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
A study of curved surfaces by means of certain associated ruled surfaces
Introduction. In this paper a point correspondence is introduced which is proving to be very helpful in the study of a general non-ruled analytic surface in ordinary space. If on a surface S tangents to the curves of an asymptotic family are constructed at the points of two curves of 5 which are not members of the family and which intersect in a point y of S, two ruled surfaces are thereby formed which have at y a common generator. The plane which is tangent to one of these ruled surfaces at a selected point of the common generator is tangent to the other at a distinct point whose location depends on the selection of the first point and on the choice of the two curves which determine the ruled surfaces. The use of this correspondence serves the following fourfold purpose: (1) to unify many of the apparently isolated topics which have been studied heretofore, (2) to interpret geometrically, by methods which are simpler than those formerly used, quantities which are intrinsically and projectively related to a surface, (3) to introduce and characterize new configurations which are covariantly related to a surface, and (4) to solve both recognized unsolved problems and new problems which present themselves in the theory. 1. Analytic basis. If the homogeneous projective coordinates y(1), • • • , y(4) of a general point y on a non-ruled surface S in ordinary space are analytic functions of two independent variables u, v, and if the parametric net on S is the asymptotic net, the functions y(i) are solutions of a system of differential equations, which by a suitable transformation can be reduced to Wilczynski's canonical form
Inhomogeneous Boundary Value Problems for Compressible Navier-Stokes Equations: Well-Posedness and Sensitivity Analysis
In this paper compressible, stationary Navier–Stokes equations are considered. A framework for analysis of such equations is established. In particular, the well-posedness for inhomogeneous boundary value problems of elliptic-hyperbolic type is shown. Analysis is performed for small perturbations of the so-called approximate solutions that take form (1.12). The approximate solutions are determined from Stokes problem (1.11). The small perturbations are given by solutions to (1.13). The uniqueness of solutions for problem (1.13) is proved, and, in addition, the differentiability of solutions with respect to the coefficients of differential operators is shown. The results on the well-posedness of nonlinear problems are interesting on their own and are used to obtain the shape differentiability of the drag functional for incompressible Navier–Stokes equations. The shape gradient of the drag functional is derived in the classical and useful for computations form; an appropriate adjoint state is introduced to ...