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,FV = p = VF, 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 ∈ 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 ∈ M, there is some n such that V na = 0. Suppose now Spf H∨ is a connected p-divisible group. Then c H ∈ Hopfκ, so we can look at DM(H) ∈ LModD(κ). As H is a p-divisible Hopf algebra, DM(H) is p-divisible, and we can noncanonically identify DM(H) =∼ W (κ)/(p∞)⊕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 ∈ LModD(κ), we have f HomW (κ)(M,W (κ)) ∈ 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 ∈ LModD(κ) given by H = W (κ){x, V x, . . . , V h−1x},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 ∈ LModD(κ) an identification ∼ h HomD(κ)(H,M) = {a ∈ M : V a = pa}, 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 }, where a ranges through W (Fph ). CLASSIFYING FORMAL GROUPS WITH DIEUDONNE´ THEORY 3

h Proof. As above, identify EndD(κ)(H) = {a ∈ H : F a = pa}. Write an arbitrary Ph−1 i element a ∈ 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 ∈ 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 ∈ 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) = {H}, 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 ∈ 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 ∈ 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 ∈ 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. Then there exists y ∈ M such that V h−1y 6= 0 in M/(p). Moreover, for any such y, we have pnM =∼ W (k){V nhy, V nh+1y, . . . , V nh+h−1y} h Ph−1 i × −1 h for each n. In particular, V y = i=0 paiV y with a0 ∈ W (k) , and p V is a φ-semilinear automorphism of M. Proof. By Lemma 1.3.1, we find y ∈ M/(p) such that y, V y, . . . , V h−1y form a basis for M/(p). Choose a lift y over y to M. Then y, V y, . . . , V h−1y form a basis for M, and V hy, . . . , V 2h−1y lie in pM. By repeating our arguments with each pnM, it is sufficient to verify that these form a basis for pM. Out of y, . . . , V h−1y, only h Ph−1 i y is not in the image of V ; thus when we expand V y = i=0 paiV y, the element h+i a0 ∈ W (k) must be unit. We learn that for 0 ≤ i ≤ h − 1, the expansion of V y 2 i h−1 in our basis mod p is of the form pbiV y + ··· + pbh−1V y with bi a unit. It is easily seen from this that V hy, . . . , V 2h−1y project to a basis for pM/p2M, and thus form a basis of pM.  1.3.3. Corollary. Let k and M be as above, and moreover let N be of the same type as M. Suppose given a W (k)-linear map f : M → N which commutes with r V . Then for each r ≥ 0, there are unique automorphisms µ: M → M (φ ) and r η : N → N (φ ) fitting into a diagram V hr

r p µ r M M M (φ ) r f f f (φ ) r p η r N N N (φ ) V hr

r of W (κ)-linear maps that commute with V . Here, f (φ ) indicates functoriality of (φr ) (−) applied to f.  CLASSIFYING FORMAL GROUPS WITH DIEUDONNE´ THEORY 5

1.3.4. Corollary. Let κ be a perfect field, and M a free module over W (κ) of finite rank h equipped with a σ−1-semilinear topologically endomorphism V satisfying ∼ v M/V M = κ. Then M can be uniquely upgraded to an object of LModD(κ). Proof. As VF = p = FV and M is torsion-free, we see that there is at most one choice of F . Choose y ∈ M such that M =∼ W (κ){y, V y, . . . , V h−1y}. We are forced to define FV iy = pV i−1y for 1 ≤ i ≤ h − 1. As M/V M =∼ κ, necessarily py = V a for some a ∈ M, and we are forced to define F y = a. This description on the basis h−1 y, V y, . . . , V y extends by semilinearity to a description on all of M.  1.3.5. Proposition. Let κ be an algebraically closed field, and U a finite-dimensional vector space over κ. Let T : U → U be a σ−1-semilinear automorphism. Then there exists some nonzero x ∈ U which is fixed by T . Proof. By replacing T with T −1, we may instead deal with a σ-linear automor- phism. Let us work for now with a general φ-semilinear automorphism T : U → U, where φ is some endomorphism of k. Pick y ∈ U. As U is finite-dimensional, there is some minimal n such that T n+1y n n+1 Pn i is in the span of y, . . . , T y; write T y = i=0 aiT y. As T is an automorphism, i not all ai are zero. After possibly replacing y with some T y, we may suppose a0 6= n 0, so that y, T y, . . . , T y are linearly independent. For indeterminates λ0, . . . , λn, write n X i x = λiT y. i=0 Then solving T x = x amounts to solving φ φ φ φ φ n (λ0 − λna0)y + (λ1 − λ0 − λna1)T y + ··· + (λn − λn−1 − λnan)T y = 0. As y, T y, . . . , T ny are linearly independent, this is equivalent to asking each coef- φ ficient in parentheses to vanish. Starting with λ0 = λna0, we can substitute each equation into the next, allowing us to reduce to solving a single equation. Writing φi λ = λn and bi = an−i, this comes out to solving φn+1 φn φ λ = λbn + λbn−1 + ··· + λb0.

