aa r X i v : . [ m a t h . AG ] S e p Purity of Crystalline Strata
Jinghao Li and Adrian VasiuSeptember 7, 2018
ABSTRACT.
Let p be a prime. Let n ∈ N ∗ . Let C be an F n -crystal over alocally noetherian F p -scheme S . Let ( a, b ) ∈ N . We show that the reducedlocally closed subscheme of S whose points are exactly those x ∈ S such that( a, b ) is a break point of the Newton polygon of the fiber C x of C at x is purein S , i.e., it is an affine S -scheme. This result refines and reobtains previousresults of de Jong–Oort, Vasiu, and Yang. As an application, we show thatfor all m ∈ N the reduced locally closed subscheme of S whose points areexactly those x ∈ S for which the p -rank of C x is m is pure in S ; the case n = 1 was previously obtained by Deligne (unpublished) and the general case n ≥ KEY WORDS: F p -scheme, F -crystal, Newton polygon, p -rank, purity. MSC 2010:
For a reduced locally closed subscheme Z of a locally noetherian scheme Y ,let ¯ Z be the schematic closure of Z in Y . We recall from [NVW], Definition1.1 that Z is called pure in Y if it is an affine Y -scheme. The paper [NVW]also uses a weaker variant of this purity which in [L] is called weakly pure :we say Z is weakly pure in Y if each non-empty irreducible component of thecomplement ¯ Z − Z is of pure codimension 1 in ¯ Z . It is well-known that if Z is pure in Y , then Z is also weakly pure in Y (for instance, cf. Proposition3 of Subsection 4.4).Let n and r be natural numbers. Let p be a prime. Let S be a locallynoetherian F p -scheme. Let Φ S : S → S be the Frobenius endomorphism of S .1et M be a crystal of the gross absolute crystalline site CRIS ( S/ Spec ( Z p ))introduced in [B], Chapter III, Example 1.1.3 and Definition 4.1.1 in locallyfree O S/ Spec ( Z p ) -modules of rank r . We assume that we have an isogeny φ M : (Φ nS ) ∗ ( M ) → M ; thus the pair C = ( M , φ M ) is an F n -crystal of CRIS ( S/ Spec ( Z p )). If the F p -scheme S = Spec A is affine, then the pair C = ( M , φ M ) is canonically identified with a σ n − F -crystal on A in the senseof [K], Subsection (2.1).Let ν : [0 , r ] → [0 , ∞ ) be a Newton polygon , i.e., a nondecreasing piecewiselinear continuous function such that ν (0) = 0 and the coordinates of all its break points are natural numbers. For x ∈ S , let ν x be the Newton polygon of the fiber C x of C at x . Let S ν be the reduced locally closed subschemeof S whose points are exactly those x ∈ S such that we have ν x = ν , cf.Grothendieck–Katz Theorem (see [K], Corollary 2.3.2); if non-empty, S ν is a stratum of the Newton polygon stratification of S defined by C .Let a, b ∈ N be such that 0 ≤ a ≤ r . Let T = T ( a,b ) ( C ) be the reducedlocally closed subscheme of S whose points are those x ∈ S such that ( a, b )is a break point of ν x . The end break point ( r, ν x ( r )) remains constant underspecializations of x ∈ S . Thus locally in the Zariski topology of S , we canassume that there exists d ∈ N such that for all x ∈ S we have ν x ( r ) = d and this implies that T is the reduced locally closed subscheme of S whichis a finite union S ν ∈ N r,d,a,b S ν of Newton polygon strata S ν indexed by theset N r,d,a,b of all Newton polygons ν : [0 , r ] → [0 , ∞ ) with the two propertiesthat ν ( r ) = d and ( a, b ) is a break point of ν .It is known that T is weakly pure in S , cf. [Y], Theorem 1.1. It is alsoknown that S ν is pure in S , cf. [V1], Main Theorem B. This last resultimplies the celebrated result of de Jong–Oort which asserts that S ν is weaklypure in S , cf. [dJO], Theorem 4.1. Strictly speaking, the references of thisparagraph work with n = 1 but their proofs apply to all n ∈ N ∗ .In general, a finite union of locally closed subschemes of S which are purein S is not pure in S . Therefore the following purity result which refines andreobtains the mentioned results of de Jong–Oort, Vasiu, and Yang comes asa surprise. Theorem 1
With the above notations, T is pure in S . In Section 2 we gather few preliminary steps that are required to proveTheorem 1 in Section 3. We have the following two direct consequences ofTheorem 1, the first one for n = 1 just reobtains [V1], Main Theorem B inthe locally noetherian case. 2 orollary 1 Each Newton polygon stratum S ν is pure in S . The p -rank χ ( x ) of C x is the multiplicity of the Newton polygon slope 0of ν x . Equivalently, χ ( x ) is the unique natural number such that (0 ,
0) and( χ ( x ) ,
0) are the only break points of ν x on the horizontal axis (i.e., whichhave the second coordinate 0). Corollary 2
Let m ∈ N . We consider the reduced locally closed subscheme S m of S whose points are exactly those x ∈ S such that the p -rank χ ( x ) of C x is m . Then S m is pure in S . If m >
0, then we have S m = T ( m, ( C ) and if m = 0, then we have S = T (1 , ( C ⊕ E ) where E is the pull back to S of the F n -crystal overSpec ( F p ) of rank 1 and Newton polygon slope 0 which has a Frobeniusinvariant global section; therefore, regardless of what m is, Corollary 2 followsfrom Theorem 1.For n = 1 Corollary 2 was first obtained by Deligne and more recently byVasiu and Li (see [D], [V3], and [L]). Corollary 2 also refines and reobtainsa prior result of Zink which asserts that S m is weakly pure in S (see [Z],Proposition 5).In Section 4 we first follow [L] to show that Corollary 1 follows directlyfrom Theorem 1 and then we follow [V3] to include a second proof of Corollary2 in the more general context provided by a functorial version of the Artin–Schreier stratifications introduced in [V2], Definition 2.4.2 which is simpler,does not rely on Theorem 1, and is based on Theorem 2 of Subsection 4.2.Theorem 1 is due to the first author, cf. [L]. While the proof of [Y],Theorem 1.1 follows the proof of [dJO], Theorem 4.1, the proof of Theorem1 presented follows [L] and thus the proofs of [V1], Main Theorem B andTheorem 6.1. It is known (cf. [NVW], Example 7.1) that in general S m isnot strongly pure in S in the sense of [NVW], Definition 7.1 and thereforeTheorem 1 and Corollary 2 cannot be improved in general (i.e., are optimal).We refer to T ( a,b ) ( C ), S ν , and S m as crystalline strata of S associatedto C and certain (basic) discrete invariants of F n -crystals. Cases of non-discrete invariants stemming from isomorphism classes are also studied inthe literature (for instance, see [V1], Subsection 5.3 and [NVW], Theorem1.2 and Corollary 1.5). Crystalline strata have applications to the study inpositive characteristic of different moduli spaces and schemes such as specialfibers of Shimura varieties of Hodge type (for instance, see [V1] and [NVW]).3 Standard reduction steps
