Pullback of regular singular stratified bundles and restriction to curves
aa r X i v : . [ m a t h . AG ] M a r PULLBACK OF REGULAR SINGULAR STRATIFIEDBUNDLES AND RESTRICTION TO CURVES
LARS KINDLER
Abstract.
A stratified bundle on a smooth variety X is a vector bundlewhich is a D X -module. We show that regular singularity of stratifiedbundles on smooth varieties in positive characteristic is preserved bypullback and that regular singularity can be checked on curves, if theground field is large enough. Introduction If X is a smooth complex variety, then it is proved in [2] that a vectorbundle with flat connection ( E, ∇) on X is regular singular if and only ϕ ∗ ( E, ∇) is regular singular for all maps ϕ ∶ C → X with C a smoothcomplex curve.In this short note we analyze an analogous statement for vector bundleswith D X / k -action on smooth k -varieties, where k is an algebraically closedfield of positive characteristic p > D X / k the sheaf of differential opera-tors of X relative to k . Vector bundles with such an action are called strat-ified bundles, see [3]. A notion of regular singularity for stratified bundleswas defined and studied in loc. cit. under the assumption of the existenceof a good compactification, and in [9] in general. We recall this definition inSection 2.The first result of this article is: Theorem 1.1.
Let k be an uncountable algebraically closed field of charac-teristic p > , X a smooth, separated, finite type k -scheme and E a stratifiedbundle on X . Then E is regular singular if and only if ϕ ∗ E is regularsingular for every k -morphism ϕ ∶ C → X with C smooth k -curve. In [9, Sec. 8] it is proved that Theorem 1.1 holds without the uncount-ability condition for stratified bundles with finite monodromy. This relieson work of Kerz, Schmidt and Wiesend, [8]. For stratified bundles witharbitrary monodromy, the author does not know at present whether theuncountability condition of k in Theorem 1.1 is necessary or not. For thisreason we have to content ourselves with the following general criterion forregular singularity, which easily follows from Theorem 1.1. Corollary 1.2. If X is a smooth, separated, finite type k -scheme, and E astratified bundle on X , then E is regular singular if and only if for every This work was supported ERC Advanced Grant 226257. algebraic closure k ′ of a finitely generated extension of k , and every k ′ -morphism ϕ ∶ C → X k ′ with C a smooth k ′ -curve, the stratified bundle ϕ ∗ ( E ⊗ k ′ ) on C is regular singular. In the course of the proof we establish a general result on pullbacks, whichis of independent interest.
Theorem 1.3.
Let k be an algebraically closed field of characteristic p > and f ∶ Y → X a morphism of smooth, separated, finite type k -schemes. If E is a regular singular stratified bundle on X , then f ∗ E is a regular singularstratified bundle on Y .In other words, if Strat ( X ) denotes the category of stratified bundles on X and Strat rs ( X ) its full subcategory with objects the regular singular stratifiedbundles, then the pullback functor f ∗ ∶ Strat ( X ) → Strat ( Y ) restricts to afunctor f ∗ ∶ Strat rs ( X ) → Strat rs ( Y ) . The difficulty in proving this theorem is the unavailability of resolution ofsingularities. Our proof relies on a desingularization result kindly commu-nicated to H. Esnault and the author by O. Gabber. In [10], Theorem 1.3was only shown in the case that f is dominant.We conclude the introduction with a brief outline of the article. InSection 2 we recall the definition of regular singularity of a stratified bundlevia good partial compactifications. In Section 3 we prove Theorem 1.3, andin Section 4 we establish Theorem 1.1. Acknowledgements
The author wishes to thank H. Esnault, M. Morrow and K. R¨ulling forenlightening discussions, and O. Gabber for communicating Lemma 3.10 tous. 2.
