It is known (Lazard [3]) that (ii) is redundant if R has no nilpotent elements. Remark 14. The group-valued functor of F can also be described using the formal group ring H of F. For simplicity we will assume that F is 1-dimensional; the general case is similar. In the smooth case, choosing coordinates is equivalent to taking a distinguished basis of the formal group ring. Description. This makes F into a functor from commutative R-algebras S to groups. So in some sense formal group laws are the "right" substitute for Lie algebras in characteristic p > 0. More generally, one can define the n-dimensional additive group: G a(X;Y) = (X 1 + Y 1;:::;X n+ Y n): 2)Let G In other words, there is a strict isomorphism f from the additive formal group to F, called the logarithm of F, so that. h << /S /GoTo /D (chapter.1) >> Authors. We will say that fhas height exactly nif it has height nand v n 2Ris invertible. For observe that for ∑ i, j a i, j x 1 i x 1 j \underset{i,j}{\sum} a_{i,j} x_1^i x_1^j an element in a formal power series algebra, then the condition that it defines a formal group law is equivalently a sequence of polynomial equations on the coefficients a k a_k. The idea of the definition is that F should be something like the formal power series expansion of the product of a Lie group, where we choose coordinates so that the identity of the Lie group is the origin. In other words we can always find a (unique) power series G such that F(x,G(x)) = 0. In characteristic zero, the closure of each point contains all points of greater height. The (non-strict) isomorphisms between formal group laws induced by change of parameters make up the elements of the group of coordinate changes on the formal group. 28 0 obj Over the rational numbers, there is an isomorphism from the additive formal group law to the multiplicative one, given by exp(x) − 1. PROBLEMS ABOUT FORMAL GROUPS AND COHOMOLOGY THEORIES ARIZONA WINTER SCHOOL 2019 VESNA STOJANOSKA 1. They were introduced by S. Bochner (1946). We will say that fhas height nif v i = 0 for i> endobj This gives an action of the ring Zp on the Lubin–Tate formal group law. This difference gives formal groups a rich geometric theory in positive and mixed characteristic, with connections to the Steenrod algebra, p-divisible groups, Dieudonné theory, and Galois representations. At first sight it seems to be incredibly complicated: the relations between its generators are very messy. For each element a in Zp there is a unique endomorphism f of the Lubin–Tate formal group law such that f(x) = ax + higher-degree terms. Remark 11. Conversely, given a Hopf algebra whose coalgebra structure is given above, we can recover a formal group law F from it. In particular we can identify the group-like elements of H⊗S with the nilpotent elements of S, and the group structure on the group-like elements of H⊗S is then identified with the group structure on F(S). Then f is either zero, or the first nonzero term in its power series expansion is stream However Lazard proved that it has a very simple structure: it is just a polynomial ring (over the integers) on generators of degrees 2, 4, 6, ... (where ci,j has degree 2(i + j − 1)). Source code. (4) Show that over a eld kof char p>0, Gb ais not isomorphic to Gb m. (5) Let F be a formal group over R. We call G a simply the additive group. Let f(x;y) 2R[[x;y]] be a formal group law over a commutative ring R. For every nonneg-ative integer n, we de ne the n-series [n](t) 2R[[t]] as follows: (1) If n= 0, we set [n](t) = 0. %PDF-1.4 Baptiste Calmès ; Viktor Petrov ; Version. A formal group law is called commutative when moreover 3) G i(X;Y) = G i(Y;X). We let. A formal group is a group object internal to infinitesimal spaces. The source code from which this documentation is derived is in the file FormalGroupLaws.m2. for nilpotent elements x. Let Abe a ring. 8 0 obj For any cocommutative Hopf algebra, an element g is called group-like if Δg = g ⊗ g and εg = 1, and the group-like elements form a group under multiplication. onal formal group law; we omit the rst two adjectives as this is the only type of formal group law we will consider. From now on we will omit \one-dimensional" and work only with one-dimensional formal group laws. Remark 3.2. A Lie group is an ndimensional manifold endowed with a group structure. 15 0 obj If in addition F(X;Y) = F(Y;X), then Fis called commutative. endobj So 1-dimensional formal group laws are essentially the same as Hopf algebras whose coalgebra structure is given above. endobj Given a smooth formal group, one can construct a formal group law and a field by choosing a uniformizing set of sections. Formal groups are intermediate between Lie groups (or algebraic groups) and Lie algebras. F(x,y) with coefficients in R, such that. Theorem The construction which sends a 1–dimensional, commutative formal group law F to the ring spectrum map F ∗ induces a natural bijection between the set of strict isomorphism classes of formal group laws over B and the set of homotopy classes of ring spectrum maps from HZ to DB. A formal group is sometimes defined as a cocommutative Hopf algebra (usually with some extra conditions added, such as being pointed or connected). 7 0 obj endobj endobj For simplicity we describe the 1-dimensional case; the higher-dimensional case is similar except that notation becomes messier. where we write F for (F1, ..., Fn), x for (x1,..., xn), and so on. For supersingular elliptic curves, this control is complete, and this is quite different from the characteristic zero situation where the formal group has no deformations. They sit between Lie algebras and finite Lie groups or algebraic groups. This package provides elementary functions to deal with commutative formal group laws of dimension one. There is no need for an axiom analogous to the existence of an inverse for groups, as this turns out to follow automatically from the definition of a formal group law. Formal Group Laws In this document, a formal group law over a commutative ring Ris always com- It is also a major ingredient in some approaches to local class field theory. a Some authors use the term formal group to mean formal group law. << /S /GoTo /D (chapter.5) >> , !/) = cp(y, x). endobj The formal group ring of a formal group law is a cocommutative Hopf algebra analogous to the group ring of a group and to the universal enveloping algebra of a Lie algebra, both of which are also cocommutative Hopf algebras. (The main theorems of Cartier theory) FORMAL GROUP LAWS Strictly speaking, such an object should be called a commutative one-dimensi- onal formal group law; we omit the rst two adjectives as this is the only type of formal group law we will consider. There is a universal commutative one-dimensional formal group law over a universal commutative ring defined as follows. It is known (Lazard) that (ii) is redundant if R has no nilpotent elements. A one dimensional formal group law over Ais a power series F(X;Y) 2A[[X;Y]] with the following properties F(X;Y) = X+ Y+ higher degree terms F(X;F(Y;Z)) = F(F(X;Y);Z) (associativity). In general cocommutative Hopf algebras behave very much like groups.