INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

ON THE THEORY OF SMOOTH STRUCTURES

Shafei Deh Abad A.

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

MIRAMARE-TRIESTE

• T " T—r

IC/92/8

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

ON THE THEORY OF SMOOTH STRUCTURES •

ShafeiDeh AbadA."

International Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

In this paper which is the first of a series of papers on smooth structures, we introducethe concepts of C-structures and smooth structures, and deduce some properties of them. Thenotion of smooth structure on integral domains is introduced. Among other things, we prove thateach integral domain which is not a field, admits a unique smooth structure. We construct a largeclass of non-polynomial smooth functions on some integral domains. We also construct a smoothfunction from TL - {0} into 7L which cannot be extended into a smooth funcrion on TL. Thewhole theory is developed without any concept from topology. By applying this theory we provethe following: Theorem - Let M and N be two finite dimensional smooth manifolds. Assumethat ip : C°°(.N) —> <.;°°( M) is a homomorphism of R-algebras. Then, there exists exactly onesmooth mapping <f> : M —> N such that ip = <j>". Proposition - Let M be as above and let wbe a point which is not in M. Assume that p : C°°(M U {w}) -> C^iM) is the restrictionhomomorphism. Then, for each section of <p, say TJ, there exists a point m £ M such that for each/ £ C°°(M), i)(/)(w) = / (m) . Proposition - Let / : R —> R be an entire function. Then /admits a smooth extension to R[ (T]].

MIRAMARE - TRIESTE

January 1992

This research is partially supported by A.E.O.I.Permanent address: Atomic Energy Organization of Iran (A.E.O.I.), Tehran, Iran;Mailing Address: Department of Mathematics, Faculty of Sciences, University of Tehran, Tehran,Iran.

1. SMOOTH STRUCTURES

In what follows R denotes a commutative ring with the identity element 1. By an R-algebra we mean an associative and commutative K-algebra with tre identity. Furthermore, forevery .R-algebra with the identity element e, we assume that {\ £ R\\e = 0 } = {0}. All algebrahomomorphisms are assume ' to preserve the identity element. Any sub-algebra contains the iden-tity element. Finally, by a submodule of an ii-algebra A we mean a submodule of the i?-moduleA.

Let A be an ii-algebra with the identity e. Assume that A and B are subsets of A Thesubmodule of .A generated by the set {xy\x e A,y e B) will be denoted by A • B. For eachn 6 IN*, let nA - {1112 ...x*\xi £ ^4}. The submodule of A (resp. the ideal of A) generatedby "A will be denoted by A" (resp. by A?). We use the convention, A0 = Re. Moreover if A is asubmodule of A, J4' = A.

1,1 Algebras of type C

This section and the next include some elementary results on C-structures which will beused later in this paper. More results on this topic will be given in the subsequent papers.

Let A be an .R-algebra with the identity e, An ideal A of .4 is called an ideal of type Cif A = Re © A. We identify Re with R and denote the projection on R = Re (resp. on A) byrA (resp. by rA). The set of all ideals of type C of A will be denoted by Ac. We say that A is anR-algebra of type C if Ac J ffl. Let A £ Ac. Then rA is clearly a character of A. On the otherhand if a is a character of A, then kero e Ac. The set of all characters of .4 will be denoted byA"1- The mapping A1 —»Ac given b y a n kera is clearly bijective.

The proof of the following lemma is straightforward.

Lemma 1.1.1. Let A and A' be iL-algebras. Assume that ip : A -* A' is an H-algebra homo-morphism. Suppose that A' is an .R-algebra of type C- Then ,4 is also an .R-algebra of type C.Moreover, if A' £ _4'c and a' g A''1, then p"1 (A) e Ac and ip'(a') € A"1.

Let A, A' and yj be as above. The mapping _4'c —» Ac given by A' t-» tp~x (A') will bedenoted by £c. We say that <£ is induced by <p.