Regular singular stratified bundles
Let k denote an algebraically closed field and X a smooth k -variety, i.e. asmooth, separated, finite type k -scheme. If k has characteristic 0, thengiving a flat connection ∇ on a vector bundle E on X is equivalent to giving E a left- D X / k -action which is compatible with its O X -structure. Here D X / k is the sheaf of rings of differential operators of X relative to k ([5, § k has positive characteristic, the sheaf of rings D X / k is still defined, but aflat connection does not necessarily give rise to a D X / k -module structure on E . Definition 2.1. If X is a smooth, separated, finite type k -scheme, thena stratified bundle E on X is a left- D X / k -module E which is coherent withrespect to the induced O X -structure. The category of such objects is denotedby Strat ( X ) . ULLBACK AND RESTRICTION TO CURVES 3
The usage of the word “bundle” is justified as a stratified bundle is auto-matically locally free as an O X -module. See, e.g., [1, 2.17]. The analogousstatement for flat connections is not true in positive characteristic. Thenotion of a stratification goes back to [6] and a vector bundle on a smooth k -scheme equipped with a stratification relative to k in the sense of loc. cit. isa stratified bundle in the sense of Definition 2.1, and vice versa.To define regular singularity of a stratified bundle, we introduce somenotation. Definition 2.2. ● If X is a smooth, separated, finite type k -schemeand X ⊆ X an open subscheme such that X ∖ X is the support ofa strict normal crossings divisor, then the pair ( X, X ) is called goodpartial compactification of X . If in addition X is proper over k , then ( X, X ) is called good compactification of X . ● If ( Y, Y ) and ( X, X ) are two good partial compactifications, then a morphism of good partial compactifications is a morphism ¯ f ∶ Y → X such that ¯ f ( Y ) ⊆ X . ● If ( Y, Y ) , ( X, X ) are good partial compactifications and if f ∶ Y → X is a morphism, then we say that f extends to a morphism of goodpartial compactifications if there exits a morphism of good partialcompactifications ¯ f ∶ ( Y, Y ) → (
X, X ) , such that f = ¯ f ∣ Y .Let X be a smooth, separated, finite type k -scheme. If k has characteristic0, then there exists a good compactification ( X, X ) according to Hironaka’stheorem on resolution of singularities. By definition, a vector bundle withflat connection ( E, ∇) on X is regular singular, if for some (and equivalentlyfor any) good compactification ( X, X ) , there exists a torsion free, coherent O X -module E extending E and a logarithmic connection ∇ ∶ E → E ⊗ O X Ω X / k ( log X ∖ X ) extending ∇ .If k has positive characteristic then it is unknown whether every smooth X admits a good compactification. We work with all good partial compact-ifications instead. Definition 2.3.
Let k be an algebraically closed field of positive character-istic and X a smooth, separated, finite type k -scheme.(a) If E is a stratified bundle on X and ( X, X ) a good partial com-pactification, then E is called ( X, X ) -regular singular if there existsa D X / k ( log X ∖ X ) -module E , which is coherent and torsion freeas an O X -module, such that E ≅ E ∣ X as stratified bundles. Here D X / k ( log X ∖ X ) is the sheaf of differential operators with logarith-mic poles along the underlying normal crossings divisor X ∖ X , asdefined in [3, Sec. 3]; see also Remark 2.4.(b) A stratified bundle E is called regular singular if E is ( X, X ) -regularsingular for all good partial compactifications ( X, X ) . We writeStrat rs ( X ) for the full subcategory of Strat ( X ) with objects theregular singular stratified bundles. LARS KINDLER
Remark 2.4.
A good partial compactification ( X, X ) gives rise to a loga-rithmic structure on X ([7]), and the associated log-scheme is a log-schemeover Spec k equipped with its trivial log-structure. Associated to this mor-phism of log-schemes is a sheaf of logarithmic differential operators D ( X,X )/ k ,which agrees with the sheaf of rings D X / k ( log X ∖ X ) from [3, Sec. 3]. Inlocal coordinates, if x ∈ X is a closed point and x , . . . , x n ´etale coordinatesaround x such that in a neighborhood of x the normal crossings divisor X ∖ X is defined by x ⋅ . . . ⋅ x r = r ≤ n , then D X / k ( log X ∖ X ) isspanned by the operators x s ∂ ( s ) x , . . . , x sr ∂ ( s ) x r , ∂ ( s ) x r + , . . . , ∂ ( s ) x n , s ∈ Z ≥ where ∂ ( s ) x i is the differential operator such that ∂ ( s ) x i ( x tj ) = ⎧⎪⎪⎨⎪⎪⎩ i ≠ j ( ts ) x t − sj i = j. We refer to [9, Sec. 3] for more details.The notion of regular singularity for stratified bundles is studied in [3,Sec. 3] for smooth varieties X which admit a good compactification and in[9] in general.We conclude this section by recalling the following fact about regularsingularity, which we will use repeatedly in the sequel. Proposition 2.5.