In our case, φ = σ. Recall also b0, . . . , bn are not all zero. So we are reduced to finding a nonzero root of the nontrivial separable equation pn+1 pn p t = t bn + t bn−1 + ··· + t b0, which can be done under our assumption that k is algebraically closed.  We would like to lift the previous proposition from κ-vector spaces to suitable W (κ)-modules. 1.3.6. Lemma. Let M be a W (κ)-module which is locally p-power torsion, and let T : M → M be a σ−1-semilinear map. Let M T = {a ∈ M : T a = a}, which is a subgroup of M. Then the resulting map W (κ) ⊗ M T → M is injective. T T Proof. Let M0 ⊂ M be a finitely generated subgroup; it will suffice to show that T W (κ) ⊗ M0 → M is injective. Write

T M ti M0 = Z/(p ), 1≤i≤k 6 W. BALDERRAMA

T so that M0 ⊂ M is given by the choice of a1, . . . , ak ∈ M satisfying T ai = ai and ti p ai = 0. Suppose towards a contradiction we had

λ1a1 + ··· + λkak = 0

ti for some λi ∈ W (κ)/(p ) with not all λi zero. We can suppose this relation is chosen so that each λi is nonzero, and k is minimal so that such a relation holds. c Moreover, we may suppose λ1 = p for some integer c. Applying T to the above gives c σ−1 p a1 + ··· + λk ak = 0. Subtracting these equations yields σ−1 σ−1 (λ2 − λ2 )a2 + ··· + (λk − λk )ak = 0, σ−1 so that λi = λi for each i by minimality of k. But then λ1a1 + ··· + λkak ∈ T T T Zp ⊗ M = M , and certainly M ⊂ M is injective.  1.3.7. Lemma. Suppose κ is algebraically closed. Let M be an Artinian W (κ)- module, and T : M → M a σ−1-semilinear injection. Then M is generated by M T = {x ∈ M : T x = x} over W (κ).

−1 Proof. By considering the induced W (κ)-linear injection M → M (σ ) between Artinian W (κ)-modules of equal length, we find that T is an isomorphism. Let M 0 ⊂ M be the W (κ)-submodule generated by M T , and let M 00 = M/M 0. Consider the diagram

0 M 0 M M 00 0

T 0−1 T −1 T 00−1 . 0 M 0 M M 00 0 As κ is algebraically closed, σ−1 − id: κ → κ is surjective. By induction on composition series, we learn that σ−1 − 1: W (κ) ⊗ A → W (κ) ⊗ A is a surjec- tion for any finite abelian group A. In case A = M T , under the isomorphism W (κ)⊗M T =∼ M 0 given by Lemma 1.3.6 we find that T −1: M 0 → M 0 is surjective. As ker(T 0 − 1) =∼ ker(T − 1) by construction, the long exact sequence associated to the above diagram tells us that T 00 − 1 is an injective endomorphism of M 00. So T 00 is a σ−1-semilinear automorphism of M 00 with no fixed points. The same then must be true of T 00 acting on the p-torsion submodule of M 00, contradicting Proposition 1.3.5.  Putting the previous two lemmata together, we learn that under the assumptions and notation of the previous lemma, the map W (κ)⊗M T → M is an isomorphism. 1.3.8. Lemma. Let j1 j2 j3 I1 I2 I3 ··· be a tower of p-complete Hausdorff abelian groups, and suppose jn(In+1) ⊂ pIn for each n. Then 1 lim In = 0 = lim In. n n CLASSIFYING FORMAL GROUPS WITH DIEUDONNE´ THEORY 7

1 Proof. The assertion that limn In = 0 is immediate, so let us verify limn In = 0. 1 Recall that limn In can be identified as the cokernel of Y Y In → In, (an)n 7→ (an − jn(an+1))n, n n Q so we must show this map is surjective. Indeed, fix an element (bn)n ∈ n In. Then (bn)n = (an − jn(an+1))n has the solution X an = (jn ◦ · · · ◦ jk−1)(bk), k≥n which converges under our assumptions. 