Lemma 1.1.2. Let A and A' be two ii-algebras of type C Assume that ip : A —> A' is an R-algebra homomorphism. Let A' £ A'c and A = p~'( A'). Then, rA = r^iop and rA* 040

Proof. Lety £ A. Then y - rA(y)e + rA(y) andp(;/) = rA<(<p(y)e'+ rA.(p(r/)). Hence,<(fo(i/)) = p(y) = ^[rA(y)e + rA(y)] = rA(j/)e' + p(rA(j0). 3r, ( V o+ (rA .op(v) - <P 0 rA(j/)) = 0. But ip( A) C A' and A' 6 A'c. Therefore

o Viy) = »A(V) and rA< o <p(y) = po rA(!/). Since y e A is arbitrary. f A i o y rA and

"T 1—

A submodule A oiA is called asubmodule of type C if A € Ac. The set of all submodulesof type C of A will be denoted by C( A). A submodule A e C{ A) is called a submodule of typeD" if for each k < n, Ak n A**1 = {0 }. We say that A is a submodule of type D if for each n,AE D". The set of all submodules of type IT (resp. of type D) will be denoted by Dn(A) (resp.by DM)) .

Let A e C(A) and k £ IN. Then A* = Ak • A = Ah • (A0 + A) = Ak - (Av + A A) =^ t + A**1 . ,4 = J4* + A.t+1. Therefore we have the followii.g simple lemma.

Lemma 1.1.3. Let A be an K-algebra and n £ IN. Then

i) If A G C M ) , then ,4 = *£to A* + A""1 .

ii) If A e Dn(A), then A = ffiJUo A* © A"*1.

Let A be a submodule of type Dn. By the above lemma, A = Gj»0 Ak 0 A"+1 • T he

projection of .4 onto A* (resp. onto A1*1) in the above decomposition will be denoted by ak

{resp.

Let A and B be in C( A). We say that A is weaWy e<?mva/t!nr to £? if A = £ . Two weaklyequivalent submodules A and B are called equivalent if they are isomorphic. The submodule A £C(.4) is called weakly compatible (resp. C-compatible) with B g C U ) if for some B' in CM)weakly equivalent to B (resp. equivalent to f?) and for some A' £ C(A) weakly equivalent to A(resp. equivalent to A), rA> (B') andA' are equal. The submodule A £ D ' M ) is called stronglycompatible with B € Dl(A) if <*i |B : S —> A is an isomorphism. We say that C € DHA)is compatible with B £ D}{A), if there exist B' € Dl(A) equivalent to B and A 6 D1 U )equivalent to C such that A is strongly compatible with B'.

Lemma 1.1.4.ble with B.

Let A, B £ Dl(A) bt two equivalent submodules. Then A is strongly compati-

Proof. By (l.l,3.ii)Heffi A® A2 = A = Re(B B ® S 2 . Assume that i e B andQ](i) = 0.The equality z = ai ( i ) + Oj ( i ) implies that 1=02(1) £ A2 = B2 . Since B e DU.A), X = 0.Thus ai is injective. Now assume that y £ A. The equality y = 0\{y) + /?2(y) implies thaty = ai ({/) = as (/3i ( JI)). Therefore a\ is surjective.

D

Let M be a set. Assume that A is a subalgebra of RM. Let u f l f . Clearly the set of allelements of A which are zero at w is an ideal of type C. It will be denoted by /„. If the elementsof .A separate the points of M, then the mapping M ~* Ac given by ui —> Iw is injective. In thissituation we identify M with its image under the above mapping.

1.2 C-Structures

Let A be an it-algebra of type C. Assume that X is a subset of C(A). The set {A_ e^4C|J4 e £ } will be denoted by So. Let £0 be a subset of Io. The set £ will denote the subset ofI defined by £ = {A € X \A € £0 }•

A C-triple over R is a triple (A, I ; Io), where A is an /J-algebra of type C, S is a subsetof C( A) and £0 is a subset of Io.