The above notations p , S , Φ S , ¯ Z , n , r , C = ( M , φ M ), C x , ν x , ( a, b ) ∈ N , T = T ( a,b ) ( C ), S ν , m , S m , χ ( x ), and E will be used throughout the paper.For a fixed Newton polygon ν , let S ≥ ν be the reduced closed subscheme of S whose points are exactly those x ∈ S such that the Newton polygon ν x isabove ν , cf. [K], Corollary 2.3.2.In what follows by an ´etale cover we mean a surjective finite ´etale mor-phism of schemes. For basic properties of excellent rings we refer to [M],Chapter 13. If V → Y is a morphism of F p -schemes and if F (or F Y ) is an F n -crystal over Y , let F V be the pull back of F (or F Y ) to an F n -crystalover V , i.e., of CRIS ( V /
Spec ( Z p )). Let k ( y ) be the residue field of a point y ∈ Y . If V = Spec ( k ( y )) → Y is the natural morphism, then we denote F V = F Spec ( k ( y )) simply by F y (the fiber of F at y ).For an F p -algebra R , let W ( R ) be the ring of p -typical Witt vectors withcoefficients in R . Let W ( R ) = (Spec R, Spec ( W ( R )) , can) be the thickeningin which ‘can’ stands for the canonical divided power structure of the kernelof the epimorphism W ( R ) → W ( R ) = R . For s ∈ N ∗ , let W s ( R ) be the ringof p -typical Witt vectors of length s with coefficients in R . Let W s ( R ) =(Spec R, Spec ( W s ( R )) , can) be the thickening defined naturally by W ( R ).Let Φ R be the Frobenius endomorphism of either W ( R ) or W s ( R ).The property of a reduced locally closed subscheme being pure in S islocal for the faithfully flat topology of S , and thus until the end we willalso assume that S = Spec A is an affine F p -scheme and that there exists d ∈ N such that for all x ∈ S we have ν x ( r ) = d . As the scheme S is locallynoetherian and affine, it is noetherian. To prove Theorem 1, we have to provethat T is an affine scheme. Let M ( W s ( R )) be the abelian category whose objects are pairs ( O, φ O ) com-prising from a W s ( R )-module O and a Φ nR -linear endomorphism φ O : O → O (i.e., φ O is additive and for all z ∈ O and σ ∈ W s ( R ) we have φ O ( σz ) =Φ nR ( σ ) φ O ( z )) and whose morphisms f : ( O , φ O ) → ( O , φ O ) are W s ( R )-linear maps f : O → O satisfying f ◦ φ O = φ O ◦ f . If t ∈ { , . . . , s − } ,then by a quasi-isogeny of M ( W s ( R )) whose cokernel is annihilated by p t we mean a morphism f : ( O , φ O ) → ( O , φ O ) of M ( W s ( R )) which has thefollowing two properties: (i) both O and O are projective W s ( R )-modules4hich locally in the Zariski topology of Spec ( W s ( R )) have the same positiverank, and (ii) the cokernel O /f ( O ) is annihilated by p t . An object ( O, φ O )of M ( W s ( R )) is called divisible by t ∈ { , . . . , s − } if O is a projective W s ( R )-module such that Im( φ O ) ⊆ p t O .For l ∈ N ∗ we have a natural functor M ( W s + l ( R )) → M ( W s ( R )) to bereferred by abuse of language as the reduction modulo p s functor.If Y is a Spec ( F p )-scheme, in a similar way we define the scheme W s ( Y ),its Frobenius endomorphism Φ Y , and the abelian category M ( W s ( Y )) andspeak about quasi-isogenies of M ( W s ( Y )) whose cokernels are annihilatedby p t with t ∈ { , . . . , s − } , about objects of M ( W s ( Y )) divisible by t ∈ { , . . . , s − } , and about reduction modulo p s functors M ( W s + l ( Y )) →M ( W s ( Y )). We have canonical identifications M ( W s ( R )) = M ( W s (Spec R )).For homomorphisms R → R and morphisms Y → Y we have naturalpull back functors M ( W s ( R )) → M ( W s ( R )) and M ( W s ( Y )) → M ( W s ( Y )).To prove that T is an affine scheme, we can also assume that the evalua-tion M of M at the thickening W ( A ) is a free A -module of rank r . The eval-uation of φ M at this thickening is a Φ nA -linear endomorphism φ M : M → M .In what follows we will apply twice the following elementary general factwhich can be also deduced easily from the elementary divisor theorem. Fact 1
Let D be a discrete valuation ring and let π ∈ D be a uniformizer ofit. Let s, t ∈ N be such that s > t . Let D s = D/ ( π s ) . Let g s : D rs → D rs bea D s -linear endomorphism such that its cokernel is annihilated by π t . Thenfor each x ∈ D rs − πD rs , we have g s ( x ) ∈ D rs − π t +1 D s . Proof:
Let g : D r → D r be a D -linear endomorphism which lifts g s . Let E = Im( g ) + π s D r (one can easily check that E = Im( g ) but we will not stopto argue this). It is a free D -module of rank r which (as π t Coker( g s ) = 0)contains π t D r . Thus π s D r ⊆ pE and therefore Im( g ) surjects onto the D -vector space E/πE of rank r . Hence a D s -basis of D rs maps via g to a D -basis of E . From this and the fact that π t +1 D r ⊆ πE we get that noelement of a D s -basis of D rs is mapped by g to π t +1 D r . Thus the fact holds. ( a, b ) If ( a, b ) is (0 ,
0) or ( r, d ), then T = S . If a = 0 and b > a = r and b = d , then T = ∅ . Thus, to prove that T is an affine scheme we can assumethat 1 ≤ a ≤ r −
1. 5 emma 1
Let k be a field of characteristic p . Let ν : [0 , r ] → [0 , ∞ ) be theNewton polygon of an F n -crystal F over k of rank r . Let a, b ∈ N be suchthat ≤ a ≤ r − . Then ( a, b ) is a break point of ν if and only if (1 , b ) isa break point of the Newton polygon V a ( ν ) of the F n -crystal over k of rank (cid:0) ra (cid:1) which is the exterior power V a ( F ) of F . Proof:
Let α ≤ · · · ≤ α r be the Newton polygon slopes of ν . Let β ≤· · · ≤ β ( ra ) be the Newton polygon slopes of V a ( ν ). We have β = a X i =1 α i and β = ( a − X i =1 α i ) + α a +1 = β + α a +1 − α a . Thus β < β if and only if α a < α a +1 . Moreover, ( a, b ) is a break point of ν if and only if we have α a < α a +1 , and (1 , b ) is a break point of the Newtonpolygon V a ( ν ) if and only if we have β < β . The lemma follows from thelast two sentences. (cid:3) Based on Lemma 1, to prove that T is an affine scheme by replacing C with its exterior power V a ( C ) we can assume that a = 1. T Let q ∈ N ∗ be such that for each x ∈ S the Newton polygon slopes of the F nq -crystal over Spec ( k ( x )) which is the q -th iterate of C x are all integers.For instance, as each Newton polygon slope of C x is a rational number whosedenominator is a natural number at most equal to r , we can take q = r !.Thus by replacing n by nq and C by its q -th iterate, we can assume that foreach x ∈ S the Newton polygon slopes of C x are natural numbers.We consider the Newton polygon ν : [0 , r ] → [0 , ∞ ) whose graph is:6 b b b xy s l o p e b s l o p e b + (0 ,
0) (1 , b ) ( r − , ( r − b + r − r, d )If x ∈ T , then as all Newton polygon slopes of C x are natural numbers,these Newton polygon slopes are α = b , α ≥ b + 1, α r − ≥ b + 1, and α r = d − P r − i =1 α i ≥ b + 1. Therefore, if x ∈ T then we have x ∈ S ≥ ν .This implies that T is a subscheme of the closed subscheme S ≥ ν of S . Byreplacing S with S ≥ ν we can assume that S = S ≥ ν . Thus S is reduced.If r ( b + 1) > d , then S = S ≥ ν = S ν = T and thus T is affine. Thus wecan assume that r ( b + 1) ≤ d and therefore there exists a Newton polygon ν : [0 , r ] → [0 , ∞ ) whose graph is: b bb b b xy s l o p e b + (0 ,
0) (1 , b + 1)(1 , b ) ( r − , ( r − b + 1))( r, d )7f x ∈ S − T = S ≥ ν − T , then all Newton polygon slopes of C x are naturalnumbers α ≥ b + 1, α ≥ b + 1, α r − ≥ b + 1, and α r = d − P r − i =1 α i ≥ b + 1and thus ν x is above ν . If ν x is not above ν , then as ν x is above ν (as S = S ≥ ν ) we get that we have α = b and α i ≥ b + 1 for i ∈ { , . . . , r } .From the last two sentences we get that we have identities T = T (1 ,b ) = S − S ≥ ν = S ≥ ν − S ≥ ν . Thus, under all the above reduction steps, T is an open subscheme of S . S The statement that T is an affine scheme is local in the faithfully flat topologyof S and therefore until the end of Section 3 we will assume that A is acomplete local reduced noetherian ring. Thus A is also excellent and thereforeits normalization in its ring of fractions is a finite product of normal completelocal noetherian integral domains. Based on [V1], Lemma 2.9.2 which is astandard application of Chevalley’s theorem of [G1], Chapter II, (6.7.1), toprove that T is an affine scheme we can replace A by one of the factors ofthe last product. Thus we can assume that A is a normal complete localnoetherian integral domain. We can also assume that T is non-empty andtherefore it is an open dense subscheme of S . Let K be the field of factionsof A and let ¯ K be an algebraic closure of it. In this section we complete the proof of Theorem 1, i.e., we prove that T isan affine scheme when a = 1 < r , for each x ∈ S all Newton polygon slopesof C x are natural numbers, we have S = S ≥ ν = Spec A with A a normalcomplete local noetherian integral domain, and T = T (1 ,b ) = S − S ≥ ν is opendense in S . Let E b = ( M b , φ M b ) be the pull back to S of the F n -crystalover Spec ( F p ) of rank 1 and Newton polygon slope b defined by the pair( Z p , p b Z p ). Let η be the generic point Spec K → S of S . Let s, l ∈ N ∗ .In Subsection 3.1 we consider commutative affine group schemes H s over S of morphisms between certain evaluations of E b and C . In Sections 3.2we glue morphisms between different such evaluations in order to introducegood sections above T of the morphisms H s → S in Subsection 3.3. InSubsection 3.4 we complete the proof of Theorem 1. The key idea (the plan)8an be summarized as follows: under suitable reductions, for s >> T we can identify T with a closed subscheme of H s andtherefore we can conclude that T is an affine scheme.If R is a reduced perfect ring of characteristic p , following [K] we saythat an F n -crystal F over Spec R is divisible by b if its evaluation at theendomorphism Φ nR of the thickening W ( R ) is defined by a Φ nR -linear endo-morphism whose q -th iterate for all q ∈ N ∗ is congruent to 0 modulo p bq .Thus if y ∈ Spec R , then the Hodge polygon slopes of F y are all greater orequal to b . For an A -algebra B and an F n -crystal F over B , let E s ( F ) be the evaluationof F at the thickening W s ( B ); it is an object of the category M ( W s ( B )). Inparticular, we write E s ( C B ) = ( M s,B , φ M s,B ) and let E s ( E b,B ) = ( N s,B , φ N s,B ).Thus we have M = M ,A , φ M = φ M ,A , and N s,B = W s ( B ). Moreover φ N s,B : N s,B → N s,B is the Φ nB -linear endomorphism which maps 1 to p b and φ M s,B : M s,B → M s,B is a Φ nB -linear endomorphism and we have M s,B = W s ( B ) ⊗ W s ( A ) M s,A . The kernel of the epimorphism W s ( B ) → W ( B ) = B is a nilpotent ideal. Based on this and the fact that M is a free A -module ofrank r , we get that each M s,B is a free W s ( B )-module of rank r .We consider the commutative affine group scheme H s over S which rep-resents the following functor: for an A -algebra B , the abelian group H s ( B ) = Hom M ( W s ( B )) ( E s ( E b,B ) , E s ( C B ))is the group of all W s ( B )-linear maps f : N s,B → M s,B which satisfy theidentity f ◦ φ N s,B = φ M s,B ◦ f . The S -scheme H s is of finite presentation (for n = 1, see [V1], Lemma 2.8.4.1; the proof of loc. cit. applies to all n ∈ N ∗ ).Let x ∈ S be a point of codimension 1. Thus the local ring D x := O S,x of S at x is a discrete valuation ring. Let E x be a complete discrete valuationring which dominates D x and has a residue field which is algebraically closed.Let P x be the perfection of E x . We recall that C P x is the pull back of C via thenatural morphism Spec P x → S . As S = S ≥ ν , the Newton polygon slopesof the two fibers of C P x are greater or equal to b . Thus from [K], Theorem2.6.1 we get the existence of an F n -crystal D over Spec P x which is divisibleby b and which is equipped with an isogeny ψ x : D → C P x p t for some t ∈ N . Based on the proof ofloc. cit. we can assume that t = ( r − b depends only on r and b . Proposition 1
We assume that the point x ∈ S of codimension belongs to T . Then there exists a unique F n -subcrystal D b of D which is isomorphic tothe pull back E b,P x of E b . Moreover, D b has a unique direct supplement in D . Proof:
We know that for y ∈ Spec P x all Hodge polygon slopes of D y are atleast b . If all Hodge polygon slopes of D y are at least b + 1, then all Newtonpolygon slopes of D y are at least b + 1. As under the morphism Spec P x → S ,the point y maps to either x ∈ T or η ∈ T and as ψ x is an isogeny, (1 , b ) is abreak point of the Newton polygon of D y . From the last three sentences weget that (1 , b ) is a point of the Hodge polygon of D y .Thus for each point y ∈ Spec P x , (1 , b ) is a break point of the Newtonpolygon of D y and is a point of the Hodge polygon of D y . Due to this, from[K], Theorem 2.4.2 we get that there exists a unique direct sum decomposition D = D b ⊕ D >b into F n -crystals over Spec P x , where D b is of rank 1 and each fiber of it at apoint y ∈ Spec P x has all Hodge and Newton polygon slopes equal to b andwhere D >b is of rank r − y ∈ Spec P x hasall Newton polygon slopes greater than b (and has all Hodge polygon slopesgreater or equal to b ).As D is divisible by b , D b and D >b are also divisible by b .As P x is perfect, for each l ∈ N ∗ we have W ( P x ) / ( p l ) = W l ( P x ) and themodule of differentials Ω W l ( P x ) is 0. Thus, from [BM], Proposition 1.3.3 weget that an F n -crystal over Spec P x is uniquely determined by its evaluationat the thickening W ( P x ). The evaluation of E b,P x at the thickening W ( P x ) iscanonically identified with ( W ( P x ) , p b Φ nP x ) and the evaluation of D b at thethickening W ( P x ) can be identified with ( W ( P x ) , p b Φ b ), where Φ b : W ( P x ) → W ( P x ) is a Φ nP x -linear endomorphism such that Φ b (1) generates W ( P x ).As P x is the perfection of E x and as E x is complete and has an al-gebraically closed residue field, the rings W ( P x ) and W l ( P x ) are strictly10enselian and p -adically complete. We check that these properties implythat there exists a unit υ of W ( P x ) such that we haveΦ b ( υ ) = Φ nP x ( υ )Φ b (1) = υ. If n = 1, then from [BM], Proposition 2.4.9 we get that for each l ∈ N ∗ thereexists a unit υ l ∈ W ( P x ) such that we have Φ b ( υ l ) − υ l ∈ p l W ( P x ) and theproof of loc. cit. checks that we can assume that υ l +1 − υ l ∈ p l W ( P x ). Thusfor n = 1 we can take υ to be the p -adic limit of the sequence ( υ l ) l ≥ . Thisargument applies entirely for n > u defines an isomorphism( W ( P x ) , p b Φ nP x ) → ( W ( P x ) , p b Φ b )which defines an isomorphim E b,P x → D b . (cid:3) From now we will assume that x ∈ T . We consider a composite morphism j x [ s ] : E s ( E b,P x ) → E s ( D b ) → E s ( D ) = E s ( D b ) ⊕ E s ( D >b )in which the first arrow is an isomorphism and the second arrow is the splitmonomorphism associated to the direct sum decomposition.Let i x ( s ) : E s ( E b,P x ) → E s ( C P x )be the composite of j x [ s ] with the morphism ψ x [ s ] : E s ( D ) → E s ( C P x ) whichis the evaluation of the isogeny ψ x at the thickening W s ( P x ) (i.e., which isthe reduction modulo p s of ψ x ). From now on, we will take s > t = ( r − b .We note that ψ x [ s ] is a quasi-isogeny whose cokernel is annihilated by p t andwhose domain is divisible by b . For each point x ∈ T of codimension 1 (i.e., whose local ring D x is a discretevaluation ring), we follow [V1], Subsection 2.8.3 to show the existence of afinite field extension K x of K and of an open subset T x of the normalizationof T in Spec K x such that T x has a local ring which is a discrete valuationring D + x that dominates D x and moreover we have a morphism i T x ( s ) : E s ( E b,T x ) → E s ( C T x )11f the category M ( W s ( T x )) which is the composite of a split monomorphismwith a quasi-isogeny whose cokernel is annihilated by p t and whose domainis divisible by b .To check this, with the notations of Subsection 3.1 we consider four identi-fications E s ( C D x ) = ( W s ( D x ) r , φ s,x ), E s ( E b,D x ) = ( W s ( D x ) , p b Φ nD x ), E s ( D b ) =( W s ( P x ) , p b Φ nP x ), and E s ( D >b ) = ( W s ( P x ) r − , p b φ s,>b,x ). Now, the W s ( P x )-linear map ψ s,P x : W s ( P x ) r → W s ( P x ) r defining ψ x [ s ] and the Φ nP x -linear map φ s,>b,x : W s ( P x ) r − → W s ( P x ) r − involve a finite number of coordinates ofWitt vectors of length s and therefore are defined over W s ( B x ), where B x is a finitely generated D x -subalgebra of P x . We can choose B x such thatthe resulting W s ( B x )-linear map ψ s,B x : W s ( B x ) r → W s ( B x ) r has a coker-nel annihilated by p t . The faithfully flat morphism Spec B x → Spec D x hasquasi-sections (cf. [G2], Corollary (17.16.2)) and therefore there exists a finitefield extension K x of K and a discrete valuation ring D + x of the normalization T in K x which dominates D x and for which we have a D x -homomorphism B x → D + x . The W s ( D + x )-linear map ψ s,D + x : W s ( D + x ) r → W s ( D + x ) r which isthe natural tensorization of ψ s,B x induces (via restriction to the first factor W s ( D + x ) of W s ( D + x ) r ) a morphism i D + x ( s ) : E s ( E b,D + x ) → E s ( C D + x ) of the cat-egory M ( W s ( D + x )) which is the composite of a split monomorphism with aquasi-isogeny whose cokernel is annihilated by p t and whose domain is di-visible by b . It is easy to see that there exists an open subset T x of thenormalization of T in K x which has D + x as a local ring and for which thereexists a morphism i T x ( s ) : E s ( E b,T x ) → E s ( C T x ) of the category M ( W s ( T x ))that has all the desired properties and that extends the morphism i D + x ( s ) ofthe category M ( W s ( D + x )).By working with s + l instead of s , we can assume that there exists l ∈ N , l >> i T x ( s ) : E s ( E b,T x ) → E s ( C T x ) is the reduction modulo p s ofa morphism i T x ( s + l ) : E s + l ( E b,T x ) → E s + l ( C T x )of the category M ( W s + l ( T x )).Let I s be the set of morphisms E s ( E b, ¯ K ) → E s ( C ¯ K ) which lift to morphisms E s + l ( E b, ¯ K ) → E s + l ( C ¯ K ) for some l >>
