Two Lectures on Regular Singular Stratified Bundles

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1 Two Lectures on Regular Sngular Stratfed Bundles Lars Kndler Free Unverstät Berln Web: Emal: Abstract These are the notes for two 90 mnute lectures gven at the Vetnam Insttute for Advanced Studes n Mathematcs n August 202. The author thanks VIASM and n partcular Prof. Dr. Phung Ho Ha for the knd nvtaton and hosptalty. Contents Lecture - Dfferental Operators 2. Overvew Dfferental operators Defntons Composton Functoralty The smooth case Logarthmc dfferental operators Defntons Exponents Functoralty Lecture 2 - Stratfed Bundles 2. Stratfed bundles Defntons and frst propertes Restrcton to an open subset (X, X-regular sngular stratfed bundles Exponents Canoncal extensons Proof of the Man Lemma wth respect to (X, X Regular sngular stratfed bundles n general slghtly revsed verson from March 6, 203

2 Lecture - Dfferental Operators. Overvew The goal of these lectures s to gve an elementary ntroducton nto the theory of stratfed bundles n postve characterstc, wth a specal emphass on regular sngularty as defned n [Ge75], and, more generally, the author s thess [Kn2]. One of the man theorems of loc. ct. s. Theorem. Let k be an algebracally closed feld of characterstc p > 0, X a smooth, connected, separated, fnte type k-scheme and E a stratfed bundle on X,.e. an O X -coherent D X/k -module. Then the followng are equvalent: (a E s regular sngular wth fnte monodromy group. (b E s trvalzed on a fnte, tame coverng of X. The noton of tameness used here wll be defned n the lectures, and t s studed n [KS0]. The proof of Theorem. splts nto two parts: An argument usng more or less standard facts from the theory of Tannakan categores, and the followng Man Lemma:.2 Theorem (Man Lemma. Let f : Y X be a fnte galos étale morphsm. Then f O Y s a stratfed bundle on X,.e. an O X -coherent D X/k -module, and f s tame f and only f f O Y s regular sngular. In the course of the lectures, we wll defne the terms used n the statement of the Man Lemma, and develop the theory far enough to gve a proof. We try to avod Tannakan and log-geometrc language..2 Dfferental operators We gve a bref summary of the relevant facts from [EGA4,.6] and [BO78, Ch. 2], wthout proofs..2. Defntons Let f : X S be a separated (for smplcty morphsm of schemes and dag X/S : X X S X the dagnoal. Ths s a closed mmerson by assumpton. Let I X/S O X S X be the assocated sheaf of deals and n X/S the closed subscheme defned by I n+ X/S, whch s called the n-th nfntesmal neghborhood of X. Note that the closed mmerson n : X n X/S s a homeomorphsm on topologcal spaces..3 Defnton. We defne P n X/S := n O n = dag X/S X/S O X S X/I n+ X/S and call t the sheaf of n-th prncpal parts. Note that the two projectons X S X X nduce maps p n 0, p n : n X/S X, and accordngly, the sheaf of rngs PX/S n carres a left- and a rght-o X-structure va the correspondng rng morphsms d n 0, d n : O X PX/S n. In fact, for n > 0, 2

3 one always has P n X/S = (O X f O S O X /J n+, where J s the kernel of the multplcaton map O X f O S O X O X, and wth ths dentfcaton we have d n 0 (a = a, d n (a = a..4 Defnton. Let E, F be two O X -modules. A dfferental operator E h F of order n s a f O S -lnear morphsm whch factors as E h F d 0 P n X/S O X E h wth h beng O X -lnear. Here the tensor product P n X/S O X E s constructed wth respect to the rght-o X -structure of P n X/S and consdered as an O X-module va the left-o X -structure of P n X/S. In some sense a dfferental operator s hence a f O S -lnear map, whch s almost O X -lnear. Such a factorzaton s unque: If a b e PX/S n E, then h(a b e = a h( b e = a h( be = ah(be. Thus we see that the set of dfferental operators E F of order n s n fact Dff n X/S (E, F := Hom O X (P n X/S O X E, F, whch s an O X (X-bmodule, and the sheaf of dfferental operators E F of order n s Dff n X/S (E, F = Hom O X (P n X/S O X E, F, whch s an O X -bmodule..5 Example. A dfferental operator O X O X of degree 0 s just multplcaton wth an element of O X (X. Note that there are canoncal surjectons... P n X/S Pn X/S..., and that they nduce morphsms Dff n X/S (E, F Dff n+ X/S (E, F, so t makes sense to defne.6 Defnton. If E, F are O X modules, then Dff X/S (E, F := lm Dff n n X/S (E, F. If E = F = O X, we also wrte D X/S := Dff X/S (O X, O X. 3

