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Barsotti-Tate Groups and Dieudonné Crystals BARSOTTI–TATE GROUPS AND DIEUDONNÉ CRYSTALS ALEXANDRE GROTHENDIECK TRANSLATED BY ERIC PETERSON FOREWORDS This is an English translation of Grothendieck’s Groupes de Barsotti–Tate et Cristaux de Dieudonné, published in 1974 by Les Presses de l’Université de Montréal in their Séminaire de Mathématiques Supérieures sequence. While the French original is actually quite legible even to an anglophone, I find it preferable to be able to read this interesting lay-of-the-land document without even the minor distraction of having to mentally pair the cognates “cohomologie” and “cohomology”. I hope that this translation encourages a wider audience to become familiar with its contents, rather than (as was the case for me) skimming the main for the biggest ideas and relegating the rest to a rainy day. I’ve taken some liberties with the prose, but I’ve done my best to preserve the letter of the mathematics. Corrections are warmly welcomed at [email protected]. Eric Peterson These notes do not include certain material treated by Prof. Grothendieck in his 1970 Avertissement summer Montreal course. We hope to include in a later edition those materials not published here. Among the items omitted here, the first is a chapter intended to cover F –crystals, which would have been inserted between the final two. As a compromise, one will instead find a letter from Grothendieck to Barsotti in an appendix at the end of the notes which covers the intended applications of F –crystals. One may also consult the notes of Demazure on p–divisible groups [Dem72], here called Barsotti–Tate groups. The second omission is two other chapters which were to sit at the end of the text and which were to cover the definition of generalized Dieudonné functors and their relationship with the classical functors. We will instead refer the reader to the article, presently in preparation, of Mazur and Messing [MM74]. The existence of these notes is primarily due to the efforts of Monique Hakim and Jean- Pierre Delale, who drafted the majority of the chapters. We sincerely thank them. We must also mention that they are not the ones responsible for our incapacity to reproduce all the materials outlined in the course; we hope nonetheless that these notes, even if incomplete, will give to readers an introduction to the theory of crystals and to Barsotti–Tate groups. Finally, we thank Ms. Thérèse Fournier for her excellent work in typing these notes. J.P. Labute, October 1973 1 2 ALEXANDRE GROTHENDIECK CONTENTS Forewords 1 1. Preliminaries on Witt vectors 3 1.1. Reminders on Witt vectors 3 1.2. The ind-schemes Wc and Wc0 4 1.3. The ring W 5 1.4. The Frobenius and Verschiebung of a group scheme 7 2. Locally free finite groups and classical Dieudonné theory 8 2.1. General reminders on locally free finite groups 9 2.2. Particular types of groups 11 2.3. Decomposition of a finite locally free p–group over a point s of char. p 13 2.4. Dieudonné theory over a perfect field 14 2.5. Corollaries 17 2.6. Construction of a quasi-inverse to D∗ 18 3. Barsotti–Tate groups 21 3.1. Notation, preliminary definitions, and flatness and representability criteria 21 3.2. Barsotti–Tate groups, in full and truncated at stage n 23 3.3. Facts about Barsotti–Tate groups 25 3.4. Examples and particular types of Barsotti–Tate groups 26 3.5. Composition series of a Barsotti–Tate group 29 4. Crystals 30 4.1. Reminders on divided powers 30 4.2. The crystalline site of a scheme 34 4.3. Relation between crystals and Witt vectors 35 4.4. The case of a perfect scheme 38 4.5. Case of a relatively smooth scheme 42 4.6. Crystals on a scheme over a perfect field of characteristic p 44 4.7. Indications on crystalline cohomology 44 5. Main course 46 5.1. The Dieudonné functor 47 5.2. Filtrations associated to Dieudonné crystals 48 5.3. Admissible F –V –crystals in characteristic p 49 5.4. Deformation theory for abelian schemes 49 5.5. Relations between the two Dieudonné theories 51 6. Infinitesimal properties and deformations of Barsotti–Tate groups 52 6.1. Infinitesimal neighborhoods and formal Lie groups 52 6.2. Results special to characteristic p 54 6.3. Formal Lie groups in characteristic p 57 6.4. The Formal Lie group of a Barsotti–Tate group where p is locally nilpotent 57 6.5. Infinitesimal deformations of Barsotti–Tate groups (announcements) 58 6.6. The relative cotangent complex 58 6.7. Example deformation problems 60 Appendix A. A letter from M. A. Grothendieck to Barsotti 61 References 65 BARSOTTI–TATE GROUPS AND DIEUDONNÉ CRYSTALS 3 1. PRELIMINARIES ON WITT VECTORS To lead off, we describe a variety of results stemming from the study of Witt vectors Chapitre I which will either be of specific use to us later on or which serve as motifs which we will Preliminaires sur Witt emulate. Throughout this section, p will denote a fixed prime number. 