Let M , £ ; £0) and M,X'; £(j) be two C-triples over i t A C-homomorphism <p :M , £ ; I o ) —» M' , £';£,;) is a homomorphism of H-algebras ip : A —».A' such that for all A' 6£g, p"1 (A') e Zo - Clearly, the restriction of <gf to £jj is a map from £Q into £0. This map will bedenoted by y. It is also clear that the map id^ : M , £ ; £ o ) - > M , £ ; £ o ) i s a C-homomorphism.Let M,, £.,£(), i = 1,2,3 be C-triples over R. Assume that p, : M , , ! , ; ^ ) -> M i , £2; £20)and tp2 : (A2,^i',^ao) —* (A3 ,£3 ;£JO) are C-homomorphisms. Then, clearly ipi aip^ isaC-homomorphism, and tp2 ° 'Pi — *P\ ° ^2 •

The above observations can be summarized in the following proposition.

Proposition 1,2.1. The class of all C-triples over R together with C-homomorphisms betweenthem form a category. This category will be denoted by R-CT.

A C-homomorphism p : (,4;£,£o) —» M' ,£ ;£d) is called injective {resp, surjective,resp. bijective) if <p : A - * A' is injective (resp bijective, resp. surjective).

A C-triple over R, (A , £, £0) is called separated if nA e j ^ = {0 }.

Assume that the C-triple (A, £,£0) is separated. Then the mapping .4 -^> R^ given byJ H ( A i-t rA(y)) is clearly an injective homomorphism of .R-algebras. In this situation weidentify A with its image under the mapping 8.

Lemma 1.2.2. Let <p : (A, £•',%>) —* (A',V; E,J) be aC-homomoiphism. Then

i) If f is injective and M ' , £ ' , £ Q ) is separated, then (A, 2.: £0) is separated,

ii) If M , £ , £0) is separated and £ is surjective, then ip is injective.

iii) If <p is surjective ip is injective.

Proof. i)Assumethaty6nA€£(,A.ThenyenA,ei i !¥!-1(A') = ip-HnA,£t,A') = p" l({0}) ^{0 }. Therefore (A, £; £0) is separated.

ii) Let j) e A- Assume that ¥>(y) = 0. By Lemma 1.1.2 for each A' £ Io, VA') (JJ) =

^\'(^(y)) = 0 . Since ^ : £0 —* £0 is surjective, for all A £ £0, ?A(I/) = 0 . But (A,!•;%>) isseparated. Therefore y = 0 and p is injective.

iii) Let A',A" £ £(J and^r ' (A' ) = A = " ) . Since <p is surjective A" =

= <p{A) = tp{<p~\A')). Therefore<p is injective.

Let CA,I;£o) be a C-triple. We say that (A,!,; So) is complete (resp. C-completeresp weaWy complete) if £o = .4C (resp. Eo = .4C, resp. Zo = Eo) •

Clearly completeness implies C-corr.pleteness and weakly completeness.

Proposition 1.2.3. Let(.A,E,Io) and(.4',E,E^) be two C -triples over R. Assume that p :_4 —> ,4' is a homomorphism of J?-algebras. Suppose that (.A,E; Eo) i s complete. Then p :(,4,E;Eo) -> ( ^ ' , ^ ' ; ^ ) is a C-homomorphism.

Proof. Let A' e ZjJ. Then p- ' (A ' ) 6 .4'. Since (,4,E; £c) is complete tp~\ A') £ %>.Therefore <p is a C-homomorphism.

a

Assume that p : (.A,E;Eo) -* ( .A ' ,E ' ;ZQ) is an injective C-homomorphism. Wesay that ip is a wea/t domination, or (.4, Z; Eo) 's wea% dominated by (.A', E'; Eo') wider p or(.4, E';E,J) weat/y(iwM(nafM(>l,S;£o) under p, if for each A e I , p(A) G £'.

Lemma 1.2.4. Let \p : (.A,E;Eo) -> ( .A' .Z' ;^) be a weak domination. Assume that

(.A,I;Io) is C-complete then the C-triple (^', I ' ; IQ) is also C-complete.

Proof. Let A' £ A'c. Then p - ^ A ' ) £ Ac. Since the C-triple (<4,1; to) is C-complete,p~!( A') 6 Eo. Thus there exists A £ Z such that A. = p - ' (A ' ) . As p is a weak domination,<p{A) £ I ' . Clearly ^(yl) C A'. Hence A' £ I£. Therefore (A1,1.'; IQ) is C-complete.