4 .2.2 Composton Now we defne composton of dfferental operators. Consder the followng dagram: n+m X/S X S X X! pr 0,2 n X/S X n+m X/S (X S X X (X S X One checks that the dotted arrow exsts. Ths gves morphsms δ n,m : P n+m X/S P n X/S O X P m X/S. It s nduced by a b a b. Fnally, f E φ F, F ψ G are two dfferental operators, then we construct: F φ G ψ H d n+m 0 d n 0 P n X/S O X F P n+m X/S O X F δm,n φ d 0 P m X/S O X G d φ P m X/S O X P n X/S O X F ψ We see that ψ φ Dff n+m X/S (F, H. Accorngly, Dff X/S(F, F becomes a sheaf of b-o X -algebras. Note that ths sheaf s noncommutatve n general: If a O X (X and ψ : F F a dfferental operator, then aψ ψa..2.3 Functoralty Now let g : Y X be an S-morphsm of S-schemes..7 Proposton. The followng statements are true: g D X/S := O Y g O X g D X/S s a (D Y/S, g D X/S -bmodule. There exsts a canoncal left-d Y/S, rght-g D X/S map D Y/S g D Y/S. If E s a left-d X/S -module, then g E s somorphc to g D X/S g D X/S g E and carres a canoncal left-d Y/S -structure. 4

5 .2.4 The smooth case Assume there are global functons x,..., x n O X (X such that dx,..., dx n are a bass for Ω X/S,.e. such that the defne a factorsaton X étale A n S f S..8 Proposton. In ths stuaton D m X/S s free (as left- and as rght-o X- module on generators of the form (a x... (am x n, a m. Here x (a acts on O X va x (a (x r j = ( r a x r a followng composton rules hold true: (a x and x (b j commute for all, j, a, b. f = j and = 0 otherwse. The x (a x (b (a x = ( a+b a f = r+s=a r,s 0 (a+b x. x (r (f x (s for f O X (X..9 Corollary. If f : Y X s an étale morphsm of smooth S-schemes, then the morphsm D Y/S f D X/S from Proposton.7 s an somorphsm. If S s defned over Q, then (a x = a! a / x a. If S s defned over F p, then x (a cannot be wrtten as product of lower order operators, but t nontheless behaves lke a! a / x a, so we wll use ths notaton, wthout t makng lteral sense. ( Evaluate n characterstc 0, notce that the coeffcent s dvsble by a suffcently hgh power of p, cancel the p-power, and take the result mod p. If S s defned over F p, we compute ( (a x p = p r= ( ra x (ap a = 0 because ( pa a = 0. To prove the last statement, and for later reference, we state the followng:.0 Lemma. Let p be a prme number. (a Ths s called Lucas Theorem: For a 0,..., a n, b 0,..., b n ntegers n [0, p ], a := a 0 + a p a n p n, b := b 0 + b p b n p n we have ( a ( ak mod p. b b k k 5

6 (b If N, n, m, k 0 are ntegers such that p k > n, then ( ( N + mp k N mod p. n n Hence the term ( b a Z/pZ s well-defned for a, b Z/p k Z, and t can { be computed by dentfyng the set Z/p k Z wth the set of ntegers 0,,..., p k }. Smlarly, ( b a s well-defned for b Zp and a Z/p k Z. (c If α Z p, then α = n=0 ( α p n p n, where a means the unque nteger n [0,..., p ] congruent to a. (d If α, β Z p,d 0, then ( αβ p d a+b=d a,b 0 ( α p a ( β p b mod p. Proof. Everythng follows from (a, whch s easly proven by computng the coeffcent of x b of a ( a n x k = ( + x a ( + x pk a k mod p. k k=0 k=0 Just lke n characterstc 0 one proves:. Proposton. Let S = Spec k for k a feld. If X s smooth, separated, fnte type over k, then a D X/k -module whch s coherent as O X -module, s a locally free O X -module. Proof. The characterstc 0 proof goes through mutats mutands: Wthout loss of generalty we may assume that k s algebracally closed. Then t suffces to check that Ê := E O X,x Ô X,x s free for all closed ponts x X, because X s of fnte type over k. After choosng local coordnates, we can wrte Ô X,x = kx,..., x n. Let e,..., e r be a mnmal system of generators of Ê and assume that there s a relaton r = a e = 0 wth a kx,..., x n. Then all the a le n the maxmal deal m := (x,..., x n, by the mnmalty of the system of generators. Thus there exsts some N > 0, such that all the a le n m N, but (after renumberng a m N+. Then a = λx l... xln n + m N+, wth λ k, and at least one l 0. We apply the operator to the gven relaton and obtan D l := (l x... (ln x n 0 = D l (a e + D l (a 2 e D l (a n e n + mê. But by constructon D l (a kx,..., x n, whch s a contradcton due to Nakayama s lemma..2 Remark. In fact Ê s even trval as a D kx,...,x n/k-module, but we do not need ths fact. 6