1.3.9. Lemma. Suppose given W (κ)-modules N1,N2,... fitting into exact sequences pn Nn+1 Nn+1 Nn 0 , and suppose N1 is a finite dimensional vector space over κ. Let N = limn Nn. Then n N is a finitely generated W (κ)-module, and N/(p ) = Nn. Proof. The assumptions give for all n an exact sequence of towers whose m’th level is pn Nn+m Nn+m Nn 0 . The leftmost towers consist of surjections, so by the Mittag-Leffler condition for vanishing lim1 we obtain in the limit an exact sequence

pn N N Nn 0

n n telling us that N/(p ) = Nn. We learn that N = limn N/(p ) is a p-complete Hausdorff W (κ)-module such that N/(p) is a finite-dimensional κ-vector space. It follows by the Nakayama lemma for complete Hausdorff modules that N is finitely generated over W (κ).  1.3.10. Proposition. Let κ be an algebraically closed field. Let M be an W (κ)- module free of finite rank h and T : M → M a σ−1-semilinear automorphism. Let T T ∼ M = {a ∈ M : T a = a}. Then W (κ) ⊗b M = M, where ⊗b is taken with respect to the p-adic topology. T n Proof. For each n ≥ 1, let Mn = {a ∈ M : T a ≡ a (mod p )}. We obtain a tower T T T T n T M1 ← M2 ← · · · of inclusions, and M = limn Mn . Let In = p M ∩ Mn , and T n T observe Mn /In = M/(p ) . Combining this with Lemma 1.3.6 and Lemma 1.3.7 we learn T ∼ n W (κ) ⊗ Mn /In = M/(p ). In particular, this gives us exact sequences

n T p T T W (κ) ⊗ Mn+1/In+1 W (κ) ⊗ Mn+1/In+1 W (κ) ⊗ Mn /In 0, and as Zp ⊂ W (κ) is faithfully flat this passes to exact sequences

n T p T T Mn+1/In+1 Mn+1/In+1 Mn /In 0. 8 W. BALDERRAMA

1 T T By Lemma 1.3.8, we know limn In = limn In = 0, so that M = limn Mn = T T n T limn Mn /In. By Lemma 1.3.9, we learn M /(p ) = Mn /In. So we can identify T W (κ) ⊗b M → M as W (κ) ⊗ M T = lim W (κ)/(pm) ⊗ M T /(pn) b m,n m T = lim W (κ)/(p ) ⊗ Mn /In m,n T = lim W (κ) ⊗ Mn /In n = lim M/(pn) = M. n  Finally, Theorem 1.2.3 is an immediate consequence of the following. 1.3.11. Proposition. Let κ be an algebraically closed field, and M a free module over W (κ) of finite rank h equipped with a σ−1-linear topologically nilpotent en- domorphism V : M → M such that M/V M =∼ κ. Then there exists x ∈ M such that M =∼ W (κ){x, V x, . . . , V h−1x} and V hx = px. Proof. By Proposition 1.3.2, we know that p−1V h : M → M is a σ−1-semilinear automorphism, and moreover there is y ∈ M such that y, V y, . . . , V h−1y forms a basis for M. On the other hand, Proposition 1.3.10 tells us that M admits a −1 h Ph−1 basis x0, . . . , xh−1 with p V xi = xi for each i. If we write y = i=0 aixi and choose n such that an is nonzero in M/(p), then Proposition 1.3.2 tells us that h−1 xn, V xn,...,V xn is a basis for M. So x = xn has the stated properties.  2. Classification of formal groups 2.1. Galois cohomology. For our purposes, a topological group will be a group G equipped with topology arising from an inverse limit construction G = limi G/Ui, where Ui ⊂ G is a normal subgroup not necessarily of finite index. Let Γ and S be topological groups, and suppose we have an action of Γ on S. We can define the semidirect product group S o Γ with multiplication (s, γ) · (t, δ) = (s · γ t, γ · δ). This is equipped with a projection π : S o Γ → Γ, and we define the set of S-valued cocycles on Γ by 1 Z (Γ,S) = {α:Γ → S o Γ: π ◦ α = idΓ}. Equivalently, Z1(Γ,S) = {α:Γ → S : α(γ · δ) = α(γ) · γ α(δ)} ⊂ Hom(Γ,S). This set is equipped with an S-action by conjugation: for s ∈ S and α ∈ Z1(Γ,S), we define αs(γ) = s−1 · α(γ) · γ s. 1 1 Let H (Γ,S) = Z (Γ,S)S be the set of orbits of this action. If S is abelian, then H1(Γ,S) agrees with the usual continuous group cohomology of Γ with coefficients in S, as can be seen directly from a suitable bar resolution. In general, H1(Γ,S) is a set pointed at the zero map Γ → S. Observe that H1(Γ,S) = 0 precisely when for every α ∈ Z1(Γ,S) there exists some s ∈ S such that α(γ) = s−1 · γ s for all γ ∈ Γ. CLASSIFYING FORMAL GROUPS WITH DIEUDONNE´ THEORY 9