Let E be a stratified bundle on a smooth, separated, finitetype k -scheme X . (a) If ( X, X ) is a good partial compactification then E is ( X, X ) -regularsingular if and only if there exists an open subset U ⊆ X with codim X ( X ∖ ( X ∪ U )) ≥ , such that E ∣ U ∩ X is ( X ∩ U , U ) -regular singular. (b) If there exists a dense open subset U ⊆ X such that E ∣ U is regularsingular, then E is regular singular.Proof. (a) This is [9, Prop. 4.3].(b) Assume that E ∣ U is regular singular. It follows from the first partof this proposition that all we have to show is that for any goodpartial compactification ( X, X ) there exists an open subset U ⊆ X containing all generic points of X ∖ X , such that ( U, U ) is a goodpartial compactification. Write η , . . . , η d ∈ X for the codimension 1points not contained in X . Let U ′ i be an open neighborhood of η i and Z i the closure of ( U ′ i ∩ X ) ∖ U in X . Defining U i ∶ = ( U ∪ U ′ i ) ∖ Z i ,the open subset U ∶ = ⋃ di = U i ⊆ X does the job (note that Z i ∩ U = ∅ ). (cid:3) ULLBACK AND RESTRICTION TO CURVES 5
Remark 2.6.
Once we have proved Theorem 1.3, we will also know thatif E is a regular singular stratified bundle on X then E is regular singularwhen restricted to any open subset.3. Pullback of regular singular stratified bundles
In this section we construct the pullback functor for regular singular strat-ified bundles. We first recall a basic fact, which is obvious from the perspec-tive of log-schemes.
Proposition 3.1 ([9, Prop. 4.4]) . Let k be an algebraically closed field ofpositive characteristic and ¯ f ∶ ( Y, Y ) → (
X, X ) a morphism of good partialcompactifications over k (Definition 2.3). Write f ∶ = ¯ f ∣ Y ∶ Y → X . If E is an ( X, X ) -regular singular stratified bundle on X , then f ∗ E is ( Y, Y ) -regularsingular. Next, we show that in order to prove Theorem 1.3, it suffices to studydominant morphisms and closed immersions separately.
Proposition 3.2.
Theorem 1.3 is true, if and only if it is true for all closedimmersions and all dominant morphisms.Proof.
Without loss of generality we may assume that X and Y are con-nected. Let f ∶ Y → X be as in Theorem 1.3, and let i ∶ Z ↪ X be the closedimmersion given by the scheme theoretic image of f . Since Y is reduced, sois Z . We factor f as Y g Ð→ Z i Ð→ X . Note that g is dominant. Since Z isreduced, there exists an open subscheme U ⊆ X , such that U ∩ Z is regular.Define V ∶ = f − ( U ) , and consider the sequence of maps V g ∣ V ÐÐ→ Z ∩ U i ∣ Z ∩ U ÐÐÐ→ U j Ð→ X where j ∶ U ↪ X is the open immersion. The two outer maps are dominant,the middle map is a closed immersion and all four varieties are regular. Weapply the assumption of this proposition from right to left.Let E be a regular singular stratified bundle on X . By assumption E ∣ U is regular singular on U , then ( i ∣ Z ∩ U ) ∗ E ∣ U is regular singular on Z ∩ U and finally g ∣ ∗ V ( i Z ∩ U ) ∗ E ∣ U = f ∣ ∗ V E ∣ U is regular singular on V . According toProposition 2.5 this means that f ∗ E is regular singular. (cid:3) From this proposition together with Proposition 3.1 we see directly thatto prove Theorem 1.3, it suffices to prove the following statement.
Proposition 3.3.