7 .3 Logarthmc dfferental operators.3. Defntons From now on we set S := Spec k wth k an algebracally closed feld of characterstc p > 0. Let X be a smooth, connected, fnte type, separated k-scheme and X X an open dense subscheme, such that D := X \ X s a strct normal crossngs dvsor. Recall what ths means:.3 Defnton. D s a strct normal crossngs dvsor f X can be covered by open sets U, such that there exst x,..., x n Γ(U, O U such that Ω U/k s free wth bass dx,..., dx n, and such that D U = (x... x r, wth r n. In ths stuaton, we defne:.4 Defnton ([Ge75]. D X/k (log D s defned to be the subsheaf of algebras of D X/k, generated by all dfferental operators whch locally fx all powers of the deal of D. If X admts coordnates x,..., x n, such that D = (x... x r, then D X/k (log D s generated by δ (a x.5 Remark. := x a (a x for r, a 0 and (a x, for > r, a 0. Ths sheaf of rngs can be constructed more generally n a sutable category of schemes wth logarthmc structures. As n the classcal case, one constructs t from (an approprate noton of thckenngs of the dagonal n ths category, see [Kn2, Ch. 2]. In Proposton. we saw that an O X -coherent D X/k -module s locally free. For O X -coherent D X/k (log D-modules E, the stuaton s more complcated: E s not necessarly torson free, and even f t s torson free, then t s not necessarly locally free. We wll see a partal remedy later on, once we have the noton of exponents at our dsposal..3.2 Exponents.6 Proposton ([Ge75, Lemma 3.8]. Assume that X \ X s a smooth dvsor and that E s an O X -locally free module of fnte rank carryng a D X/k (log D-structure. Then there exsts a unque decomposton E D = α Z p E α such that locally, after a choce of coordnates x,..., x n wth D = (x, f e + x E E α, then δ x (m := x m x (m (e = ( α m e + x E for all m 0. Proof. Geseker gves a trcky coordnate free descrpton of ths decomposton, but we only do the local constructon, wthout provng that t glues. Frst some prelmnares: For d, wrte C d := Maps(Z/p d Z, F p for the set of maps of sets Z/p d F p. Ths s an F p -algebra of dmenson p d+. For 7

8 a Z/p d Z defne h a C d by h a (b = ( b a. Recall that ths s well-defned by Lemma.0. Also defne χ a as the characterstc functon of a. Then t s easy to see that { h a a Z/p d Z } and {χ a a Z/p d Z} are two F p -bases for C d, and that the χ a are commutng, orthogonal dempotents. Now back to geometry: We may assume that X = Spec A wth global coordnates x,..., x n and that D = (x. Then X = Spec A[x ]. We may also assume that E s free on A, because E s torson free by assumpton. Then the operators δ x (a act lnearly on E/x E = E D by defnton. Thus we get a map φ d : C d End A/(x(E/x E by ( defnng φ d (h a = δ x (a. Ths s n fact a map of rngs, snce the δ x (a (x r = r a x r. But then the elements φ d (χ a (or a subset of the set of these elements are orthogonal, commutng dempotents n End A/x (E/x E, whch means that we get a decomposton E/x E = a Z/p d E a such that χ a acts trvally on E b wth b a and dentcally on E a. Snce h a = ( b b Z/p d a χb C d, we see that h a acts as ( b a on Ea. Fnally we have to let d vary. Let ρ : C d C d+ be the map comng from the projecton Z/p d+ Z Z/p d Z. For a Z/p d Z we have ρ(χ a = χ b. b Z/p d+ b a mod p d So the decomposton for C d+ refnes the decomposton for C d, and ths process has to stop snce E/x E s fntely generated. The δ x (n -acton can be computed for any d wth p d > n, and then by Lemma.0 ( ( α n = α mod p d n..7 Defnton. In the stuaton of Proposton.6, wrte Exp D (E = {α Z p rank E α > 0}, and call t the set of exponents of E along D. Now let X \X be an arbtrary strct normal crossngs dvsor wth D = D, and D smooth. If E s a torson free O X -coherent D X/k (log D-module, then we defne Exp D (E to as Exp D (E U where U X s an open set on whch E s locally free, and whch ntersects D j f and only f = j. One easly checks that ths defnton does not depend on the choce of U. We can now remedy the problem wth the local freeness of D X/k (log D- modules:.8 Theorem (reformulaton of [Ge75, Thm. 3.5]. If E s an O X -coherent and O X -torson free D X/k (log D-module such that Exp D (E maps njectvely to Z p /Z for all,.e. such that the exponents along D do not dffer by ntegers, then E s locally free f and only f E s reflexve. 8