2.1.1. Lemma. Suppose S = limn≥1 S/Un is a complete Hausdorff topological 1 group, and set U0 = S. Suppose H (Γ,Un/Un+1) = 0 for each n ≥ 0. Then H1(Γ,S) = 0.

1 −1 γ Proof. Let α ∈ Z (Γ,S). We find that α(γ) ≡ s1 · s1 (mod U1) for some fixed s−1 s1 ∈ S. Replacing α with α 1 , we may suppose α lands in U1 ⊂ S. Repeating this −1 (sn···s1) argument, we find elements s1, s2,... ∈ S such that sn ∈ Un−1 and f lies in Un for each n. As S is complete Hausdorff with respect to the filtration by the s−1 T subgroups Ui, we can take s∞ = limn→∞ sn ··· s1 and find f ∞ lies in Un = 0. 1 So f is zero in H (Γ,S).  Fix now a field k of positive characteristic p, and let k ⊂ K be a Galois extension with Galois group Γ. Write W (K) for the ring of p-typical Witt vectors on K, and consider this as equipped with the p-adic topology. We obtain a continuous action of Γ on the Witt components of W (K). The quotients W (K) → W (K)/(pn) pass to n surjections GLn(W (K)) → GLn(W (K)/(p )), and these make GLn(W (K)) into a topological group. The action of Γ on the coordinates of GLn(W (K)) is continuous with respect to this topology. The following can be seen as an analogue of Hilbert’s Satz 90. 1 2.1.2. Theorem. H (Γ, GLn(W (K))) = 0. n Proof. Let Un be the kernel of the quotient map GLn(W (K)) → GLn(W (K)/(p )). One can show that × n ⊕n U0/U1 = (K ) ,Un+1/Un = K . So by Lemma 2.1.1, we are reduced to verifying H1(Γ,K×) = 0 = H1(Γ,K). The first is exactly Hilbert’s Satz 90. For the second, let α ∈ Z1(Γ,K), realized as a map α:Γ → K. As K is discrete and Γ is profinite, this factors through α0 : G → L, where k ⊂ L ⊂ K is some finite Galois extension and G = Autk(L). It is then sufficient to show α0 = 0 in H1(G, L). This follows from the normal basis theorem, telling us that L =∼ k[G] as k[G]-modules, and the identification of H1 with ordinary group cohomology in this case.  2.2. The classification theorem. Fix a perfect field κ of positive characteristic, and fix an algebraic closure κ ⊂ κ. Let Γ = Autk(κ) be the absolute Galois group v v of κ. For M ∈ LModD(κ), let Mκ = W (κ) ⊗b W (κ) M ∈ LModD(κ). The continuous action of Γ on W (κ) gives rise to a continuous action of Γ on Mκ by left D(κ)- Γ module maps, and we can identify Mκ = M. Fix a positive integer h, and write H = W (κ){x, V x, . . . , V h−1x} for the object of v,h LModD(κ) considered in Section 1.2. We have an action of Γ on HomD(κ)(Hκ,Mκ) defined as follows. Recall that we can identify a map f : Hκ → Mκ as the element h a = f(x) ∈ Mκ subject to V a = pa. For γ ∈ Γ we have γ(a) ∈ Mκ, and we claim γ this is associated to some other f : Hκ → Mκ. Indeed, as Γ acts by D(κ)-linear maps, we have V hγ(a) = γ(V ha) = γ(pa) = pγ(a).

