Weak Kolchin Irreducibility for Arithmetic Jet Spaces

Transcription

1 Weak Kolchin Irreducibility for Arithmetic Jet Spaces Taylor Dupuy (with James Freitag and Lance E. Miller)

2 Kolchin (1970s) X/C irreducible =) J 1 (X) irreducible (singular) Claim: X/W p,1 (F alg p ) bx irreducible +" =) J p,1 (X) weakly irreducible

3 9Z J p,1 (X) Z is closed irreducible subset. Z contains an open. Claim: Z has all of the closed points. X/W p,1 (F alg p ) bx irreducible +" =) J p,1 (X) weakly irreducible 9h : Y! X Y smooth, b Y irreducible. X =Spec(A) Y (W p,1 (F alg p ))! X(W p,1 (F alg p )) surjective

4 9h : Y! X Y smooth, b Y irreducible. Y (W p,1 (F alg p ))! X(W p,1 (F alg p )) surjective Includes X/R generically smooth. X =SpecR[x, y]/(y 2 x 2 (x 1)) Excludes X =SpecR[x, y]/(y 2 x 2 (x + p) X =SpecR[x, y, z]/(x p = zy p ) X =SpecR[x.y]/(xy p)

5 Background

6 Let D 1 : CRing! CRing be the functor A 7! A[t]/(t 2 ). A derivation A! A is the same as a section of D 1 (A)! A. Generalize to ring schemes: R : CRing! CRing with natural transformations R! Id. An action of R on a ring C is a section of R(C)! C

7 Functor Operation D 1 W p,1 Derivation p-derivation A 7! A A Ring Endo W big witt -rings

8 (Borger-Weiland 00s, Tall-Wraith 70s) When R is an a ne ring scheme R =Spec(Q) there exists a left adjoint CRing(Q A, B) =CRing(A, R(B)). The bifunctor is called the composition product For X a scheme define functor of jets J Q (X) :=X(R( )) : CRing! Set If R a comonad, then call it a functor of arcs. When functor representable, we call it a jet or arc space.

9 Jet Functor J Q (X)(A) :=X(R(A))

10 O O O O There exists a relative version of this construction as well. For a fixed action Q C! C on a base we let J Q (X/C, ) denote the relativized version. Prolongations Q A / B A s 1 / R(B) alg map R( alg map ) Q C / C C s 0 / R(C)

11 O O / O alg map A C s 1 s 0 / / R(B) R( alg map ) R(C) Spec R(B) Spec(s 1) X Spec R(C) Spec(s 0) / Spec(C) prolongation pt Relative Jet Functors: J Q (X/C, ) J Q (X/C, )(B) ={P 2 X(R(B)) : prolongation pt }

12 J Q (X/C, ) Notations. J n (X/C, D) = nth order classical jet spaces J 1 (X/C, D) = classical arc spaces J p,r (X/C, ) =J p,r (X) truncated p-jet spaces J p,1 (X/C, ) =J p,1 (X) p-arc spaces J b p,r (X) and J b p,1 (X), Buium s p-formally completed version

13 Example. J 1 (X/C, D) J 1 (X/C) = = classical first order tangent space ( T X/C, twisted T X/C, D = trivial D = not trivial

14 Let X/C[[t]] be defined by xy = t. Consider C[[t]] as having a trivial derivation. The equations for J 1 (X/C[[t]]) Spec C[[t]][x, y, x 0 y 0 ] are xy = t and x 0 y + y 0 x = 0. Let X/C[[t]] be defined by xy = t. Consider C[[t]] with its nontrivial derivation D = d/dt. The equations of J 1 (X/C[[t]]) are then xy t = 0 ẋy + xẏ 1 = 0 ẍy +2ẋẏ + xÿ = 0.

15 Let f(x, y) =xy t. Then we look to satisfy the equation 0=(x 0 + x 1 " + )(y 0 + y 1 " + ) exp(t) where exp(t) =t + " given the system of equations x 0 y 0 t = 0 x 0 y 1 + y 0 x 1 1 = 0 x 0 y 2 +2x 1 y 1 + x 2 y 0 = 0.

16 Special fiber of J p,1 (X) example: Let X =SpecR[x, y]/(xy p). Let x =(x 0,x 1,...) y =(y 0,y 1,...) p =(0, 1, 0,...) Multiplication by p acts by translating to the right and pth powering x 0 y 0 = 0 x p 0 y 1 + y p 0 x 1 = 1 m 2 = 0.. and one can trivially see that Gr 1 (X) =V (x 0 ) [ V (y 0 ).

