CLASSIFYING FORMAL GROUPS WITH DIEUDONNE´ THEORY WILLIAM BALDERRAMA Abstract. Dieudonn´etheory gives an equivalence between one-dimensional commutative formal groups over a perfect field and certain modules over a certain ring. One can then attempt to develop the classic theory of such formal groups purely within the category of these modules, as this note does. At a technical level, this replaces power series manipulations with Frobenius- semilinear algebra. The motivation for this exercise was to obtain a clean formulation and proof of Theorem 3.2.3. Contents 1. Basic properties of formal groups 1 1.1. Dieudonn´emodules 1 1.2. The Honda module 2 1.3. Semilinear algebra 3 2. Classification of formal groups 8 2.1. Galois cohomology 8 2.2. The classification theorem 9 2.3. Case of a finite field 11 3. Determinants 12 3.1. Exterior powers 12 3.2. Classical determinant 13 1. Basic properties of formal groups 1.1. Dieudonn´emodules. Fix a perfect field κ of positive characteristic p. Let W (κ) denote the ring of p-typical Witt vectors on κ, and let D(κ) be the Dieudonne ring of κ. This is the noncommutative ring obtained from W (κ) by adjoining two symbols F and V subject to −1 F λ = λσF; V λ = λσ V; F V = p = V F; where λ ranges over W (κ) and (−)σ is the canonical Frobenius automorphism v of W (κ). Let LModD(κ) be the category of left D(κ)-module M subject to the following conditions: (1) As a W (κ)-module, M is free of finite and positive rank, k (2) The action of V on M is topologically nilpotent, and M = limk M=V M, (3) The quotient M=V M is a simple W (κ)-module, i.e. M=V M ' κ. Date: January 1, 2019. 1 2 W. BALDERRAMA v Observe that the condition FV = p = VF tells us that for M 2 LModD(κ), the operators F and V determine each other. In fact, we need not have introduced F at all; see Corollary 1.3.4. The main Dieudonn´etheorem relevant to us is the following. v 1.1.1. Theorem. The category LModD(κ) is equivalent to the category of one- dimensional commutative formal groups over Spec κ. We will not need the details of the construction of this equivalence as we will be v working entirely within LModD(κ), but let us say something to establish where our c conventions sit. Let Hopfκ denote the category of Hopf algebras H over κ such _ c that Spf H is a connected formal group. There is an equivalence DM from Hopfκ to the category of left D(κ)-modules M such that for any a 2 M, there is some n such that V na = 0. Suppose now Spf H_ is a connected p-divisible group. Then c H 2 Hopfκ, so we can look at DM(H) 2 LModD(κ). As H is a p-divisible Hopf algebra, DM(H) is p-divisible, and we can noncanonically identify DM(H) =∼ W (κ)=(p1)⊕h; ker[V : DM(H) ! DM(H)] =∼ κ⊕d; where h is the height and d is the dimension of Spf H. Let DM(H)[pn] ⊂ DM(H) be the pn-torsion subgroup. Multiplication by p gives maps DM(H)[pn] ! DM(H)[pn−1], n and we define DM(d H) = limn DM(H)[p ]. Under the above noncanonical isomor- phism, we identify DM(H)[pn] ! DM(H)[pn−1] as the quotient map W (κ)=(pn)⊕h ! W (κ)=(pn−1)⊕h, and can then identify DM(d H) =∼ W (κ)⊕h; DM(d H)=V DM(d H) =∼ κ⊕d: Specializing to the case where d = 1, the functor Spf H_ 7! DM(d H) is our equiva- lence. v f 1.1.2. Remark. The Serre dual of LModD(κ) is the category LModD(κ), defined v in the same way except with F in place of V . For M 2 LModD(κ), we have f HomW (κ)(M; W (κ)) 2 LModD(κ), where we give HomW (κ)(M; W (κ)) the D(κ)- module structure (V f)(a) = f(F a); (F f)(a) = f(V a) v f for f : M ! W (κ). This gives a duality between LModD(κ) and LModD(κ). / v;h v 1.2. The Honda module. Fix a positive integer h and let LModD(κ) ⊂ LModD(κ) be the full subcategory on those objects free of rank h over W (κ). There is a h;v distinguished object H 2 LModD(κ) given by H = W (κ)fx; V