Let k be an algebraically closed field and f ∶ Y → X amorphism of smooth, separated, finite type k -schemes. Assume that f iseither dominant or a closed immersion. If ( Y, Y ) is a good partial compactification, then there exist ● an open subset V ⊆ Y containing all generic points of Y ∖ Y , and LARS KINDLER ● a good partial compactification ( X, X ) , such that f induces a mor-phism of good partial compactifications ¯ f ∶ ( V ∩ Y, V ) → (
X, X ) . The remainder of this section is devoted to the proof of Proposition 3.3.We first treat the dominant case (Lemma 3.4), then the case of a closedimmersion (Proposition 3.9).
Lemma 3.4.
Proposition 3.3 is true for dominant morphisms f ∶ Y → X .Proof. This is essentially [11, Ex. 8.3.16]. Without loss of generality, we mayassume X and Y to be irreducible. If ( Y, Y ) is a good partial compactifica-tion, we may assume that Y ∖ Y is a smooth divisor, say with generic point η . If X ′ is a normal compactification of X , then after removing a closedsubset of codimension ≥ Y , f extends to a morphism f ′ ∶ Y → X ′ .If f ′ ( η ) ∈ X there is nothing to do; we may take X = X . Otherwise, [11,Ex. 8.3.16] tells us that there is a blow-up X ′′ → X ′ of X ′ in { f ′ ( η )} suchthat f ′ extends to a map f ′′ ∶ Y → X ′′ such that f ′′ ( η ) is a normal codimen-sion 1 point of X ′′ . We define X to be a suitable neighborhood of f ′′ ( η ) tofinish the proof. (cid:3) Let k be an algebraically closed field and X a normal, irreducible, sepa-rated, finite type k -scheme. We write k ( X ) for the function field of X . Werecall a few basic definitions: Definition 3.5.
Let v be a discrete valuation on k ( X ) . ● We write O v ⊆ k ( X ) for its valuation ring, m v for the maximal idealof O v and k ( v ) for its residue field. ● If X ′ is a model of k ( X ) , then a point x ∈ X ′ is called center of v , if O X ′ ,x ⊆ O v ⊆ k ( X ) , and m v ∩ O X ′ ,x = m x . ● v is called geometric if there exists a model X ′ of k ( X ) such that v has a center ξ ∈ X ′ which is a normal codimension 1 point. In thiscase O v = O X ′ ,ξ . Remark 3.6.
Recall that if X ′ is separated over k , then v has at most onecenter on X ′ and if X ′ is proper, then v has precisely one center on X ′ . Proposition 3.7 ([11, Ch. 8, Thm. 3.26], [11, Ch.8 , Ex. 3.14]) . Let X be a normal, irreducible, separated, finite type k -scheme and v a discretevaluation on k ( X ) . (a) We have the inequality trdeg k k ( v ) ≤ dim X − where k ( v ) is the residue field of O v . (b) The discrete valuation v is geometric if and only if equality holds in (1) . ULLBACK AND RESTRICTION TO CURVES 7 (c)
Let X ′ be a normal, proper compactification of X and x the centerof v on X ′ . If k ( v )/ k ( x ) is finitely generated, we can make (b) more precise:Define X ′ ∶ = X ′ . Inductively define ϕ n ∶ X ′ n → X ′ n − as the blow-up of X ′ n − in the reduced closed subscheme defined by { x n − } , where x n − is the center of v on X ′ n − . If (1) is an equalityfor v , then for large n , ϕ n is an isomorphism, i.e. for large n , x n isa codimension point of X ′ n . Now we prove Proposition 3.3 in the case where f is a closed immersion. Lemma 3.8.
Let i ∶ Y ↪ X be a closed immersion of normal, irreducible,separated, finite type k -schemes, and let v be a discrete geometric valuationof k ( Y ) with center y on Y such that k ( v )/ k ( y ) is finitely generated. Write X ∶ = X , Y ∶ = Y , y ∶ = y , and inductively define X n → X n − as the blow-up of X n − in { y n − } , Y n as the proper transform of Y n − , and y n as the center of v on Y n . Then for large n , the center y n of v has codimension in Y n .Proof. This follows directly from Proposition 3.7, using that the inducedmap Y n → Y n − between the proper transforms is naturally isomorphic tothe blow-up of Y n − in { y n − } . (cid:3) Proposition 3.9.