17 Suppose now we are working ever R. Then p is not p 1 in a ring where we replace everything by the witt vectors exp p (p) =(p, 1 p p 1,...) which this means that m i (x, y) =exp p (p) i whose reduction modulo p recover the previous ones. It is nontrivial to see that this scheme is irreducible.

18 example: y 2 = x 2 (x +(3x2p +2x p )ẋ + p(3x p + 1)ẋ 2 + p 2 ẋ 4 =2y p ẏ + = f(xp,y p ) p f(x, y) p 1 1 (0, 0) ẏ2 =ẋ 2 (1 + pẋ)

19 For the rest of the talk assume X is a ne. (this deals with representability issues) Moosa-Scanlon, Bhatt-Lurie, Borger

20 Classical Jet Spaces and Singularities

21 Why do we care about jet spaces?

22 Why do we care about jet spaces? Example. X : x 4 + y 4 + z 4 =0 x = x 0 + x 1 t + x 2 t 2 + x 3 t 3 mod t 4 y = y 0 + y 1 t + y 2 t 2 + y 3 t 3 mod t 4 z = z 0 + z 1 t + z 2 t 3 + z 3 t 3 mod t 4 x 0 = y 0 = z 0 =0

23 x = x 0 + x 1 t + x 2 t 2 + x 3 t 3 mod t 4 y = y 0 + y 1 t + y 2 t 2 + y 3 t 3 mod t 4 z = z 0 + z 1 t + z 2 t 3 + z 3 t 3 mod t 4 x 0 = y 0 = z 0 =0 X : x 4 + y 4 + z 4 =0 (x 1 t + x 2 t 2 + x 3 t 3 ) 4 +(y 1 t + y 2 t 2 + y 3 t 3 ) 4 +(z 1 t + z 2 t 3 + z 3 t 3 ) 4 0 dim 4 1 (0, 0, 0) = 9 (4 + 1) dim(x) =5 2 = 10 4 dim(x) =8

24 Why do we care about jet spaces?

25 Why do we care about jet spaces? Mustata: lct(x, D) =dim(x) sup r 0 dim J r (D) r +1 x 5 + y 5 + z 5 x 3 + y 3 + z x y 99

26 Why do we care about jet spaces? Kolchin (1970s) X/C irreducible =) J 1 (X) irreducible (singular) lim n J n (X) Gillet, Mustata, de Fernex, Loeser-Sebag, Kolchin, Nicaise- Sebag, Ishii-Kollar, (Chambert-Loir)-Nicaise-Sebag

27 Proof of Kolchin Irreducibility Step 1: Deformations = Irreducibility (general). J r (X) Step 2: Smooth case. Step 3: Reduction to Smooth Case ex X X h Sing(X) =Y 1 2J 1 (Y )

28 Arc Deformations and Irreducibility

29 Step 1: Deforming Arcs = Irreducibility Kolchin (1970s) X/C irreducible =) J 1 (X) irreducible (singular) Claim: Arc deformability Irreducibility

30 & Step 1: Deforming Arcs = Irreducibility Arcs: P 2 J Q (X)(A) $ 2 X(R(A)) Spec(A) P / J(X) Spec(R(A)) / X

31 & Spec(A) P / J(X) Spec(R(A)) / X J 1 (X) P 2 J Q (X)(A) P X Y (= X sing ) 2 X(R(A))

32 & Step 1: Deforming Arcs = Irreducibility Arcs: Spec(A) P / J(X) P 2 J Q (X)(A) $ 2 X(R(A)) Spec(R(A)) / X Deformations: 0 2 X(R(A 0 )) = generic of (Spec(R(A))) { 0}3

33 Step 1: Deforming Arcs = Irreducibility Deformations: X 0 2 X(R(A 0 )) e Y (= X sing ) 2 X(R(A))

34 & Step 1: Deforming Arcs = Irreducibility Arcs: Spec(A) P / J(X) P 2 J Q (X)(A) $ 2 X(R(A)) Spec(R(A)) / X Deformations: 0 2 X(R(A 0 )) = generic of (Spec(R(A))) { 0}3