9 The proof of the theorem s a rather complcated applcaton of local cohomology, whch we wll skp, snce we wll not use the result. In fact, for the our purposes, we may always remove closed subsets of codmenson 2 from X, so we can always assume that E s actually locally free..9 Corollary. Assume we have open mmersons X U and j : U X, such that U D for every. If E s a locally free O U -module wth D U/k (log U D- acton such that Exp D U (E Z p/z, then j E s a locally free D X/k (log D- module. Proof. It suffces to note that j E s reflexve..3.3 Functoralty We contnue to denote by k an algebracally closed feld of postve characterstc p. In ths secton we consder the followng stuaton: Let X X and Y Y be open mmersons of smooth, fnte type, separated, connected k-schemes, such that D Y := Y \ Y and D X := X \ X are strct normal crossngs dvsors. Furthermore let f and f be morphsms, fttng n the dagram Y Y f f X X,.e. such that f = f X..20 Remark. The readers famlar wth logarthmc structures ([Kat89] notce that n ths stuaton f nduces a morphsm of the log-schemes assocated wth X X and Y Y. We have the smlar functoralty results as n Proposton.7:.2 Proposton. (a f D X/k (log D X := O Y f O X f D X/k (log D X s a (D Y /k (log D Y, f D X/k (log D X -balgebra. (b There exsts a canoncal morphsm fttng n the commutatve dagram D Y /k (log D Y f f D X/k (log D X D Y /k (log D Y f D X/k (log D X D Y /k f D X/k where the lower horzontal morphsm s the one from Proposton.7. 9

10 Now assume that f s fnte, and f étale. Then f s fathfully flat. Moreover, (c f s an somorphsm f f s tamely ramfed wth respect to the strct normal crossngs dvsor D X. (d Let D Y, be a component of D Y mappng to the component D X, of D X. If E s an O X -coherent, O X -torson free D X/k (log D X -module, then the exponents of f E along D Y, are the exponents of E along D X, multpled by the ramfcaton ndex of the extenson of dscrete valuaton rngs assocated wth D Y, and D X,. Proof. We only gve an explct proof for the case that f s fnte and f étale. Then all of ths s essentally a queston about fnte extensons of dscrete valuaton rngs. Let A B be such an extenson, x A and y B unformzers. Then x = uy e for some e and u B. We know that K(B D B/k (log y = K(B A D A/k (log x s an somorphsm, and we clam that δ (pm y = c+d=m c,d 0 d+c=m c,d 0 ( e p c δ (pd x + y(b A D A/k (log x ( Lets assume the truth of ( for a mnute. Then (a follows drectly, and (d s also easy to derve: Let E be an A-module wth D A/k (log x-acton. If a E A B s such that δ (pm x (a = ( α p a + x(e B for some α m Zp, then δ (pm y (a = ( ( ( e α eα p c p d a + y(e B = p m a + y(e B whch proves (d. Lets now prove that ( holds. We compute: δ (pm y (x r = δ (pm y (u r y er = δ y (a (u r δ y (b (y er = = a+b=p m a,b 0 ( er p m ( er p m x r + x r + x r a+b=p m a>0,b 0 a+b=p m a>0,b 0 δ (a y (u r ( er b y er ( (a er δ y (u r b u r } {{ } (y (2 Ths shows that δ (pm y c+d=m ( e p c δ (p d x to fnsh, we note that ( er p m x r = c+d=m y(b D A/k (log x as clamed, and ( e (p p δ d c x (x r. For (c assume that A B s tamely ramfed. It suffces to show that δ (pm x s n the mage of f for every m 0, because f s njectve by the separablty of 0

11 the resdue extensons. Consder the completons of A and B: Â B. Replacng B by an étale extenson does not change dfferental operators, so we may assume that u = v e n B. Indeed, by Hensel s Lemma, u has an e-th root n B, f and only f t has an e-th root modulo y, and snce e s prme to p by assumpton, the extenson of the resdue felds obtaned by adjonng an e-th root s separable. Replacng y by vy, we may assume that x = y e. Then (2 shows In partcular, δ y ( = eδ x (, so δ x ( δ (pm y = c+d=m c,d 0 ( e p c δ (pm y = eδ (pm x + c+d=m c>,d 0 δ (pd x s n the mage of f. We proceed nductvely: ( e p c δ (pd x } {{ } m f whch completes the proof..22 Corollary. Let f be étale and f fnte and tamely ramfed wth respect to D X, and let E be an O Y -coherent, O Y -torson free D Y /k (log D Y -module. Then f E s an O X -coherent, O X -torson free D X/k (log D X -module. 2 Lecture 2 - Stratfed Bundles 2. Stratfed bundles 2.. Defntons and frst propertes We contnue to denote by k an algebracally closed feld of postve characterstc p. Agan let X be smooth, fnte type, connected and separated over k. 2. Defnton. A stratfed bundle s an O X -coherent D X/k -module. A morphsm of stratfed bundles s a D X/k -lnear morphsm of O X -modules. We wrte Strat(X for the category of stratfed bundles. A stratfed bundle s called trval, f t s somorphc to OX n n Strat(X. 2.2 Remark. We make two remarks about the name stratfed bundle : Recall that by Proposton. a stratfed bundle s n partcular a locally free O X -module, so the word bundle s justfed. Grothendeck defnes n [Gro68] the term stratfcaton as an nfntesmal descent datum. In our stuaton wth X a smooth k-scheme, the datum of a stratfcaton on an O X -module s equvalent to the datum of a D X/k -acton on an O X -module. For detals see [BO78, Ch. 2]. 2.3 Example. Let f : Y X be a fnte étale morphsm, then the coherent O X -module f O Y carres a canoncal D X/k -structure, because D Y/k = f D X/k.

