regular k-curve.
Proof. Let ω be a neutral fiber functor for E ⊗ , and let h E,ω : X E,ω → X be the Picard-Vessiot torsor for E and ω. Clearly E is regular singular if and only if (h E,ω ) * O X E,ω is regular singular, which by Theorem 7.6 is equivalent to h E,ω being tame. By [KS10,Thm. 4.4], h E,ω is tame if and only if h E,ω × X φ : X E,ω × X C → C is tame for all φ : C → X as in the claim. But by Lemma 8.1, h E,ω × X φ is isomorphic to (a disjoint union of copies of) the Picard-Vessiot torsor associated with E| C and the fiber functor ω φ constructed in Lemma 8.1. This shows that E is regular singular if and only if h E| C ,ω φ is tame for all φ : C → X, if and only if E| C is regular singular for all φ : C → X.
Remark 8.4. It is unknown to the author whether Theorem 8.3 remains true without the finiteness assumption on the monodromy of E. There are partial results assuming resolution of singularities, see [Kin12,Sec. 3.4].