This action restricts to an action of Γ on IsoD(κ)(Hκ,Mκ). In case M = H, this gives an action of Γ on Sh. Our goal for the rest of this section is to demonstrate the following: v,h ∼ 1 π0LModD(κ) = H (Γ, Sh). 10 W. BALDERRAMA

v,h Fix M ∈ LModD(κ). Choose an isomorphism f : Hκ → Mκ; the existence of such is guaranteed by Theorem 1.2.3. Define then −1 γ αM,f :Γ → Sh, αM,f (γ) = f · f. 1 2.2.1. Claim. We have αM,f ∈ Z (Γ, Sh). Proof. We must verify the cocycle condition and continuity. For the cocycle condi- tion, we directly calculate for γ, δ ∈ Γ that −1 γδ αM,f (γδ) = f · f = f −1 · γ f · γ f −1 · γδf −1 γ γ −1 δ
γ = f · f · f · f = αM,f (γ) · αM,f (δ). P For continuity, write f(x) = i λi ⊗ ai with λi ∈ W (κ) and ai ∈ M and let κ ⊂ K be the field extension obtained by adjoining the first n Witt components of γ δ n+1 the coefficients λi. If γ|K = δ|K , then f ≡ f (mod p ), and thus αM,f (γ) ≡ n+1 αM,f (δ) (mod p ). So we learn that αM,f is continuous.  v,h We have now assigned to every M ∈ LModD(κ) and choice of isomorphism 1 f : Hκ → Mκ an element αM,f ∈ Z (Γ, Sh). 1 2.2.2. Claim. Upon passing to H (Γ, Sh), the above assignment factors through v,h π0LModD(κ).   Proof. Suppose given M ∈ LModD(κ) with chosen isomorphisms f : Hκ → Mκ for  ∈ {1, 2}. Let g : M 1 → M 2 be an isomorphism of left D(κ)-modules, and define h ∈ Sh by the diagram

1 f 1 Hκ Mκ

h g . 2 f 2 Hκ Mκ As g is defined over W (κ), we can calculate for γ ∈ Γ that −1 γ −1 γ f2 · f2 = (g · f1 · h) · (g · f1 · h) −1 −1 −1 γ γ −1 −1 γ γ = h · f1 · g · (g · f1) · h = h · f1 · f1 · h. h 1 Thus α 1 = α 2 , so that α 1 and α 2 agree in H (Γ, ). M ,f1 M ,f2 M ,f1 M ,f2 Sh  We have now constructed v,h 1 α: π0LModD(κ) → H (Γ, Sh). 2.2.3. Theorem. The map α is an isomorphism.

 v,h Proof. Let us first check that α is injective. Suppose given M ∈ LModD(κ) and  choose isomorphisms f : Hκ → Mκ for  ∈ {1, 2}. Suppose that for some h ∈ Sh h h we have α 1 = α 2 . Then for all γ ∈ Γ, the diagram M ,f1 M ,f2 CLASSIFYING FORMAL GROUPS WITH DIEUDONNE´ THEORY 11