12 Even more, the coverng s trval f and only f the stratfed bundle f O Y s trval. Indeed, f the coverng s trval, then clearly the assocated D X/k -structure s trval. Conversely, assume that f O Y = O n X as D X/k -modules. We have to show that ths s actually an somorphsm of O X -algebras. Let e,..., e r be a horzontal bass,.e. a bass, such that ψ(e = 0 for all dfferental operators ψ D X/k (U wth ψ( = 0 for U X open. Let U X be an open wth global coordnates x,..., x n. By the constructon of the D X/k -acton on f O Y, we have the equaton x (m l (e e j = x (a l (e x (b l (e j = 0 a+b=m a,b 0 over U, for any m, l,, j. If e e j = r t= λ te t, wth λ t O X (U, then 0 = r t= x (m l (λ t e t for any m > 0. Ths shows that λ t k for all t, so the étale O X -algebra f O Y comes va base-change from an étale k-algebra. But snce k s algebracally closed, any such algebra s trval. Ths completes the proof. We gve some functoral propertes of the category Strat(X: 2.4 Proposton. If f : Y X s a morphsm of smooth, separated, fnte type k-schemes, then: (a Pull-back of O X -modules along f nduces a functor f : Strat(X Strat(Y. (b If f s fnte and étale, then push-forward of O Y -modules along f nduces a functor f : Strat(Y Strat(X whch s rght adjont to f. (c If E, F are stratfed bundles, then E OX F and Hom OX (E, F are stratfed bundles n a bfunctoral way. Proof. The frst two statements follow from Proposton.7 and Corollary.9. The last statement s entrely analog to characterstc 0. In fact we know more: 2.5 Theorem ([SR72]. After the choce of a fber functor Strat(X Vectf k (Vectf k = the category of fnte dmenson k-vector spaces, the category Strat(X s neutral tannakan over k. We avod Tannaka theory n these lectures, but we need the followng defnton: 2

13 2.6 Defnton. Let E be a stratfed bundle. Then we denote by E the full subcategory of Strat(X wth objects all subquotents of stratfed bundles of the form P (E, E, for P (x, y N[x, y]. Here, f P (x, y = mx a y b, then P (E, E := m E... E E }{{}... E. }{{} a tmes b tmes = 2..2 Restrcton to an open subset In ths secton we analyze how the category Strat(X s related to the category Strat(U for U X an open dense subset. 2.7 Proposton. Let U X be an open dense subset and ρ : Strat(X Strat(U the restrcton functor. Then the followng statements are true: (a The restrcton functor ρ : Strat(X Strat(U s fully fathful. (b If E Strat(X, then the restrcton functor ρ E : E E U s an equvalence. (c If codm X X \ U 2, then the restrcton functor ρ : Strat(X Strat(U s an equvalence. Proof. We sketch the proof and begn wth (c,.e. we assume that codm X (X \ U 2. Then j E s an O X -coherent D X/k -module restrctng to E, so ρ s essentally surjectve. Note that j E, t s locally free. If E s any other locally free extenson of E, then there s a short exact sequence 0 E j E G O of O X -modules, and G s supported on X \ U. By the assumpton on the codmenson, Hom OX (G, O X = Ext O X (G, O X = 0. Thus E = (j E = j E. Fnally, the D X/k -acton on j E s unquely determned by the D U/k -acton on E, so the bjecton Hom OX (j E, j F = Hom OU (E, F nduces a bjecton Hom Strat(X (j E, j F = Hom Strat(U (E, F, so ρ s an equvalence. Now for (a, by what we just proved, we may assume that X \ U s a dsjont unon of codmenson closed subsets, and then that X \ U s a smooth dvsor. We may also shrnk X around the generc pont of ths dvsor, to assume that X = Spec A has global coordnates x,..., x n such that X \ U = (x. Then U = Spec A[x ]. It suffces to show that Hom Strat(X(O X, E Hom Strat(U (O U, E U s bjectve for every E Strat(X. Shrnkng X further around η, we may assume that E s free wth bass e,..., e r. Assume that φ : O U E U s gven by φ( = φ e, wth φ A[x ]. Then 0 = ( x (φ( = ( x (φ e + φ ( x (e. Ths shows that x ( (φ φ A A[x ]. By nducton we see that n fact x (m φ φ A for all m 0. But ths means that the pole order of φ along x s the same as the pole order of x (m for all m > 0, so φ A and Hom Strat(X (O X, E Hom Strat(Y (O U, E U s surjectve. 3