Proposition 3.3 is true for closed immersions.Proof.
Let i ∶ Y ↪ X be a closed immersion of smooth, connected, separated,finite type k -schemes. Let ( Y, Y ) be a good partial compactification. With-out loss of generatlity we may assume that Y ∖ Y is irreducible, and hence(the support of) a smooth divisor. To prove the lemma, we may replace Y by open neighborhoods V of the generic point η of Y ∖ Y and Y by V ∩ Y .Let X ′ be a normal, proper k -scheme containing X as a dense open sub-scheme. After possibly removing a closed subset of codimension ≥ Y we may assume that i extends to a morphism i ′ ∶ Y → X ′ . Note that i ′ ( η ) ∈ X ′ ∖ X : otherwise we would have i ′ ( η ) ∈ i ′ ( Y ) , as i ′ ( Y ) is irreducible,and then the valuation on k ( Y ) associated with η would have two centers,which is impossible as Y is separated over k .By Lemma 3.8 there exists a modification X ′′ → X ′ , which is an isomor-phism over X , such that (perhaps after again removing a closed subset ofcodimension ≥ Y ) i extends to a map i ′′ ∶ Y → X ′′ such that i ′′ ( η ) is a codimension 1 point of the closure of i ′′ ( Y ) in X ′′ . Thus, replacing X ′ by X ′′ we may assume that i ∶ Y ↪ X extends to a closed immersion i ′ ∶ Y ↪ X ′ . LARS KINDLER
It remains to show that we can replace X ′ by a chain of blow-ups withcenters over X ′ ∖ X , and Y with its proper transform, such that i ′ ( η ) is aregular point of X ′ and a regular point of X ′ ∖ X .For this we use a desingularization result kindly communicated to us byOfer Gabber. Lemma 3.10 (Gabber) . (a) Let O be a noetherian local integral domainand p ⊆ O a prime ideal such that O/ p is of dimension and suchthat the normalization of O/ p is finite over O/ p . Write X ∶ = X ∶ = Spec O , C ∶ = C ∶ = Spec O/ p . For n ≥ let X n + be the blow-up of X n at the closed points of C n , and C n + the proper transform of C n in X n + . For large n , C n is regular, and at every closed point ξ of C n we have that if p n ∶ = ker (O X n ,ξ → O C n ,ξ ) , then for every m ∈ N the O C n ,ξ -module p mn / p m + n is torsion-free. (b) If O p is regular, then for large n so is O X n ,ξ .Proof. The normalization of O/ p is finite, and it is obtained by blowingup singular points repeatedly. Thus we may assume that O/ p is a discretevaluation ring.After this reduction, the map C n + → C n induced by the blow-up X n + → X n is an isomorphism. In particular, C n = C n − = . . . = C = Spec O/ p .Write O ′ for the local ring of X in the closed point of C , and π ∈ O fora lift of a uniformizer of the discrete valuation ring O/ p . To ease notationwe will also write π for its image in O/ p . Then O ′ is the localization of thering ∑ i ≥ π − i p ⊆ Frac (O) at a suitable maximal ideal. Moreover, p is thelocalization of the ideal generated by π − p . From this it is not difficult tosee that we get a surjective O/ p -linear morphism ϕ ∶ π − m O/ p ⊗ O/ p p m / p m + ↠ p m / p m + , defined by π − m ⊗ x ↦ π − m x .The O/ p -module ker ( ϕ ) is torsion. Indeed, if x ∈ p m is an element suchthat π − m x ∈ p m + , then π − m x ∈ π −( m + ) p m + , so πx ∈ p m + . This impliesthat ϕ induces a surjective map on torsion submodules.Now let e m ≥ π e m kills the torsion submodule of p m / p m + or equivalently of π − m O/ p ⊗ O/ p p m / p m + . Similarly, let e ′ m ≥ π e ′ m kills the torsion submodule of p m / p m + . Since ϕ induces a surjectivemap on torsion submodules, it follows that e m ≥ e ′ m . If e m >