0. From [V1], Theorem 5.1.1 (a)(applied for l >> b and r ) we get that each elementof I s is the evaluation at the thickening W s ( ¯ K ) of a morphism of F n -crystals E b, ¯ K → C ¯ K (strictly speaking loc. cit. is stated for n = 1 but its proof worksfor all n ∈ N ∗ ). This implies that I s is a finite set whose elements are all pullbacks of morphisms of M ( W s ( L )), where L is a suitable finite field extension12f K contained in ¯ K . By replacing S with its normalization in L , we canassume that L = K . As inside K x we have an identity D + x ∩ K = D x , inside W s ( K x ) we have an identity W s ( D + x ) ∩ W s ( K ) = W s ( D x ). From the lastthree sentences we get that the pull back i D + x ( s ) of i T x ( s ) to a morphism of M ( W s ( D + x )) is the pull back of a morphism of M ( W s ( D x )). Based on thiswe can assume that there exists an open subscheme U x of T which contains x and which has the property that there exists a morphism i U x ( s ) : E s ( E b,U x ) → E s ( C U x )of the category M ( W s ( U x )) such that i T x ( s ) is the pull back of it.We consider an identification C ¯ K = ( Q, φ Q ), where Q = W ( ¯ K ) r and φ Q : Q → Q is a Φ n ¯ K -linear endomorphism. The Newton polygon ν η of C ¯ K has theNewton polygon slope b with multiplicity 1 and therefore there exists a uniquenon-zero direct summand Q b of Q such that we have φ Q ( Q b ) = p b Q b . Therank of the W ( ¯ K )-module Q b is 1. Let z b ∈ Q b be such that Q b = W ( ¯ K ) z b and φ Q ( z b ) = p b z b ; it is unique up to multiplication by units of W ( F p n ).We have a canonical identification E b, ¯ K = ( W ( ¯ K ) , p b Φ nK ). The morphism E s ( E b, ¯ K ) → E s ( C ¯ K ) defined by i T x ( s ) is an element of I s and therefore it isthe reduction modulo p s of a morphism λ x : ( W ( ¯ K ) , p b Φ nK ) → ( Q, φ Q ) of F n -crystals over ¯ K . Clearly λ x (1) ∈ Q b and thus there exists a unique element τ x ∈ W ( F p n ) such that we have λ x (1) = τ x z b . As i T x ( s ) is the composite of a split monomorphism with a quasi-isogenywhose cokernel is annihilated by p t from Fact 1 applied with D = W ( ¯ K ) weget that τ x modulo p t +1 is a non-zero element of W t +1 ( F p n ). Therefore wecan write τ x = p t x u x , where u x ∈ W ( F p n ) is a unit and where t x ∈ { , . . . , t } .From now on, we will take s > t . We consider the morphism θ x := p t − t x u − x i U x ( s ) : E s ( E b,U x ) → E s ( C U x )of the category M ( W s ( U x )); its pull back to a morphism of M ( W s ( T x )) isthe composite of a split monomorphism with a quasi-isogeny whose cokernelis annihilated by p t + t x and thus also by p t and whose domain is divisible by b . The pull back of θ x to a morphism of M ( W s ( ¯ K )) is the reduction modulo p s of the morphism p t − t x u − x λ x : ( W ( ¯ K ) , p b Φ nK ) → ( Q, φ Q ) which maps 1 to p t z b and which does not depend on the point x ∈ T of codimension 1.13et U be the open subscheme of T which is the union of all U x ’s. From theprevious paragraph we get that the θ x ’s glue together to define a morphism θ : E s ( E b,U ) → E s ( C U )of the category M ( W s ( U )).By replacing S with its normalization in anyone of the finite field exten-sions K x of K , we can assume that there exists an open dense subscheme U of U such that the pull back θ U : E s ( E b,U ) → E s ( C U ) of θ to a morphism of M ( W s ( U )) is the composite of a split monomorphism with a quasi-isogenywhose cokernel is annihilated by p t and whose domain is divisible by b : undersuch a replacement, we can take U to be T x itself. H s We have codim T ( T − U ) ≥ θ is defined by a section θ : U → H s denoted in the same way.Let I s be the schematic closure θ ( U ) of θ ( U ) in H s . As the scheme H s isaffine and noetherian and as U is an integral scheme, the scheme I s is alsoaffine, noetherian, and integral. We have a commutative diagram: I s affine (cid:15) (cid:15) U (cid:31) (cid:127) / / open θ ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ T (cid:31) (cid:127) / / S. We consider the pullback J s of I s to T : J s (cid:31) (cid:127) open / / ξ affine (cid:15) (cid:15) I s affine (cid:15) (cid:15) U (cid:31) (cid:127) open / / /(cid:15) open ? ? ⑧⑧⑧⑧⑧⑧⑧ T (cid:31) (cid:127) open / / S. Lemma 2
The affine morphism ξ : J s → T is an isomorphism. Proof:
To prove that ξ is an isomorphism, we can assume that T = S =Spec A is an affine scheme. As ξ is an affine morphism, J s = Spec B is alsoan affine scheme. Since U is open dense in both T and I s , T and J s havethe same field of fractions K . As codim T ( T − U ) ≥ U is an open14ubscheme of both T and J s , we have A p = B p for each prime p ∈ S = T ofheight 1. As A is a noetherian normal domain, inside K we have A ⊆ B ⊆ \ q ∈ Spec B of height B q ⊆ \ p ∈ Spec A of height A p = A (cf. [M], (17.H), Theorem 38 for the equality part; the first inclusion isdefined by ξ ). Therefore A = B . (cid:3) This Lemma 2 allows us in what follows to identify T itself with an opendense subscheme of I s (i.e., with J s ). In this subsection we will show that for s >>
0, we have T = I s . This willcomplete the proof of Theorem 1 as I s is an affine scheme.We are left to show that the assumption that for s >> T = I s leads to a contradiction. This assumption implies that there ex-ists an algebraically closed field k of characteristic p and a morphism ζ :Spec ( k [[ X ]]) → I s with the properties that under it the generic point ofSpec ( k [[ X ]]) maps to U and its special point maps to I s − T .Let P = k [[ X ]] perf be the perfection of k [[ X ]], let κ be the perfect fieldwhich is the field of fractions of P , and let ζ : Spec P → I s be the morphismdefined naturally by ζ . To the composite of ζ with the closed embedding I s → H s corresponds a morphism ω : E s ( E b,P ) → E s ( C P )of the category M ( W s ( P )) whose pull back ω κ to a morphism of M ( W s ( κ ))is equal to the pull back θ κ : E s ( E b,κ ) → E s ( C κ ) of θ .We have a natural identification E s ( E b,P ) = ( W s ( P ) , p b Φ nP ) and we con-sider an identification E s ( C P ) = ( W s ( P ) r , φ ). Thus we have a W s ( P )-linearmap ω : W s ( P ) → W s ( P ) r such that ω ◦ p b Φ nP = φ ◦ ω . We consider an isogeny D → C P whose cokernelis annihilated by p t and with D divisible by b , again cf. [K], Theorem 2.6.1(here t = ( r − b is as before Proposition 1). Thus we also have an isogeny ι : C P → D whose cokernel is annihilated by p t . We consider its evaluation ι [ s ] : E s ( C P ) → E s ( D )15t the thickening W s ( P ). Under an identification E s ( D ) = ( W s ( P ) r , p b ϕ )with ϕ : W s ( P ) r → W s ( P ) r as a Φ nP -linear endomorphism, we get a W s ( P )-linear endomorphism ι [ s ] : W s ( P ) r → W s ( P ) r such that we have ι [ s ] ◦ φ = p b ϕ ◦ ι [ s ]. We consider the composite morphism ρ = ι [ s ] ◦ ω : E s ( E b,P ) → E s ( D )identified with a W s ( P )-linear map ρ : W s ( P ) → W s ( P ) r such that we have ρ ◦ p b Φ nP = p b ϕ ◦ ρ . Let γ = ρ (1) = ( γ , . . . , γ r ) ∈ W s ( P ) r . From the identity ρ ◦ p b Φ P = p b ϕ ◦ ρ we get that the image of ϕ ( γ ) − γ in W s − b ( P ) r is 0. Writing γ = p u δ , where u ∈ N and δ ∈ W s ( P ) r − pW s ( P ) r ,we get that the image of ϕ ( δ ) − δ in W s − b − u ( P ) r is 0. Let ¯ δ ∈ P r − P r = W ( P ) of δ (i.e., the reduction modulo p of δ ). Lemma 3 If s ≥ t + 1 , then we have u ≤ t . Therefore, if moreover wehave s ≥ t + b + 1 , then the image of ϕ ( δ ) − δ in W s − b − t ( P ) r is . Proof:
To check this we can work over W s ( κ ). As the generic point of Spec P maps to U , ω κ = θ κ : E s ( E b,κ ) → E s ( C κ ) is the pull back of the morphism θ U . The pull back ρ κ of ρ to M ( W s ( κ )) is a composite morphism ρ κ = ι [ s ] κ ◦ θ κ : E s ( E b,κ ) → E s ( D κ )and therefore it is the composite of a split monomorphism with a quasi-isogeny whose cokernel is annihilated by p t (as θ U has this property) andwith a quasi-isogeny whose cokernel is annihilated by p t (as ι is an isogenywhose cokernel is annihilated by p t ). Therefore, ρ κ is also the composite ofa split monomorphism with a quasi-isogeny whose cokernel is annihilated by p t . This implies that the image of γ in W t +1 ( κ ) is non-zero (cf. Fact 1applied with D = W ( κ )) and therefore we have u ≤ t . (cid:3) Lemma 4 If s ≥ t + b + 1 , then the image of ¯ δ in k r = W ( k ) r is non-zero. Proof:
We show that the assumption that the image of ¯ δ ∈ P r − k r = W ( k ) r is 0 leads to a contradiction. This assumption implies thatthere exists a largest positive number c of denominator a power of p suchthat we have ¯ δ ∈ X c P r ⊂ P r = ( k [[ X ]] perf ) r . ϕ : P r → P r be the P -linear endomorphism which is the reductionmodulo p of ϕ . From Lemma 3 we get that ¯ δ = ¯ ϕ (¯ δ ). Thus ¯ δ ∈ ¯ ϕ ( X c P r ) ⊆ X p n c P r and this implies that p n c ≤ c which is a contradiction. (cid:3) From the inequality u ≤ t (see Lemma 3) and from Lemma 4 we getthat for s ≥ t + b + 1 the pull back ω k of ω to a morphism of M ( W s ( k ))is such that its reduction modulo p t +1 is non-zero. For s > t + b + 1 + l with l ∈ N ∗ large enough but depending only on b and r , the reduction of ω k modulo p s − l lifts to a morphism E ,k → D k (cf. [V1], Theorem 5.1.1 (a);again loc. cit. stated for n = 1 applies to all n ∈ N ∗ ) which is non-zero. Thus D k has Newton polygon slope b with multiplicity at least 1. From this andthe existence of the isogeny ι we get that C k has Newton polygon slope b withmultiplicity at least 1. This implies that the special point of Spec ( k [[ X ]])under the composite of ζ : Spec ( k [[ X ]]) → I s with the morphism I s → S does not map to a point of S ν = S − T and therefore it maps to a point of T . Contradiction. This ends the proof of Theorem 1. (cid:3) In Subsection 4.1 we prove Corollary 1. In Subsection 4.2 we follow [V2] tointroduce generalized Artin–Schreier systems of equations and Artin–Schreierstratifications. In Subsection 4.3 we refine and reobtain Corollary 2 in thecontext of these stratifications. Subsection 4.4 contains some complements,including Proposition 3 which prove that ‘pure in’ implies ‘weakly pure in’.Until the end let A be an arbitrary F p -algebra. To prove Corollary 2, in this subsection we can assume that S = Spec A and d ∈ N are as in the paragraph before Subsection 2.1. We can alsoassume that ν ( r ) = d as otherwise S ν = ∅ is pure in S . Let l ∈ N be suchthat the Newton polygon ν has exactly l + 1 breaking points denoted as( a , b ) = (0 , , . . . , ( a l , b l ) = ( r, d ).We have obvious identities S ν = [ S ≥ ν l \ i =0 T ( a l ,b l ) ( C )] red = [ S ≥ ν × S ( T ( a ,b ) ( C )) S × · · · × S T ( a l ,b l ) ( C )] red . T ( a l ,b l ) ( C ) is an affine scheme. We recallthat S ≥ ν is a reduced closed subscheme of S . From the last three sentenceswe get that S ν is an affine scheme, i.e., is pure in S . (cid:3) Let x , x ,. . . , x r be free variables. For i, j ∈ { , . . . , r } let P i,j ( x ) ∈ A [ x ]be a polynomial which is a linear combination with coefficients in A of themonomials x q with q ∈ N either 0 or a power of p . By a generalized Artin–Schreier system of equations in r variables over A we mean a system ofequations of the form x i = r X j =1 P i,j ( x pj ) i ∈ { , . . . , r } to which we associate the A -algebra B = A [ x , . . . , x r ] / ( x − r X j =1 P ,j ( x pj ) , x − r X j =1 P ,j ( x pj ) , . . . , x r − r X j =1 P r,j ( x pj )) . Each equation of the form x i = P rj =1 P i,j ( x pj ) will be called as a generalizedArtin–Schreier equation, and its degree e i ∈ N is defined as follows. Wehave e i = 0 if and only if for all j ∈ { , . . . , r } the polynomial P i,j ( x ) is aconstant, and if e i > e i is the largest integer such that there exists a j ∈ { , . . . , r } with the property that the degree of P i,j ( x pj ) is p e i .Let e = max { e , . . . , e r } ; we call it the degree of the generalized Artin–Schreier system of equations in r variables over A . Following [V2], when e ≤ Proposition 2
The morphism ǫ : Spec B → Spec A is ´etale and surjectiveand its geometric fibers have a number of points equal to a power of p . Proof: If e i >
1, then by adding for each j ∈ { , . . . , r } such that thedegree of P i,j ( x pj ) is p e i an extra variable y i,j and an equation of the form y i,j = x pj , the generalized Artin–Schreier equation x i = P rj =1 P i,j ( x pj ) getsreplaced by several generalized Artin–Schreier equations of degrees less than e i . By repeating this process of adding extra variables and equations which(up to isomorphisms between Spec A -schemes) does not change the morphism18 : Spec B → Spec A , we can assume that e ≤
1. Thus the proposition followsfrom [V2], Theorem 2.4.1 (a) and (b). (cid:3)