14 It s also njectve, snce f ψ, ψ 2 Hom Strat(X (O X, E, then (ψ ψ 2 A[x ] = 0 means that ψ = ψ 2. Fnally, lets prove (b. By what we have seen, we only need to prove that ρ : E E U s essentally surjectve. Frst note that for all P (x, y N[x, y], P (E U, E U are n the essental mage of the restrcton functor ρ : E E U, so we just have to check that for F Strat(X, all subquotents of F U n Strat(U lft to X. Let F F U be a subobject. Then j F j F and F j F are sub-d X/k -modules, and j F F s a stratfed bundle extendng F. Now f G s a quotent of F U, and F the kernel of F U G, then F/(j F F s a lft of G. 2.2 (X, X-regular sngular stratfed bundles Now we fx X X two smooth, connected, separated, fnte type k-schemes, wth X X open dense and such that D := X \ X s a strct normal crossngs dvsor. 2.8 Defnton. A stratfed bundle E Strat(X s called (X, X-regular sngular f there exsts an O X -coherent, O X -torson free D X/k (log D-module E extendng the D X/k -module E. We wrte Strat rs ((X, X for the full subcategory of Strat(X wth objects the (X, X-regular sngular stratfed bundles. 2.9 Remark. The extenson E s far from unque. 2.0 Proposton. For E Strat(X, the followng are equvalent: (a E s (X, X-regular sngular. (b There s some open dense subset U X contanng all generc ponts of X \ X, such that E X U s (X U, U-regular sngular. Proof. s clear. Conversely, Let E be an O U -coherent, torson free D U/k (log U \ X-module extendng E U X. Then we can glue t to E over X to see that E s (X, X U-regular sngular, say wth extenson E. Fnally, f j : U X s the open mmerson, then j E s an O X -coherent, torson free D X/k (log D-module, extendng E, by the assumpton on the codmenson Exponents For an (X, X-regular sngular stratfed bundle E there are many extensons to an O X -coherent D X/k (log D-module. In the frst lecture, we defned the noton of exponents of such an D X/k (log D-module. Luckly, we understand very well how these exponents for dfferent extensons of E are related: 2. Proposton. Let E be an (X, X-regular sngular stratfed bundle, and E, E two O X -coherent, O X -torson free D X/k (log D-extensons of E. If D s a smooth component of D, then the set of exponents of E and E along D has the same mage n Z p /Z. 4

15 Proof. We clearly may assume that D = X \ X s a smooth dvsor, say wth generc pont η, and we may shrnk X around η, so that we can assume that X = Spec A s affne, E, E are free, and that there are global coordnates x,..., x n, such that D = (x. Wrte j : X X. Then E, E j E are D X/k (log D-submodules, and E E s also an O X -coherent, torson free D X/k (log D-module extendng E. Thus, to prove the proposton, we may assume that E E s a D X/k (log D-submodule. We may now consder the stuaton over O X,η, whch s a dscrete valuaton rng wth unformzer x. For some n N we have x n E η E η E η, and t s not dffcult to compute that Exp(x n E η = {α + n α Exp(E η}. Thus t suffces to show that f α Z p s an exponent of E η, then α + N α s an exponent of E η for some N α Z. Let α,..., α r be the exponents of E η. If e E η \ x E η s an element such that δ x (m (e = ( α m e+x E η, then α also s an exponent of E η. If there s no such e, let e,..., e r be a lft of a bass of E η /x, such that δ x (m (e = ( α m e +x E η for all m 0, and defne E as the submodule of E η spanned by x e, e 2..., e r. Note that E η E. It s readly checked that E s stable under the δ x (m, and we show that α s an exponent of E. Snce O X,η /(x s a feld, and snce e x E for >, one fnds t E, such that t, e 2,..., e r s a lft of a bass of E /x E, and such that δ x (m (t = ( β m t + x E, for some β Z p. We compute: δ x (m (x e = ( α m x e + f + x E for some f e 2,..., e r. On the other hand, wrtng x e = λ t + j> λ je j wth λ O X,η, we get δ x (m (x e = ( β λ t + m j> ( αj λ j e j + x E. m Snce x e e 2,..., e r, we have λ (x. Hence, comparng coeffcents shows that β = α, whch shows that α s an exponent of E. If there s e E \ x E η such that δ x (m (e = ( α m e + x E then α s an exponent of E η. Otherwse we construct E 2 := (E E contaned n E η as above, wth exponent α 2. Snce E η/e η has fnte length, ths process has to termnate, so there s some n such that α n s an exponent of E η. 2.2 Defnton. The exponents of E Strat rs ((X, X are the exponents of an O X -coherent, torson free D X/k (log D-extenson of E modulo Z. Hence the exponents of E le n Z p /Z Canoncal extensons 2.3 Defnton. Let τ be a (set theoretc secton of the projecton Z p Z p /Z and E Strat rs ((X, X. A τ-extenson of E s an O X -locally free fnte rank D X/k (log D-module E such that the exponents of E le n the mage of τ. 5