0, we claim that e m > e ′ m . For this it is sufficient to show that for every element x ∈ p m suchthat πx ∈ p m + , we have π − m ⊗ x ∈ ker ( ϕ ) . But this is clear: ϕ ( π − m ⊗ x ) = π −( m + ) πx ∈ p m + .Repeating the argument for the blow-ups X → X , X → X , and so on,it follows that for fixed m , there exists a minimal integer N ( m ) ≥ p mn / p m + n is torsion free for all n ≥ N ( m ) . It remains to see that the sequence N ( m ) is bounded. Consider the associated graded ring ⊕ m ≥ p m / p m + . The ULLBACK AND RESTRICTION TO CURVES 9 subset of elements which are killed by a power of π is an ideal of this noe-therian ring, hence finitely generated. Thus the sequence of numbers e m isbounded, which implies that the sequence N ( m ) is bounded. This completesthe proof (a).Finally, lets prove (b). Assume that O/ p is a discrete valuation ring, that O p is regular and that p m / p m + is a free O/ p -module for every m ≥
0. Toprove that O is regular, it suffices to show that Spec O/ p → Spec O is aregular immersion, i.e. that for every m ≥ m p / p ↠ p m / p m + (2)is an isomorphism of O/ p -modules. Write K for the fraction field of O/ p .Looking at the commutative diagram O O/ p O p K = O p / p O p we see that (2) is an isomorphism after tensoring with K . In particular,Sym m p / p and p m / p m + have the same rank r . As p m / p m + is a free O/ p -module by assumption, it follows that (2) can be identified with is a surjec-tive endomorphism of a free O/ p -module of rank r and hence is an isomor-phism. (cid:3) Using Lemma 3.10 we finish the proof of Proposition 3.9. Let ϕ ∶ O X ′ ,i ′ ( η ) →O Y ,η be the morphism induced by i ′ . Then ϕ is surjective, as i ′ is aclosed immersion by construction, and if p = ker ( ϕ ) , then p is prime. As O Y ,η = O X ′ ,i ′ ( η ) / p is 1-dimensional, we can apply Gabber’s Lemma 3.10 to O X ′ ,i ′ ( η ) and p : It shows that after replacing X ′ by a chain of blow-ups withcenters over X ′ ∖ X , and Y with its proper transform, i ′ ( η ) lies in X ′′ reg .Thus, after removing a closed subset of codimension ≥ Y we have i ′ ( Y ) ⊆ X ′ reg .Moreover, as i ′ ( η ) is a regular point of i ′ ( Y ) , there is a regular system ofparameters h , . . . , h n of O X ′ ,i ′ ( η ) such that ( h , . . . , h n ) = p = ker (O X ′ ,i ′ ( η ) →O Y ,η ) , and such that h is the uniformizer of the discrete valuation ring O Y ,η = O X ′ ,i ′ ( η ) / p .Without loss of generality we may assume that X ′ ∖ X is the support ofa Cartier divisor with local equation g around i ′ ( η ) . Then g ∈ m i ′ ( η ) , where m i ′ ( η ) is the maximal ideal of O X ′ ,i ′ ( η ) , and we claim that g can be written g = uh m + n ∑ i = a i h i with u ∈ O × X ′ ,i ′ ( η ) , a i ∈ O X ′ ,i ′ ( η ) . Indeed, since i ′ ( Y ) ∩ X ≠ ∅ , we see that g hasnonzero image in O X ′ ,i ′ ( η ) /( h , . . . , h n ) = O Y ,η which is a discrete valuationring with uniformizer h . So g = ¯ uh m mod ( h , . . . , h n ) with ¯ u ∈ O × Y ,η . Anylift u of ¯ u to O X ′ ,i ′ ( η ) is a unit, so the claim follows. Moreover, m >
0, since i ′ ( Y ) / ⊆ X . If for every i > a i h i is divisible by h m we are done, becausein this case we can write g = h m ⋅ unit, so around i ′ ( η ) the reduced inducedstructure on X ′ ∖ X is V ( h ) , hence regular, and i ′ ( Y ) intersects X ′ ∖ X in i ′ ( η ) transversally.If a i h i is not divisible by h m for all i >
0, then we blow up X ′ in { i ′ ( η )} and replace X ′ by this blow-up and Y by its proper transform (note thatthis does not change Y , as i ′ ( η ) is of codimension 1 in i ′ ( Y ) ). Then thelocal ring O X ′ ,i ′ ( η ) has a regular system of parameters ( h , h / h , . . . , h n / h ) .Hence, repeating this process m -times, we can write g = h m ( u + n ∑ i = a i h i / h m ) in O X ′ ,i ′ ( η ) , and we conclude as in the previous paragraph. (cid:3) Checking for regular singularities on curves
We continue to denote by k an algebraically closed field of positive char-acteristic p , and by X a smooth, connected, separated, finite type k -scheme.To prove Theorem 1.1, we first establish the following easy lemma: Lemma 4.1.