The below definition is a natural extrapolation of [V2], Definition 2.4.2which applies to ´etale morphisms ǫ : Spec B → Spec A as in Proposition 2. Definition 1
Let ε : Spec
B →
Spec A be an ´etale morphism between affine F p -schemes. (a) We assume that A is noetherian. Then by the Artin–Schreier strati-fication of Spec A associated to ε : Spec
B →
Spec A in reduced locally closedsubschemes V , . . . , V q we mean the stratification defined inductively by thefollowing property: for each l ∈ { , . . . , q } the scheme V l is the maximalopen subscheme of the reduced scheme of ( Spec A ) − ( ∪ l − q =1 V q ) which has theproperty that the morphism ǫ V l : ( Spec B ) × Spec A V l → V l is an ´etale cover. (b) Let µ > µ > · · · > µ v be the shortest sequence of strictly de-creasing natural numbers such that each fiber of the morphism ǫ : Spec B → Spec A has a number of geometric points equal to µ l for some l ∈ { , . . . , v } .Then by the functorial Artin–Schreier stratification of Spec A associated to ε : Spec
B →
Spec A we mean the stratification of Spec A in reduced locallyclosed subschemes U , . . . , U v defined inductively by the following property:for each l ∈ { , . . . , v } the scheme U l is the maximal open subscheme of thereduced scheme of ( Spec A ) − ( ∪ l − q =1 U q ) which has the property that the mor-phism ǫ U l : ( Spec B ) × Spec A U l → U l is an ´etale cover whose all fibers have anumber of geometric points equal to µ l . The existence of the stratification V , . . . , V q of Spec A is a standard pieceof algebraic geometry. The existence of the sequence µ > µ > · · · > µ v follows from the facts that each ´etale morphism is locally quasi-finite andthat Spec B is quasi-compact. The existence of the stratification U , . . . , U v of Spec A is implied by [G2], Proposition 18.2.8 and Corollary 18.2.9 whichshow that one can define U l directly and functorially as follows: each U l isthe set of all points x ∈ Spec A such that the fiber of ε at x has exactly µ l geometric points. Theorem 2
Let ε : Spec
B →
Spec A be an ´etale morphism between affine F p -schemes. Then the functorial Artin–Schreier stratification of Spec A asso-ciated to ε : Spec
B →
Spec A in reduced locally closed subschemes U , . . . , U v is pure, i.e., for each l ∈ { , . . . , v } the stratum U l is pure in Spec A . roof: As the ´etale morphism ε : Spec B →
Spec A is of finite presentationand due to the functorial part, we can assume that A is a finitely gener-ated F p -algebra and thus an excellent ring. We follow [V3]. By replacingSpec A by its closed subscheme (Spec A ) − ( ∪ l − q =1 U q ) endowed with the re-duced structure, we can assume that l = 1 and that A is reduced. Thus U is an open dense subscheme of Spec A . Based again on [V1], Lemma 2.9.2to prove that U is an affine scheme, we can replace A by its normalizationin its ring of fractions. Thus by passing to connected components of Spec A ,we can assume that A is an excellent normal domain. Thus B = Q wl =1 B l isa finite product of excellent normal domain which are ´etale A -algebras. Let K l be the field of fractions of B l . Let L be the finite Galois extension of thefield of fractions K of A generated by the finite separable extensions K l ’s of K . By replacing A by its normalization in L (again based on [V1], Lemma2.9.2), we can assume that we have K = K = · · · = K w . This implies thateach Spec ( B l ) is an open subscheme of Spec A and thus U = ∩ wl =1 Spec ( B l ) = (Spec ( B )) × Spec A (Spec ( B )) × Spec A · · ·× Spec A (Spec ( B w ))is the affine scheme Spec ( B ⊗ A ⊗ · · · ⊗ A B w ). (cid:3) We will use Theorem 2 to obtain a second proof of Corollary 2 which issimpler and independent of Theorem 1. We can assume that S = Spec A isaffine and let φ M : M → M be as in Subsection 2.1.The identities S m = T ( m, ( C ) if m > S = T (1 , ( C ⊕ E ) show that S m is a reduced locally closed subscheme of S . Thus by replacing S by ¯ S m ,we can assume that S m is an open dense subscheme of S = ¯ S m .We consider the equation φ M ( z ) = z (1)in z ∈ M . For x ∈ S we have χ ( x ) = dim F pn ( ϑ x ), where ϑ x is the F p n -vector space of solutions of the tensorization of the Equation 1 over A withan algebraic closure of the residue field k ( x ) of S at x .From now on we will forget about C and just work with the free A -module M of rank r and its Φ nA -linear endomorphism φ M : M → M and we onlyassume that we have an open dense subset S m of S = Spec A defined by thefollowing property: for x ∈ S , we have x ∈ S m if and only if dim F pn ( ϑ x ) = m .20ith respect to a fixed A -basis { v , . . . , v r } of M , by writing z = P ri =1 x i v i the Equation 1 defines a generalized Artin–Schreier system of equations inthe r variables x , . . . , x r of the form x i = L i ( x p n , . . . , x p n r ) i ∈ { , . . . , r } , where each L i is a homogeneous polynomial of total degree at most 1. Let B = A [ x , . . . , x r ] / ( x − L ( x p n , . . . , x p n r ) , . . . , x r − L r ( x p n , . . . , x p n r )) , let ǫ : Spec B → S and let U , . . . , U v be the functorial Artin–Schreier strat-ification of S associated to ǫ : Spec B → S . Let p µ > p µ > · · · > p µ v be theshortest sequence of strictly decreasing of powers of p by natural numberssuch that for each each l ∈ { , . . . , v } every geometric fiber of the morphism ǫ U l : Spec B × S U l → U l has a number of geometric points equal to p µ l , cf.Proposition 2 and Definition 1 (b).The fact that the morphism ǫ : Spec B → S is ´etale (cf. Proposition 2)is equivalent to [Z], Proposition 3. We consider the lower semi-continuousfunction (cf. [G2], Proposition 18.2.8) µ : S → N defined by the rule: µ ( x ) = p n dim F pn ( ϑ x ) is the number of geometric points of ǫ : Spec B → S above x (i.e., is the number of elements of ϑ x ). We get that µ l is divisible by n for all l ∈ { , . . . , v } and (as S m is dense in S ) we have µ = mn . Moreover, for x ∈ S and q ∈ N we have µ ( x ) = p nq if and only if x ∈ S q . We conclude that S m = U and therefore (cf. Theorem 2) S m is anaffine scheme. (cid:3) For the sake of completeness, we include a proof of the following well-knownresult (to be compare with [V1], Remark 6.3 (a)).
