THE AMERICAN MATHEMATICAL SOCIETY.
A theory of formal schemes and groups over abitrary rings is presented. The flat formal schemes in this theory have coalgebras of distributions which behave in the usual way. Frobenius and Verschiebung maps are studied. Introduction. In this paper we provide the details of the results announced in [8]. Throughout k is a commutative ring, all algebras are commutative unless otherwise stated, &-Alg denotes the category of commutative ^-algebras, ® means ® f e , etc. Func = Func(&-Alg, Sets) is the category of set valued functors on &-Alg. The theory of formal groups over fields was classically described for group laws arising from power series algebras [6]. Later, complete local noetherian algebras over fields were treated (cf. [7], which has a comprehensive historical survey). By assuming strong topological properties—namely pseudocompactness— of the base ring, substantial generalization of the field based theory was obtained [4], but complete generality on the base ring still required only power series algebras to be treated [2]. None of these approaches permits consideration, for example, of the functor G on commutative rings given by G(R) = \(a, b) in R x R\a is nilpotent, b 2 = 0 j. This functor is formally represented by the truncated power series ring Z[[x, y]]/(y2). Even though Z has no pseudocompact topology, this ring has an extremely pleasant topology, namely the inverse limit topology arising from the system iZ[[x, y ] ] / ( x n 9 y 2 ) \ . We assume always our base ring is discrete and treat such functors with two principal tools: a topology on the affine algebra ö(G) of a functor G allows us to form its continuous linear dual, the coalgebra B(G) of distributions on G\ a topology on C*, the füll linear dual of an arbitrary coalgebra C, allows recovery of the functor when given the distributions. Our theory coincides with [4] if k is a field and so in particular coincides with the theory of cocommutative coalgebras and Hopf algebras in case k is a field. Received by the editors October 6, 1972. A M S (MOS) subject classifications (1970). Primary 14L05. (1) This research was partially supported by National Science Fonndation Grants GU 3171 and PO34003 while the second author was a visitor at SUNY at Albany. Copyright © 1974, American Mathematical Society 113 114 R. A . MORRIS AND BODO PAREIGIS Our categories of formal schemes (Funcg^) and flat formal schemes (Func^) and various related categories of algebras and coalgebras fit the following: Func z=2 U A l g c ] o p Ul * * Ul &-Coalg Func g / ^ U A l g g / ] Ul Ul Ul F u n c ^ U A l g ^ = Z ^ 1 ^ C o a l g p / B( ) A similar diagram holds for the groups in these categories, I. The affine algebra of a functor. Let G be a set valued functor on &-Alg. Let 0: Jfe-Alg Sets denote the underlying set functor and define 0 ( G ) = Func(G, 0) the set of natural transformations from G to 0.( 2) 0 ( G ) is an algebra via pointwise Operations, functorially in G. For each pair (x, K) with K a k algebra and x in G(K), denote by y x K the algebra homomorphism 0 ( G ) -» K given by X* = /(*)(*)• Topologize 0(G) with the coarsest topology making each of these maps rontinuous (each K having the discrete topology). This makes 0(G) a topological algebra. An ideal / in 0(G) is open if and only if / contains the intersection of a finite collection of kernels of \ x K's. If &-Alg c denotes the category of topological ^-algebras with continuous algebra maps, then 0: Func -* k A \ g c is a contravariant functor. By the Yoneda lemma, it follows that if G is repr^sentable by an algebra A then 0(G) ^ A, as ^-algebras. The Situation is similar for topological algebras in a sense we make precise next. Let fl be a topological algebra. We write ö : &-Alg -* Sets for the functor ö (K) = &-Alg c(S, K) with K regarded as discrete. Thus if &-Alg is considered as a füll subcategory of &«Alg c then these ö are just the restrictions of representable functors on &-Alg c. An example of such a functor which is not representable on k-Alg is the functor which assigns to each algebra K the set of nilpotent elements of K. This is &[[/]]', ''topologically represented" by the formal power series ring. 1.1 Proposition [8, § l ] , Func(G, fi") S *-Alg c(fi, 0(G)), t h a t i s ( )" a n d 0 a r e a d Joint o n t h e r i g h u ( 2) To avoid set theoretic difficulties one should cast this definition in the language of "universes" as in [3, Conventions generales']. At the expense of conventional terminology, this language can be employed throughout our work withour significant change in the Statements or proofs of the results. FORMAL GROUPS AND HOPF A L G E B R A S OVER DISCRETE RINGS 115 Proof. Let cf> be in Func(G, S ). For each a in 3 define A<f>(a) in 0(C) by A<f>(a){K)(x) = $ { K ) ( x ) ( a ) for any algebra K and any x in G(K). One easily verifies that A0(ö) is natural. To show that A<£: fl -* 0(G) is continuous it suffices to show (A0)~ lKer(y^ K ) is open for each pair U , K). But (A0)~ l K e r { X x t K ) = in ö|(A0)U)(K)(x) = 0} = \ä in Q\d>{K)(x){a) = Ol = Ker(<£(K)(x)). Since <fi(K)(x) is a continuous algebra map from (5 to the discrete algebra K, its kernel is open. Now suppose /:(?