Let S be a noetherian scheme, X → S a smooth morphism offinite type k -schemes with X = Spec A affine, U = Spec A [ t − ] , and t ∈ A a regular element. Assume that the closed subscheme D ∶ = V ( t ) ⊆ X isirreducible and smooth over S . If g ∈ A [ t − ] , then the set Pol ≤ n ( g ) ∶ = { s ∈ S ∣ g ∣ U s ∈ Γ ( U s , O U s ) has pole order ≤ n along D s } is a constructible subset of S .Proof. Note that since D → S is smooth, D s ⊆ X s is a smooth divisor forevery s ∈ S , so it makes sense to talk about the pole order of g ∣ U s along D s .Since Pol ≤ n ( g ) = Pol ≤ ( t n g ) , it suffices to show that Pol ≤ ( g ) is con-structible.The element g defines a commutative diagram of S -schemes U X A S P S . g g The image g ( X ) ⊆ P S is a constructible set, so g ( X ) ∩ ({ ∞ } × S ) is a con-structible subset of P S . If pr ∶ P S → S is the structure morphism of P S ,then pr ( g ( X ) ∩ ({ ∞ } × S )) is a constructible subset of S . Finally note that S ∖ Pol ≤ ( g ) = pr (( g ( X ) ∩ ({ ∞ } × S )) . (cid:3) We are now ready to prove Theorem 1.1 with respect to a fixed goodpartial compactification.
Proposition 4.2.
Let ( X, X ) be a good partial compactification and E astratified bundle on X . Assume that for every k -morphism ¯ ϕ ∶ C → X with ULLBACK AND RESTRICTION TO CURVES 11 C a smooth k -curve, the stratified bundle ϕ ∗ E is ( C, C ) -regular singular,where C ∶ = ¯ ϕ − ( X ) , ϕ ∶ = ¯ ϕ ∣ C .If the base field k is uncountable, then E is ( X, X ) -regular singular.Proof. Let k be an uncountable algebraically closed field of characteristic p >
0. We immediately reduce to the case where X is connected and dim X ≥ ≥ X we may assume that D ∶ = ( X ∖ X ) red is a smooth divisor; by treating its components separatedly,we may assume that D is irreducible with generic point η . Shrinking X further, we may assume that ● X = Spec A is affine, ● there exist ´etale coordinates x , . . . , x n ∈ A such that D = V ( x ) . ● E corresponds to a free A [ x − ] -module, say with basis e , . . . , e r .If we write δ ( m ) x ∶ = x m ∂ ( m ) x ∈ D X / k (see Remark 2.4), then for f ∈ A we canalso write δ ( m ) x ( f e i ) = r ∑ j = b ( m ) ij ( f ) e j , with b ( m ) ij ( f ) ∈ A [ x − ] . To show that E is regular singular, it suffices to show that the pole order ofthe elements b ( m ) ij ( f ) along x is bounded by some N ∈ N , because then the D X / k ( log D ) -module generated by ∑ ri = e i A is contained in ( ⊕ ri = x − N e i A ) ,and thus finitely generated over A .Write A as the quotient of a polynomial ring k [ y , . . . , y d ] and ¯ y i for theimage of y i in A . It then suffices to show that the pole order of b ( m ) ij ( ¯ y hc ) has a common upper bound, for 1 ≤ i, j ≤ r , 1 ≤ c ≤ d , m, h ≥ S ∶ = A n − k = Spec k [ x , . . . , x n ] . We get a commutative diagram X X A S S ´etalesmooth (3)We are now in the situation of Lemma 4.1. For N ∈ N consider the con-structible sets Pol ≤ N ( b ( m ) ij ( ¯ y hc )) ⊆ S , and define P ≤ N ∶ = ⋂ i,j,m,c,h Pol ≤ N ( b ( m ) ij ( ¯ y hc )) . This is a closed subset of S . Now since for every closed point s ∈ S the fiber X s is a regular curve over k meeting D transversally, we see that by assump-tion E ∣ X s is ( X s , X s ) -regular singular. But this means that there is some N s ≥
0, such that s ∈ P ≤ N s . In other words, the union ⋃ N ≥ P ≤ N containsall closed points of S . Since k is uncountable and since the P ≤ N are closedsubsets of S , this means there exists some N ≥
0, such that P ≤ N = S . Thedefinition of P ≤ N and Lemma 4.1 imply that the sets Pol ≤ N ( b ( m ) ij ( ¯ y hc )) are dense constructible subsets of S . But a dense constructible subset of an irre-ducible noetherian space contains an open dense subset by [4, Prop. 10.14]. This shows that the pole order of b ( m ) ij ( ¯ y hc ) along x is bounded by N , andthus that E is ( X, X ) -regular singular. (cid:3) Now we can easily finish the proof of Theorem 1.1.