Proposition 3
Let Z be a reduced locally closed subscheme of a locallynoetherian scheme Y . If Z is pure in Y , then Z is weakly pure in Y . Proof:
We can assume that Z ( ¯ Z = Y . By localizing Y at the generic pointof an irreducible component of ¯ Z − Z , we can assume that Y = ¯ Z = Spec C is a local affine scheme of dimension at least 1 and Z is the complement in21 of the closed point of Y and we have to prove that C has dimension 1.By passing to a connected component of the normalization of the reducedcompletion ˆ C red of C in the ring of fractions of ˆ C red , we can assume that C is in fact an integral normal local ring which is not a field.We show that the assumption that dim( C ) ≥ Z of Y is pure in in Y , Z is the spectrum ofa C -subalgebra of the field of fractions of C which contains C and which iscontained in the intersection of all the localizations of C at points of Y ofcodimension 1 in Y (as these points belong to Z ). As dim( C ) ≥
2, from[M], (17H), Theorem 38 we get that this intersection is C and thus we have Z = Spec C = Y . Contradiction. Thus dim( C ) = 1. (cid:3) Remark 1
Suppose A is a local noetherian F p -algebra of dimension at least . Let m be the maximal ideal of A . Suppose M = A r is equipped with a Φ nA -linear endomorphism φ M : M → M such that for each non-closed point x of S = Spec A , with the notations of Subsection 4.3 we have dim F pn ( ϑ x ) = m .Then S m = U being pure in S , it is also weekly pure in S (cf. Proposition 3)and thus S − S m cannot be m as codim S ( m ) ≥ . Therefore we have S m = S and in this way we reobtain [Z], Proposition 5. One can view Theorem 2 asa generalization and a refinement of [Z], Proposition 5. Remark 2
For q ∈ N ∗ we define recursively an A -linear map φ ( q ) M : A ⊗ F nqA ,A M → M as follows: let φ (1) M : A ⊗ F nA ,A M → M be the A -linear map defined by φ M andwe have the recursive formula φ ( q ) M = φ (1) M ◦ (1 A ⊗ F nA ,A φ ( q − M ) . Deligne provedin [D] the case n = 1 of Theorem 2 using ranks of images of φ ( q ) M for q >> at points x ∈ S = Spec A and properties of Grassmannians. Acknowledgement.
The first author would like to thank the second au-thor for the continuous support during his Ph.D. studies and his family forspiritual support throughout his life. The second author would like to thankBinghamton University for good working conditions. Both authors wouldlike to thank the referee for many valuable comments.22 eferences [B] P. Berthelot,
Cohomologie cristalline des sch´emas de car-act´eristique p > , Lecture Notes in Mathematics, Vol. ,Springer-Verlag, Berlin-Heidelberg, 1974[BM] P. Berthelot and W. Messing,
Th´eorie de Dieudonn´ecristalline III,
The Grothendieck Festschrift, Vol. I, Progr.Math., Vol. , 173–247, Birkh¨auser Boston, Boston, MA,1990[dJO] J. de Jong and F. Oort, Purity of the stratification by Newtonpolygons,
J. Amer. Math. Soc. (2000), no. 1, 209–241[D] P. Deligne, Unpublished note to A. Vasiu,
IAS, Princeton,New Jersey, U.S.A., October 2011[G1] A. Grothendieck, ´El´ements de g´eom´etrie alg´ebrique. II. ´Etudeglobale ´el´ementaire de quelques classes de morphisms,
Inst.Hautes ´Etudes Sci. Publ. Math., Vol. , 1961[G2] A. Grothendieck, ´El´ements de g´eom´etrie alg´ebrique. IV.´Etude locale des sch´emas et des morphismes de sch´emas, Qua-tri`eme partie, Inst. Hautes ´Etudes Sci. Publ. Math., Vol. ,1967[K] N. Katz, Slope filtration of F -crystals, Journ´ees de G´eom´etrieAlg´ebrique de Rennes, Rennes, 1978, Vol. I, Ast´erisque, Vol. , Soc. Math. de France, Paris, 1979, pp. 113–163[L] J. Li, Purity Results on F -crystals, Thesis (Ph.D.) – StateUniversity of New York at Binghamton, 2015, 81 pages, ISBN978-1321-90186-3, ProQuest LLC[M] H. Matsumura,
Commutative algebra. Second edition,
The Benjamin/Cummings Publ. Co., Inc., Reading, Mas-sachusetts, 1980[NVW] M.-H. Nicole, A. Vasiu, and T. Wedhorn,
Purity of level m stratifications, Ann. Sci. ´Ecole Norm. Sup. (2010), no. 6,927–957 23V1] A. Vasiu, Crystalline boundedness principle,
Ann. Sci. ´EcoleNorm. Sup. (2006), no. 2, 245–300[V2] A. Vasiu, A motivic conjecture of Milne,
J. Reine Agew. Math.(Crelle) (2013), 181–247[V3] A. Vasiu,
Talk ‘Purity of Crystalline Strata’,
Conference onArithmetic Algebraic Geometry on the occasion of Gerd Falt-ings’ 60th birthday, Max Planck Institute for Mathematics,Bonn, Germany, June 13, 2014[Y] Y. Yang,
An improvement of de Jong–Oort’s purity theorem,
M¨unster J. Math. (2011), 129–140[Z] T. Zink, On the slope filtration,
Duke Math. J.109