D

Lemma 1.2.5. Let ip : (>4,I; Io) —> (^'; ^';^o) be a weak domination. Assume thai A £and y{ A) ^ A ' e l ' f l J ) 1 (.4'). Then the following diagram is commutative.

a,

A'

i

Proof. Let y e A. Then y = <*o(y)e + a\{y) + a2(!/)> where ao(,y) £ R, ai(y) 6 A and22(y) £ A2- Thus^(y) = ao(i/)e' + yj(cvt(y)) + v(a2(v). Furthermore ply) is an element of4 ' . Hence ip(y) = ao(¥>(y))e' + a',(v>(y)) + d2(<p(y)), where aj,(ip(v)) E f i , a i W ! / ) ) £A' and a;(V(!/)) C A'2• So we have (ao(j/) - ai(f>(y)))e' + (p o ct\{y) - a\ o p ( s ) ) t

) — S2 ° ¥>(y) = 0 . But j4' G Z?'(^4') and ^(c^dO) £ A' • Therefore p o a\ (y) =. Since v £ -4 is arbitrary, f iom = a'j o >.

D

Lemma 1.2.6. Let <p : (A,T.,%>)-> {A1,2,11) be a weak domination. Then^: £,j -> Zo isinjective. If (.4', E'; Eo) is weakly complete, p_ is also surjective.

Proof. LetA\A"e£( j ,A g £„. Suppose that ip{ A') = A = <p( A"). Then, since p is a weak

domination, for any A £ E such that A = A, we have ip(A) G I ' andip(yl) C A' H A". Thus

A' = ifi(A) = A". Therefore <p is injective.

Now let (j4', E'; £5) be weakly complete. Suppose that A 6 La. Then for some A e l ,A = A- Since p is a weak domination <p(A) 6 E^. But (A', E'; Eg) is weakly complete. Hencep( J4) € EQ . Clearly tp~l(tp(A}) = A = A. Therefore > is surjective.

D

We say that a C-triple (A , E; %>) is semi-analytic, if for each A E ^ , n A " = ( 0 } .

Lemma 1.2.7. Let p : (_A,E;£c>) -* (-4'iE';£,j) be a weak domination. Assume that A £C(A) andp(yl) £ Dk(A').Th&nA £ Z>*(^). Furthermore, if (.A', I ' ; 1Q) is semi-analytic andweakly complete, (^4, E;Io) "s semi-analytic.

Proof. LetAeC(.A)andy(A) 6 D ^ X ) . Then, for each n < A,{0}. Since ^ is inject)ve A"nA""1 = (p~'(y(A))° n (y

*1 = {0}. Therefore A £ Dk(A).

n (" n

Now assume that (j4',E';So) is semi-analytic and weakly complete: By Lemma 1.2.6£ is surjective. LetA 6 Eg. Then there exists A' e E,j suchthat^'fA') =A. Hencen^1A" =n™i¥>-1(AM) =¥'"1(n^1A">) = p - ' ( { 0 } ) = {0}. Therefore (A, I ;Eo) is semi-analytic.

D

Let (j4,E; EQ) be a C-triple. A submodule A of A is called a italic submodule for( .4 ,1 ; Eo), if for each A €Eo,rA(A) <E I . The C-triple ( .4 ,1 ; £0) is called a basic C-triple ifit admits a basic submodule.

Lemma 1.2.8. Letp : (_4,E;Zo) -» {A',Z';^ be a weak domination.a basic submodule A, then <p(A) is a basic submodule for (_4',E'; Ip).

admits

Proof. LetA' £ S,j. There exists A e to such that A = p - ' (A ' ) . By Lemma 1.1.2 rA. op(A) = ip o rA(j4). Since p is a weak domination andrA(A) £l.,rx'{fp(A)) = <p( r\{ A)) <~zE'.

D

: (A,Z;to) —> (,4',E';E<S) be a weak domination. We say that ip is a dominationif ip : t^ —* Eo is surjective. Assume that ip is a domination. Then by Lemma 1.2.6 <p ih. bijective.Moreover the same lemma shows that if tp is a weak domination and if (.A', !•'; E^) is weaklycomplete, then <p is a domination.