16 Whle the extenson to X of an (X, X-regular sngular stratfed bundle s not unque, there always exsts a unque τ-extenson. Ths s completely parallel to the stuaton n characterstc Theorem ([Kn2, Sec ]. Let E be an (X, X-regular sngular stratfed bundle. Then a τ-extenson exsts and t s unque up to somorphsms restrctng to the dentty on E. Proof. We sketch the constructon of a τ-extenson, and do not dscuss the uncty. The proof s an extenson of methods of Geseker. As usual we may reduce to the followng stuaton (usng Corollary.9: X = Spec A wth global coordnates x,..., x n, such that D = (x and X = Spec A[x ]. Let E be any O X -coherent, torson free extenson of E to a D X/k (log D-module. By shrnkng X around the generc pont of D, we may assume E s free. Wrte Exp(E = {α,..., α r } Z p for the set of exponents of E along D. Let b := (b,..., b r Z r be an r-tuple, such that b = b j whenever α = α j. From E we construct an O X -coherent, torson free D X/k (log D-extenson E (b of E, such that the set of exponents of E (b s {α + b, α 2 + b 2,..., α r + b r }. Ths would prove the theorem, we move the exponents of E nto the mage of the secton τ. The constructon s done n two steps: (a If a N, then E(aD has exponents {α a,..., α r a}. (b For [, r] we construct E such that Exp(E = {α j j } {α + }. In fact, let =. Then E s defned as the submodule of E generated by x E and the mages of δ (pj x ( α j d. Ths s a DX/k (log D-submodule of E, because (m x j commutes wth δ (pj x for all m,, n, j. Now assume that α = α 2 =... = α l, and α α j for j > l. Let s,..., s r be a bass of E such that ( δ x (m α (s = s + x E, for all, m m Then defne s := x s for [, l] and s = s for > l. Then the s are a bass of E whch fnshes the proof. 2.5 Corollary. The essental mage of the restrcton functor Strat(X Strat rs ((X, X s the full subcategory of (X, X-regular sngular bundles wth exponents equal to 0 Z p /Z. Proof. By the τ-extenson theorem, we have to show that a locally free O X - coherent, O X -torson free D X/k (log D-module wth exponents 0 has a canoncal D X/k -acton. Ths s enough by Proposton 2.7. For ths we may assume that X s affne wth local coordnates x,..., x n such that D = (x and E free. Then havng exponents 0 means that for every e E, δ ( x (e = 0 e + x E. 6

17 Thus we can defne x ( (e := δ( x. In partcular, the D X/k (log D-acton defnes an honest flat connecton wth p-curvature 0 on E. Then, by Carter s Theorem ([Kat70], f ( ( denotes Frobenus twst, then E = FX/k E, where x (e E s the D X ( /k (log D ( -module obtaned as the sheaf of sectons s of E such that ( x as δ ( x ( ( x ( (s = 0 for all. Moreover, E also has exponents 0, and δ x (p acts on E. We reapply the argument, to gve meanng to the acton of = (p x on E. Then we apply Carter s Theorem agan, etc Proof of the Man Lemma wth respect to (X, X We now have all the tools we need to prove the Man Lemma Theorem.2 wth respect to a fxed partal compactfcaton X X. 2.6 Theorem ((X, X-Man Lemma. Let X X be an open mmerson of smooth, fnte type, separated, connected k-schemes, such that D X := X \ X s a strct normal crossngs dvsor. Let f : Y X be a fnte étale galos coverng. Then f s tamely ramfed wth respect to D X f and only f f O Y s (X, X-regular sngular. Proof. By takng the normalzaton f : Y X of X n k(y, and by removng codmenson 2 subsets of X f necessary, we arrve at the followng stuaton: D Y := Y \ Y has strct normal crossngs, the dagram Y Y f f X X commutes, and f s fnte. Ths mples that f s flat snce X and Y are both smooth, and the dmensons of the fber of f s constantly 0. Then t follows that f s surjectve, as t s open and closed, and X s connected. The drecton s easer: Corollary.22 shows that f O Y s an O X - coherent, O X -torson free D X/k (log D X -module extendng the stratfed bundle f O Y, so f O Y s (X, X-regular sngular. The converse drecton s more nvolved: We assume f O Y to be (X, X- regular sngular, and we construct a fnte étale morphsm h : Z X, whch s tamely ramfed wth respect to (X, X, such that Y X Z Z s the trval coverng Z Z. Then h domnates f, so f s tame wth respect to (X, X. Agan we may assume wthout loss of generalty that X \ X s a smooth dvsor wth generc pont η, and n the constructon we may shrnk X around η. We proceed n fve steps: (a Note that the exponents of f O Y are torson n Z p /Z, because by Proposton.2 pullng back an D X/k (log D X -extenson of f O Y along f multples the exponents by the ramfcaton ndces of f along D X, and clearly f f O Y = O deg f Y s trval. (b By Theorem 2.4 we fnd an O X -coherent, torson free D X/k (log D X - extenson E of f O Y wth exponents n Z Q; say a b,..., ar b wth (b, p =. 7