1.1. Reminders on Witt vectors. The scheme W of Witt vectors and the scheme Wn of 1. Rappels sur les Witt vectors truncated to order n are certain ring-schemes, each affine over Z. To give vecteurs de Witt n their definitions, let n 1 be an integer and let E = SpecZ[T1,T2,...,Tn] be affine space ≥ n n of dimension n over Z. The scheme E represents the functor A A in the category of affine schemes in the category of sets. It is thus endowed with7! a natural structure of n n a scheme in rings. We will denote the scheme E endowed with this structure. One O n then proves [Ser62, Section II.6] that there exists a unique ring scheme structure on E such that the morphism of schemes n n ' = ('1,'2,...,'n): , E !O i i j X j 1 p − 'i : (x1, x2,..., xn) p − xj 7! j=1 is a homomorphism of rings. Definition 1.1.1. Wn is the scheme En endowed with this ring scheme structure. This definition of the ring (resp. group) structure comes with a universality property: a map f : X n is a morphism of ring (resp. group) schemes if and only if for every ! W 1 i n, the map 'i f : X is a homomorphism of rings (resp. of groups). This entails≤ ≤ that the restriction◦ morphisms!O Rn : n 1 n W + ! W (x1,..., xn, xn 1) (x1,..., xn) + 7! are morphisms of ring schemes, since 'i Rn = 'i . Definition 1.1.2. The ring scheme of Witt vectors is given by Rn = lim n 1 n . W ···! W + −! W !··· The scheme underlying W is, in particular, isomorphic to EN. We define two families of maps: 2. Quelques mor- phismes remarquables V = Vn : n n, T = Tn : n n 1, W ! W W ! W + (x1, x2,..., xn) (0, x1,..., xn 1), (x1, x2,..., xn) (0, x1,..., xn). 2.1 7! − 7! The morphism T is additive, since 't T = p'i 1 for 1 i n + 1. It follows then that − ≤ ≤ Vn = RnTn = Tn 1Rn 1 is also an additive morphism. − − Definition 1.1.3. Passing to the limit, the morphisms Vn define an additive morphism V : W ! W called Verschiebung.1 4 ALEXANDRE GROTHENDIECK Definition 1.1.4. The Frobenius F : is defined as the limit of the morphisms W ! W 2.2 Fn : Wn Wn, ! p p p (x1, x2,..., xn) (x , x ,..., xn ). 7! 1 2 The Frobenius is generally not even additive, but after reducing to characteristic p it defines a ring homomorphism F : , Fp WFp ! WFp where denotes the prime field =p and denotes the scheme base-changed to Fp Z Z WFp W Fp . We may use these morphisms to study the two composition laws on W: (x1, x2,..., xn,...) + (y1, y2,..., yn,...) = (S1, S2,..., Sn,...), (x1, x2,..., xn,...) (y1, y2,..., yn,...) = (P1, P2,..., Pn,...), · where Si and Pi are integer-coefficient polynomials in the indeterminates x1, x2,..., xi , and y , y ,..., y . Using the fact that F is a ring morphism, for all –algebras we then 1 2 i Fp Fp have p p p P(x , y ) = [P(x, y)] . Proposition 2.3 Proposition 1.1.5 ([Ser62, Section II.6, Corollary of Theorem 7]). In characteristic p, one has the relations on : WFp F V = V F = p. ◦ ◦ Proof. Set h = (h1, h2,..., hn,...) = (p VF )(x1,..., xn,...). − We know hi Z[x1,..., xi ], and it suffices to show that hi is divisible by p for all i to prove the proposition.2 We calculate i 'i (h) = p'i (x) 'i (VF (x)) = p xi , − from which it follows first that h1 = px1, then that p 2 2 '2(h) = h1 + ph2 = p x p 2 where h2 = p(x2 p − x1). Continuing in this manner and inducting on i, one may − deduce that for all i one has hi = pki . Remark 1.1.6. From this it follows that p 0,1,0,... in . = ( ) WFp 1.2. The ind-schemes c and c . In addition to , we may also make the following 3. L’ind-schéma W et W W0 W ! dual definition: l’ind-schéma W0 ! Definition 1.2.1. Set Wc to be the group ind-scheme Tn c = colim n n 1 . W ···! W −! W + !··· 1This is also known as the décalage morphism. BARSOTTI–TATE GROUPS AND DIEUDONNÉ CRYSTALS 5 ( ) Remark 1.2.2. The underlying ind-scheme in sets is isomorphic to E N : for all rings A, (N) N c(A) = A = (x1, x2,..., xn,...) A xi = 0 except for finitely many indices .
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