-> 0(G) is a continuous algebra map. Define Tf: G -* (2 by r/(K)(x)(a) = /(a)(K)(x). Again it is straightforward co show T/ is natural. To show Vf(K)(x): Q -* K is continuous we need only show its kernel is open (K is discrete). This kernel is \a in Ö|/U)(K)(x) = 0! = \ a in Ö | x x > K ( / W ) = 0! = /" 1(Ker x x K ) which is open since / is continuous. That A and V are inverse to one another is immediate from their definitions. The reader may easily verify that all maps which want to be algebra maps are indeed so, and that the arguments above are natural in ö and G, completing the proof. The proposition gives rise to two natural adjunction maps: G -> 0((J)* and Ö O t ö O . W e set forth next some conditions motivated by the theory of algebraic groups and formal groups [ l , p. 101], under which these adjunction maps are isomorphisms. We consider functors G in Func equipped with a set of pairs j(D, p D ) \ D € &-Alg; p D € G(D)| satisfying the axioms below. Associated with this collection will be a diagram 3) consisting of the D*s and 3)(D, D') = ja € &-Alg(D,D')| G ( a ) ( p D ) = p D X (1.2a) Each \ D = X P d o'' -» D is surjective. (1.2b) Given any A in &-Alg and x in G(A), there is a D and an algebra map y: D -> A such that G(y)(pD) = x [ l , p. 101]. (1.3) 3) is filtered. If (1.2) holds call the p D g e n e r i c p o i n t s of G. If Q is a topological &-algebra, when we refer to one of these axioms as holding for (2 we mean that it should hold for ($ . Thus for any continuous map x: Q -* A there is a D and a unique algebra map y : D -» A making A commute (cf. [ l , p. 101]). 116 R. A. MORRIS AND BODO PAREIGIS 1.4 Remark. If G has generic points then there is at most one a in D(0, D') and it is a surjection. Note that a representable functor (or a discrete algebra) satisfies axioms (1.2)—(1.3) for the trivial diagram containing the representing algebra and the identity. The power series ring k[[t]] satisfies these axioms with respect to the diagram arising naturally from the projections k[[t\] -» &[[/]]/(/"). 1.5 Lemma. L e t (2 = proj lim 3), 3) a diagram in &-Alg, with \ p D : Q -* D\D e3)l t h e n a t u r a l projections, If 3) satisfies (1.2) then t h e adjunction m a p Q * 0 { Q ) is a n isomorphism i n k-Algc. Proof. By definition of the projective limit topology, the maps p D lie in (2 (D) = &-Alg c (ö , D). Hence by definition of the topology on 0((2 ), the maps . Xry ~ X p D D : ® ^ ) -* ̂ are continuous. It follows directly from the definition of X P D T D t h a t t h e d i a g r a m s
Some existence theorems in the calculus of variations. III. Existence theorems for nonregular problems
1. Outline of the method of proof. This note is the third of a series, the first and second of which have already appeared in these Transactions.t In the present paper we shall establish an existence theorem for singleintegral problems of the calculus of variations without requiring quasi-regularity. In order to help the reader to find his way through the many details of this proof, we first present a heuristic outline of the method used. Given an integral 7(C) =fF(z, z)dt, where z = (z1, , Zq), we seek to find a rectifiable curve (in q-dimensional space) joining two fixed points z1, Z2 for which this integral is least. It is always possible to find a minimizing sequence Cn } ; that is, a sequence for which 7(Cn) tends to the greatest lower bound ,u of 7(C) on the class of curves under consideration. Moreover, it is easy to find hypotheses on 7(C) which will ensure that a minimizing sequence exist which has a curve of accumulation Co. That is, the curves Cn have representations z = zs(t), (0 < t < 1), for which zs(t) tends uniformly to a limit function zo(t). If now we have some method of selecting the Cn so that zn! (t) tends almost everywhere to so'(t), then 7(Cn) -7(C0), and the curve Co is the one sought. Suppose that a curve C: z=z(t), (tl?<t < t2), in q-dimensional space minimizes an integral 7(C) in the class of all curves of classt D' joining two fixed points zi and Z2. As is well known, the equation
Memoir on the general theory of surfaces and rectilinear congruences
CONTENTS Introduction. 79 1. Fundamental equations for a surface referred to its asymptotic curves. 85 2. Reciprocal congruences and the relation R. 86 3. The developables of the congruences T and r". 88 4. The focal points of the lines I and I'. 90 5. The directrix congruences. 91 6. Some general properties of reciprocal congruences. 93 7. The osculating quadric, and its connection-with reciprocal congruences. The canonical quadric. 96 8. Congruences conjugate and harmonic to a surface. 99 9. Congruences whose T-curves or r'-curves are indeterminate. 103 10. Planar nets with equal invariants. 106 11. The perspectives of the asymptotic curves on the tangent plane, and their four-point and osculating conies. 108 12. The canonical congruences of the first and second kinds. Characterization and generalization of scroll directrix congruences. Ill 13. Darboux's canonical expansion in non-homogeneous coordinates. 115 14. The completed characterization of the canonical tetrahedron. 122 15. The congruence of pseudo-normals. Í25 16. General theorems on conjugate nets. 130 17. Pseudo-geodesies, union curves, and generalized surfaces of Voss. 136 18. Dual considerations; curves of Darboux and Segre; axis and ray congruences. 140 19. Projective generalization of metric theorems. 144 20. Fundamental equations of the theory of surfaces. 148