18 (c Shrnkng X around η, f necessary, we may assume that X = Spec A, wth local coordnates x,..., x n such that D X = (x. Then defne Z := Spec A[x /b ], and let h : Z X be the assocated coverng. Let Z := h (X and h = h Z. Then h s étale and h fnte and tamely ramfed wth respect to X \ X. Then h f O Y has exponents equal to 0 n Z p /Z, whch means by Corollary 2.5 there exsts a stratfed bundle E Strat(Z extendng h f O Y. (d Now we clam that there exsts a fnte étale coverng ḡ : Z Z such that ḡ E s trval. Indeed, ths s true for E Z because t s true for f O Y, and Proposton 2.7 shows that the restrcton functor E h f O Y s an equvalence. Ths uses a tny bt of Tannakan category. In other words: The Pcard-Vessot torsor assocated wth E restrcts to the Pcard-Vessot torsor of E X = h f O Y, whch s fnte. (e We can fnsh up: Wrte Z := ḡ (Z, g = ḡ Z and h = h g. Then we have the followng dagram Y X Z f Z Z h Y g Z h Y f X h and h s tamely ramfed wth respect to X \ X by constructon. But also by constructon h f O Y s trval, and snce h f O Y = f Z, h Y O Y, Example 2.3 shows that f Z s the trval coverng, so the proof s complete. 2.3 Regular sngular stratfed bundles n general Ths secton was not covered n the lectures. 2.7 Defnton. (a Let X be a smooth, connected, separated, fnte type k-scheme. A par (X, X s called good partal compactfcaton f X s smooth, separated, and of fnte type over k, X X s a dense open subset. X \ X s a strct normal crossngs dvsor. (b A fnte étale coverng f : Y X s called tame, f t s tamely ramfed wth respect to every good partal compactfcatons (X, X of X. (c A stratfed bundle E Strat(X s called regular sngular, f and only f t s (X, X-regular sngular for every good partal compactfcaton (X, X of X. 8

19 2.8 Remark. Note that X s not requred to be proper. Hence the name partal good compactfcaton. The noton of tameness from the above defnton was ntensvely studed n [KS0] under the name dvsor tameness. Fnally, Theorem 2.6 mmedately mples the Man Lemma Theorem.2 References [BO78] P. Berthelot and A. Ogus, Notes on crystallne cohomology, Prnceton Unversty Press, Prnceton, N.J., 978. MR MR (58 #0908 [EGA4] A. Grothendeck, Éléments de géométre algébrque. IV: Étude locale des schémas et des morphsmes de schémas., Publ. Math. IHES 32 (967. [Ge75] D. Geseker, Flat vector bundles and the fundamental group n non-zero characterstcs, Ann. Scuola Norm. Sup. Psa Cl. Sc. (4 2 (975, no., 3. MR MR (52 #356 [Gro68] A. Grothendeck, Crystals and the de Rham cohomology of schemes, Dx Exposés sur la Cohomologe des Schémas, North-Holland, Amsterdam, 968, pp MR (42 #4558 [Kat70] N. M. Katz, Nlpotent connectons and the monodromy theorem: Applcatons of a result of Turrttn, Inst. Hautes Études Sc. Publ. Math. (970, no. 39, MR MR02977 (45 #27 [Kat89] K. Kato, Logarthmc structures of Fontane-Illuse, Algebrac analyss, geometry, and number theory (Baltmore, MD, 988, Johns Hopkns Unv. Press, Baltmore, MD, 989, pp MR (99b:4020 [Kn2] L. Kndler, Regular sngular stratfed bundles n postve characterstc, 202, Dssertaton, Unverstät Dusburg-Essen. [KS0] [SR72] M. Kerz and A. Schmdt, On dfferent notons of tameness n arthmetc geometry, Math. Ann. 346 (200, no. 3, MR (20a:4052 N. Saavedra Rvano, Catégores Tannakennes, Lecture Notes n Mathematcs, Vol. 265, Sprnger-Verlag, Berln, 972. MR (49 #2769 9