h Hκ Hκ

γ γ f2 f1

2 1 Mκ Mκ

f2 f1 h Hκ Hκ commutes, telling us −1 γ −1
1 ∼ 2 f2 · h · f1 = f2 · h · f1 : Mκ = Mκ for all γ ∈ Γ. So we may take fixed points with respect to Γ to obtain an isomor- phism M 1 =∼ M 2. 1 Let us now check α is surjective. Fix η ∈ Z (Γ, Sh), which we can push for- 1 −1 γ ward to Z (Γ, GLn(W (κ))). By Theorem 2.1.2, we know that η(γ) = T · T in GLn(W (κ)) for all γ ∈ Γ and some matrix T ∈ GLn(W (κ)). Let N = W (κ) ⊗W (κ) W (κ){y0, . . . , yh−1}, and using the distinguished bases treat T as an isomorphism ∼ T : Hκ = N of W (κ)-modules. We can give N the structure of a left Dκ-module such that this is an isomorphism of left D(κ)-modules; explicitly, if V = VH : Hκ → Hκ, −1 then VN = T ◦ VH ◦ T . Observe now that for all γ ∈ Γ we have γ γ −1
VN = T · VH · T γ γ γ −1 = T · VH · T −1 −1 = T · η(γ) · VH · η(γ) · T −1 = T · VH · T = VN , the last equality as η(γ) ∈ Sh commutes with FH . As VN is Γ-equivariant, we can take fixed points with respect to Γ to obtain the left D(κ)-module M = N Γ. Evidently αM,T = η, so this concludes the proof.  2.3. Case of a finite field. We maintain notation from the previous section. Sup- −r pose now κ = Fpr is a finite field. Then Γ = Zb is cyclically generated by σ . We learn that evaluation on σ−r gives an isomorphism 1 ∼ Z (Γ, Sh) = Sh. 1 Thus H (Γ, Sh) is a quotient of Sh, where two elements g1, g2 ∈ Sh are equivalent precisely when there is some h ∈ Sh such that −1 σ−r g2 = h · g1 · h.

If κ = Fphr , so that Fph ⊂ κ, then Γ acts trivially on Sh and this description yields an isomorphism v,h ∼ α: π0LModD(κ) = Sh/(conj). v,h −hr Let M ∈ LModD(κ) with κ = Fphr . Observe that over Fphr , the σ -semilinear hr map V is in fact linear. It thus induces a D(κ)-linear map Mκ → Mκ, but this hr hr is not the map V associated Mκ; when treating V in this way, we will write hr V = BM . Choose an isomorphism f : Hκ → Mκ. We can give now another description of the conjugacy class α(M) associated to M. By Corollary 1.3.3, we have a commutative diagram 12 W. BALDERRAMA

BH

r p = Hκ Hκ Hκ

−hr f f σ f . pr µ Mκ Mκ Mκ

BM −1 σ−hr −1 −1 This tells us that f · f = f ·µ·f. So f ·µ·f ∈ Sh is a representative of the conjugacy class associated to M. Moreover, µ is defined in W (Fphr ); in particular, V hrs = µsprs, so that the conjugacy class associated to M is f −1 · µs · f. Fphrs As the conjugacy class of the identity of Sh is a singleton, we learn that M ∈ LModv,h is isomorphic to H if and only if µ = 1 in the above, i.e. V hr = pr on D(Fphr ) M.

2.3.1. Example. Consider the case h = 1. In this case, the action of Γ on S1 × v,1 is trivial and 1 = is abelian, so an object M ∈ LMod is classified by S Zp D(Fpr ) × an element of Zp . Such an object M can be written as M = W (Fpr ){y} with × V y = apy for some a ∈ W (Fpr ) . The above description tells us that the invariant Qr−1 σi of M is the norm N(a) = i=0 a ./ 3. Determinants v n 3.1. Exterior powers. For M ∈ LModD(κ), consider Λ M, the exterior power taken over W (κ). 3.1.1. Theorem. The W (κ)-module ΛnM admits a natural left D(κ)-module struc- ture determined by

V (a1 ∧ · · · ∧ an) = V a1 ∧ · · · ∧ V an, −(n−1) F (a1 ∧ · · · ∧ an) = p F a1 ∧ · · · ∧ F an. Proof. This is evidently natural; we must only verify that F can be defined this way. Choose y ∈ M so that M =∼ W (κ){y, V y, . . . , V h−1y}. Then a basis for ΛnM I i1 in is given by the elements V y = V y ∧ · · · ∧ V y with I = {i1 < ··· < in} ⊂ {0, . . . , h − 1}. As FV = p in M, we find that with the above definition F (V I y) n n lives in Λ M ⊂ Q ⊗ Λ M.  3.1.2. Example. Consider the case M = H. Then ΛnH has a basis consisting of the elements V I x as above, and we can identify

 i1+1 in+1 i1 in V x ∧ · · · ∧ V x, in 6= h − 1, V (V x ∧ · · · ∧ V x) = n−1 i1+1 in−1+1 (−1) px ∧ V x ∧ · · · ∧ V x, in = h − 1;