x; : : : ; V h−1xg;V hx = px; F x = V h−1x: This is the free D(κ)-module on a generator x subject to the condition V hx = px. v We obtain for any M 2 LModD(κ) an identification ∼ h HomD(κ)(H; M) = fa 2 M : V a = pag; f 7! f(x): 1.2.1. Proposition. Suppose that κ contains all (ph − 1)'th roots of unity, and let Fph ⊂ κ be the subfield generated by these. Then there is a canonical identification h σ−1 EndD(κ)(H) = W (Fph )hV i=(V = p; V a = a V g; where a ranges through W (Fph ). CLASSIFYING FORMAL GROUPS WITH DIEUDONNE´ THEORY 3 h Proof. As above, identify EndD(κ)(H) = fa 2 H : F a = pag. Write an arbitrary Ph−1 i element a 2 H as a = i=0 aiV . Then h−1 h−1 h X σ−h h+i X σ−h i F a = ai V x = ai pV x: i=0 i=0 h σ−h We find that F a = pa precisely when ai = ai for each i, which holds precisely when each ai 2 W (Fph ). 1.2.2. Remark. The object more commonly seen in the study of formal groups is h σ E = W (Fph )hF i=(F = p; F a = a F ). There is an isomorphism h−1 h−1 op X i X i EndD(κ)(H) ! E ; aiV 7! F ai: i=0 i=0 This corresponds to Remark 1.1.2. / Let Sh = AutD(κ)(H). As underlying H is a free module over W (κ) of rank h and with distinguished basis, we have canonical inclusions ⊕h EndD(κ)(H) ⊂ EndW (κ)(W (κ) ); Sh ⊂ GLh(W (κ)): We can give EndD(κ)(H), and thus Sh, the topology arising from these inclusions and the p-adic topology on W (κ). As h is finite, this is equivalent to the topology induced from the filtration on EndD(κ) by powers of V , and makes EndD(κ)(H) into a profinite ring and Sh into a profinite group. We will prove the following theorem in the next section. v;h 1.2.3. Theorem. Suppose κ is algebraically closed. Then every M 2 LModD(κ) is isomorphic to H. We can rephrase this as follows. For a category C, let π0C denote the set of isomorphism classes of objects of C. Then the previous theorem states v;h π0LModD(κ) = fHg; so long as κ is algebraically closed. We will later leverage this into an identification v;h of π0LModD(κ) for a general perfect field κ. 1.3. Semilinear algebra. Let R be a ring, let φ: R ! R an endomorphism, and let M and N be R-modules. For r 2 R, we may write φ(r) = φr.A φ-semilinear map f : M ! N is defined as a function satisfying f(m + n) = f(m) + f(n); f(rm) = φ)f(m): (φ) Given such f, define N = Rφ ⊗R N. Then M ! N (φ); m 7! 1 ⊗ f(m) is an R-linear map. Of particular interest is the case where R = κ is a perfect field, φ = σ±1, and M = N is a finite-dimensional vector space. It turns out that in this case M is the sum of an f-nilpotent subspace and a subspace generated by f-fixed elements; we will not need this exact fact, but mention it as it provides some motivation for the following results. We first deal with the case of certain nilpotent operators. 4 W. BALDERRAMA 1.3.1. Lemma. Let k be a field, and φ an endomorphism of k. Let U be a rank h vector space over k, and V : U ! U a φ-semilinear operator. Suppose that V is nilpotent, and U=V U =∼ k. Then there is x 2 U such that x; V x; : : : ; V h−1x give a basis for U. (φn) Proof. Set U0 = U, and inductively define Un to be the image of Un−1 ! U . Our assumptions give us a sequence U0 ! U1 !···! Uh−1 ! Uh = 0 of surjective k-linear maps, each with one-dimensional kernel. In particular, the φ-semilinear operator V satisfies V h−1 6= 0 and V h = 0. Let x 2 U be such that V h−1x 6= 0. It is sufficient to show x; V x; : : : ; V h−1x are linearly independent. Ph−1 i Indeed, a linear relation of the form i=l λiV x = 0 yields, upon application h−1−l φh−1−l h−1 of V , the identity λlV x = 0, so that λl = 0 as φ is necessarily injective. 1.3.2. Proposition. Let k be a field, φ an endomorphism of k, and M a free module over W (k) of finite rank h equipped with a φ-semilinear topologically nilpotent endomorphism V satisfying M=V M =∼ k.