35 Step 1: Deforming Arcs = Irreducibility Arcs: P 2 J Q (X)(A) $ 2 X(R(A)) Deformations: 0 2 X(R(A 0 )) = generic of (Spec(R(A))) { 0}3 Arc Deformability: 8 2 X(R(A))), 8Y ( X, X(R(A 0 )) 0 deforms 0 generically outside Y

36 Step 1: Deforming Arcs = Irreducibility J 1 (X) P e Arc Deformability: P X 0 deforms e Y (= X sing ) 0 generically outside Y

37 Step 1: Deforming Arcs = Irreducibility 8 2 X(R(A))), 8Y ( X, X(R(A 0 )) 0 deforms 0 generically outside Y

38 Step 1: Deforming Arcs = Irreducibility Deformation Idea Arc deformability Irreducibility Simple Case: 1 (Sm(X)) nonempty. A a domain =) R(A) a domain.

39 Classical Kolchin Irreducibility ex J r (X) X h X Sing(X) =Y 1 2J 1 (Y )

40 X/C Step 2: Smooth Case (Classical) Theorem. X/C smooth, irreducible =) J r (X) irreducible Lemma. X/C smooth, irreducible =) J r (X) affine bundle J r (X) X

41 X/C Step 2: Smooth Case (Classical) Theorem. X/C smooth, irreducible =) J r (X) irreducible J r (X) proof assuming lemma: r 1 (U) = U A (r+1) dim(x) O( 1 r (U)) = O(U)[ variables ] domain X

42 X/C Step 3: Reduction to Smooth Case (classical) = J 1 (Sm(X)) J 1 (X) irreducible X 1 Sing(X) =Y 1 2 J 1 (Y ) (Y )

43 ex Step 3: Reduction to Smooth Case (classical) X/C J 1 (Sm(X)) J 1 (X) X h 1 Sing(X) =Y 1 2 J 1 (Y )

44 Step 3: Reduction to Smooth Case (classical) ex X/C X h Sing(X) =Y 1 2 J 1 (Y )

45 X/C Step 3: Reduction to Smooth Case (classical) Sing(X) =Y X J 1 (Y ) 1 2 J 1 (Sm(X)) = J 1 (X)

46 Recap of Classical Step 1: Deformations = Irreducibility (general). J r (X) Step 2: Smooth case (classical). Step 3: Reduction to Smooth Case X (classical) ex X h Sing(X) =Y 1 2J 1 (Y )

47 Step 2: Smooth Case (formal arithmetic) Theorem. (Buium) bj p,r (X) X/R smooth R = W p,1 (F alg p ) bj p,r (X)! b X an affine bundle bx Corollary. bx irreducible b Jp,r (X) =) irreducible

48 Corollary. Step 2: Smooth Case (formal arithmetic) bx irreducible b Jp,r (X) =) irreducible bj p,r (X) bx

49 Step 3: Reduction to Smooth Case Alterations?? (Introduces Ramification) Spec(K) / e X Spec(R) / X Neron Smoothenings (Sebag-Loeser,Nicaise-(Chambert- Loir)): 9h : Y! X Y smooth, b Y irreducible. Y (W p,1 (F alg p ))! X(W p,1 (F alg p )) surjective

50 9Z J p,1 (X) Z is closed irreducible subset. Z contains an open. Claim: Z has all of the closed points. X/W p,1 (F alg p ) bx irreducible +" =) J p,1 (X) weakly irreducible 9h : Y! X Y smooth, b Y irreducible. Y (W p,1 (F alg p ))! X(W p,1 (F alg p )) surjective

51 THANK YOU

52

53 h 1 (D) K = ex Spec K[[T ]] Spec L L = Frac(K) h 1 (Sing(X)) h C((T )) alg Y =Sing(X) K h : X e =! X

54 x p = zy p y 2 = x 2 (x + p) y 2 = x 2 (x 1) R = W p,1 X/R ex! X x 2 J p,1 (X) apple(x) W p,1 (k) char(k) 6= p ex(r)! X(R)

55 O : Spec R(B) s 1 / X s 0 B B s 1 Spec B

56 Spec(R(C)) lim n J n (X) Spec(R(B)) Q Q A C J Q (X) B C J 1 (X) p,1 J Q (X) R = CRing(Q, )

57 x a x a n n/a lct(x, D)

58 Step 3: Reduction to Smooth Case (arithmetic) exp : D 1! D 1 D 1 exp A : A[[t]]! A[[T,S]] t 7! T + S f(t) 7! X n 0 f (n) (T ) n! S n exp : W p,1! W p,1 W p,1