 i1−1 in−1 i1 in pV y ∧ · · · ∧ V y, i1 6= 0, F (V y ∧ · · · ∧ V y) = n−1 i2−1 in−1 h−1 (−1) V x ∧ · · · ∧ V x ∧ V x, i1 = 0. / v, h v,h n (n) Fix M ∈ LModD(κ). For general n, we cannot say that Λ M lives in LModD(κ) ; rather, it turns out that ΛnM is associated to a connected p-divisible group of CLASSIFYING FORMAL GROUPS WITH DIEUDONNE´ THEORY 13

h
h−1
height n and dimension n−1 . In the case h = n, one can check directly that h v,1 Λ M ∈ LModD(κ). We obtain a functor h v,h v,1 Λ :LModD(κ) → LModD(κ), and by Theorem 2.2.3 this passes to a map 1 1 λ: H (Γ, Sh) → H (Γ, S1), where Γ is the absolute Galois group of κ. By Proposition 1.2.1, we can identify × 1 S1 = Zp , and this has trivial action by Γ. We can thus identify H (Γ, S1) = × × Hom(Γ, Zp ), so that λ gives HomΓ(Γ, S o Γ) → Hom(Γ, Zp ). h Say now κ = r is a finite field, and fix M ∈ LMod . We have now Fp D(Fpr ) 1 × × H (Γ, S1) = Zp . Let us describe the resulting element λ(M) ∈ Zp . As in Ex- ample 2.3.1, we can write V r = ΛhV r :ΛhM → ΛhM as V r = pra for some × Qr−1 σi a ∈ W (Fpr ) , and then λ(M) = N(a) = i=0 a . Choose now y ∈ M such ∼ h−1 h Ph−1 i that M = W (Fpr ){y, V y, . . . , V y}, and write V y = i=0 paiV y with a0 ∈ × W (Fpr ) . 3.1.3. Proposition. With notation as above, we have r−1 r(h−1) r(h−1) Y σi a = λ(M) = (−1) N(a0) = (−1) a0 . i=0 h ∼ h−1 Proof. Continuing the previous discussion, we have Λ M = W (Fpr ){y∧· · ·∧V y} h Ph−1 i and V y = i=0 aiV y. Observe that V y ∧ V y ∧ · · · ∧ V h−1y
= V y ∧ V 2 ∧ · · · ∧ V h−1y ∧ V hy h−1 X h−1 i = V y ∧ · · · ∧ V y ∧ paiV y i=0 h−1 = V y ∧ · · · ∧ V y ∧ a0y h−1 h−1 = (−1) pa0y ∧ V y ∧ · · · ∧ V y. Continuing in this manner, we learn r h−1 r r(h−1) σ−(r−1) σ−1 h−1 V (y ∧ V y ∧ · · · ∧ V y) = p (−1) a0 ··· a0 a0y ∧ V y ∧ · · · ∧ V y r r(h−1) h−1 = p (−1) N(a0)y ∧ V y ∧ · · · ∧ V y, r(h−1) so that a = (−1) N(a0) as stated.  3.2. Classical determinant. Fix a perfect field κ containing all (ph − 1)’th roots of unity, and write Fph ⊂ κ for the subfield generated by these elements. Write still h−1 h H = W (κ){x, V x, . . . , V x} with V x = px. Then EndD(κ)(H) was identified in Proposition 1.2.1, and is independent of the choice of such κ. Moreover, we ⊕h have an injection EndD(κ)(H) → EndW (κ)(W (κ) ) which lands in the image of ⊕h ⊕h the induction map End (W ( h ) ) ⊂ End (W (κ) ). Explicitly, for W (Fph ) Fp W (κ) Ph−1 i a = i=0 aiV in EndD(κ)(H), we have ai ∈ W (Fph ) for each ai and the matrix 14 W. BALDERRAMA

⊕h associated to the image of a in End (W ( h ) ) is W (Fph ) Fp  σ−1 σ−2 σ−(h−1)  a0 pah−1 pah−2 ··· pa1 −1 −2 −(h−1)  a aσ paσ ··· paσ   1 0 h−1 2   σ−1 σ−2 σ−(h−1)   a.2 a1. a0. ··· pa3 .   . . . .   . . . ··· .   σ−1 σ−2 σ−(h−1)   ah−2 ah−3 ah−4 ··· pah−1  σ−1 σ−2 σ−(h−1) ah−1 ah−2 ah−3 ··· a0 Define now ⊕h det d: End (H) → End (W ( h ) ) −−→ W ( h ). D(κ) W (Fph ) Fp Fp

This is a homomorphism, with W (Fph ) given its multiplicative monoid structure.