Proof of Theorem 1.1.
We have proved Theorem 1.3 which shows that if E is a regular singular stratified bundle on X , then ϕ ∗ E is regular singular forevery map ϕ ∶ C → X with C a smooth k -variety.For the converse, assume that ϕ ∗ E is regular singular for every morphism ϕ ∶ C → X with C a smooth k -curve. We have to show that E is ( X, X ) -regular singular with respect to every good partial compactification ( X, X ) .But the assumptions of Proposition 4.2 are satisfied, so E is ( X, X ) -regularsingular. (cid:3) Finally, we give a proof of Corollary 1.2.
Proof of Corollary 1.2.
We need to show that E is ( X, X ) -regular singularfor every good partial compactification ( X, X ) , whenever the condition ofthis corollary is satisfied. We may assume that X ∖ X is irreducible. Asin the proof of Theorem 1.1, we reduce to X affine and E free, so thatshowing that E is ( X, X ) -regular singular boils down to showing that thepole order of a certain set of functions in k ( X ) has a common bound. Thisis independent of the coefficients, so we may base change to a field K ⊇ k , K algebraically closed and uncountable (e.g. K ∶ = k (( t )) ). Then we can applythe theorem, to see that E K is regular singular if it is regular singular alongall smooth K -curves. But every such curve is defined over a subextension k ′ of K / k , finitely generated over k . The corollary follows. (cid:3) References [1] P. Berthelot and A. Ogus, Notes on crystalline cohomology, Princeton UniversityPress, Princeton, N.J. (1978), ISBN 0-691-08218-9.[2] P. Deligne, ´Equations diff´erentielles `a points singuliers r´eguliers, Lecture Notes inMathematics, Vol. 163, Springer-Verlag, Berlin (1970).[3] D. Gieseker,
Flat vector bundles and the fundamental group in non-zero characteris-tics , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) (1975), no. 1, 1–31.[4] U. G¨ortz and T. Wedhorn, Algebraic geometry I, Advanced Lectures in Mathemat-ics, Vieweg + Teubner, Wiesbaden (2010), ISBN 978-3-8348-0676-5. Schemes withexamples and exercises.[5] A. Grothendieck, ´El´ements de g´eom´etrie alg´ebrique. IV: ´Etude locale des sch´emas etdes morphismes de sch´emas. , Publ. Math. IHES (1967).[6] ———, Crystals and the de Rham cohomology of schemes , in Dix Expos´es sur laCohomologie des Sch´emas, 306–358, North-Holland, Amsterdam (1968).[7] K. Kato,
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Regular singular stratified bundles and tame ramification , arXiv:1210.5077,to appear in Trans. Amer. Math. Soc. (2012).[10] ———, Regular Singular Stratified Bundles in Positive Characteristic (2012). Disser-tation, Universit¨at Duisburg-Essen.[11] Q. Liu, Algebraic Geometry and Arithmetic Curves, Oxford University Press (2002).
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