3.2.1. Proposition. The image of d lies in W (Fp) ⊂ W (Fph ). −1 Proof. One can check that p EndD(κ)(H) is a well defined algebra, and we have a commutative diagram

−1 EndD(κ)(H) p EndD(κ)(H)

d

⊕h ⊕h End (W ( h ) ) End ( ⊗ W ( h ) ) W (Fph ) Fp Q⊗W (Fph ) Q Fp of injections. Fix a ∈ EndD(κ)(H). Observe that inverting p also inverts V , and we −1 −1 −1 can identify V aV −1 = aσ . Thus d(a) = d(V aV −1) = d(aσ ) = d(a)σ , so that d(a) ∈ W (Fp).  ⊕h 3.2.2. Remark. Let T ∈ EndW (κ)(W (κ) ) be associated to some a ∈ EndD(κ)(H). Then the coefficients of the characteristic polynomial q(t) of T all lie in W (Fp). Indeed, we can write h X q(t) = (−1)i Tr(ΛiT )th−i, i=0 and as trace is invariant under cyclic permutations we can argue as above. v,h More generally, fix M ∈ LModD(κ) and a D(κ)-linear endomorphism g : M → M. Let q(t) be the characteristic polynomial of g considered as a W (κ)-linear endomorphism. Then the coefficients of q(t) lie in W (Fp) ⊂ W (κ). Indeed, g ∼ induces a D(κ)-linear endomorphism Mκ → Mκ, and we know Mκ = Hκ; so we can appeal to the fact that the characteristic polynomial of a linear map is independent of choice of basis. / × × We obtain the determinant mapping d: Sh → W (Fp) = Zp . v,h 3.2.3. Theorem. Fix M ∈ LMod , and let α(M) ∈ h represent the invariant D(Fpr ) S 1 × in H (Γ, Sh) associated to M. Let λ(M) ∈ S1 = Zp be the invariant associated to ΛhM. Then λ(M) = (−1)r(h−1)d(α(M)). CLASSIFYING FORMAL GROUPS WITH DIEUDONNE´ THEORY 15

Proof. Write κ = Fp and choose an isomorphism f : Hκ → Mκ so that we can say −r α(M) is represented by f −1 · σ f. By the definition of ordinary determinants, we −1 σ−r h −1 h σ−r h ∼ can identify d(f · f) = Λ (f )·Λ ( f) as automorphisms of Λ Hκ = W (κ). By Corollary 1.3.3 and Proposition 3.1.3, the diagram

r r(h−1) h p h (−1) h Λ Hκ Λ Hκ Λ Hκ

−r −r f Λh(f) Λh(σ f) = σ Λh(f) r h p h λ(M) h Λ Mκ Λ Mκ Λ Mκ commutes. This yields the result.  3.2.4. Corollary. Fix notation as above. Choose y ∈ M giving an isomorphism ∼ h−1 h Ph−1 M = W (Fpr ){y, V y, . . . , V y}, and write V y = i=0 pa0y. Then d(α(M)) = Qr−1 σi N(a0) = i=0 a0 . Proof. Combine Theorem 3.2.3 and Proposition 3.1.3.  3.2.5. Remark. The proof of Theorem 3.2.3 did not rely on Proposition 3.2.1, and × × so gives a second proof of the fact that d: Sh → W (Fph ) lands in Zp . / h−1 3.2.6. Example. Suppose h is even, and write H = W (Fp){x, V x, . . . , V x} with h h V x = px. Let M = Λ H. Then we can identify M = W (Fp){y} with V y = −py. 1 1 Let H = W (Fp){x} with V x = px. Then an isomorphism f : Hκ → Mκ is given by a ∈ W (κ)× satisfying aσ + a = 0.

We have a Galois extension W (Fp) ⊂ W (Fp2 ), and the resulting trace map is σ W (Fp2 ) → W (Fp), a 7→ a + a. The kernel is a free one-dimensional W (Fp)-module, and any element therein not 1 ∼ 1 divisible by p thus gives an isomorphism H → M 2 . We learn that M r = H r Fp2 Fp Fp Fp if and only if r is even. /

Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, IL 61801 Email address: [email protected]