Lemma 1,2,9. Letp : (-4,E;Eo) -» (^ ' .Z ' ; !^) be a domination. Assume that (A',?.';!^)is semi-analytic. Then C4,E; Eo) is semi-analytic.

The proof is as the proof of Lemma 1.2.7.

6

A C-triple (A, X; So) is called of polynomial type if for each A £ So there exists A

such that A = A, and each element y € -4 can be written as

Jfc

y = y^yn, where y. e A".

1.3 Smooth Structures

By a D"-triple (resp. r>-triple) we mean aC-triple (.4,X; Xo) such thai I C Dn(A)(resp. £ C D(A)) and each two elements of 2 are compatible with each other. Let(.A|2;Xo)be a Dn-triple (resp. D-triple). We say that 2 is maximal if for each A £ D"(A) - X (respJ4 £ Z?M) — X) there exists an element B 6 2 such that A and S are not compatible with eachother.

Definition 1.3.1 A smooth triple is a D-triple (A, X; 2o) such that I is maximal.

Let ( .4 ,1; So) be a smooth triple. Iflo = So, for simplicity we write (.4,S) instead of(A , 2 ; Id) and call (A , X) a smooth algebra. We also say that X is a smooth structure on A •

Let A be an K-algebra of type C. Assume that A admits a submodule A of type D. ByZorn's lemma, there exists a smooth structure S on A which contains A.

Assume that ( .4 ,1; So) and (.A',2'; S,J) are two smooth triples over R Ahomomorphism between them is a morphism in the calegory R-CT. It is clear that the ciass of allK-smooth triples together with smooth homomorphisms between them is a fuSl subcategory of thecategory R-CT. This category will be denoted by R-ST. The full subcategory of the categoryR-ST whose objects are .R-smoofh algebras will be denoted by R-SA.

Lemma 1.3.2 A necessary and sufficient condition for a smooth triple (A, X; So) to be of poly-nomial type is that for any A g l o , there exists A 6 X such that A = A and A = @^0 ^" •

The proof is clear.

Let (A, X) be a smooth algebra. We say that S is a complete smooth structure on A if{A, X; £Q ) is a complete smooth triple.

Lemma 1.3.3 Let A be an ii-algebra. Assume that .A admits a complete smooth structure X.Then A does not admit any other smooth structure.

Proof. Let 2 ' be another smooth structure on A. Since X is complete, Xo' C So. Assume thatA' el.'. Then i ' e l o . The submoduleA' is a submodule of type D and generates A' £ So- ThusA' e X. Therefore X' c S. Since X' is maximal X = X'.

The ring R is clearly an i?-algebra. The singleton {{0}} is the unique smooth structureon R. This structure will be denoted by \R° \.

Here we have the following simple lemma.

7

Lemma 1.3.4. Assume that (.4,2) is an i?-smooth algebra. For A e Ac, we have A 6 So ifand only ifrA : (A,Z) ->(R,\R°\) is smooth.

Proposition 1,3.5. The R-algebra C ^ F " ) , n > 1, admits a complete separated non-semi-analytic smooth structure, which is the unique smooth structure on it.

Proof. Let i ' : R " -> R denote the i-th projection and let x : R n -» R" denote the identitymapping. For \ = (X1 X2 ... A") £ R n we set Mx = £ £ , , R( xk - **e) and h = E L i Mxk -\ke). Clearly I\ is generated by M>. By a well-known lemma whose statement will follow, onededuces that Ij £ Ac. Suppose that

|i|=t

where„, e Rm&ViG.A. Let; = (;, ; 2 ...;„) £ I N M J |

Then there exists a non-zero constant C, £ R such that

= 0

Therefore /i;- = 0 and for each k £ IN , Mj n Jjf+1 = {0}. Hence MA is a submodule of type D.Let X, Y £ R. Then, clearly M\ and Mv, are compatible with each other. Therefore there existsa smooth structure on A which contains the set {M\\\ £ R"} . This structure will be denotedby [R»], Let A £ | R ° ] 0 and X = (?A( i I) ?A(i2) ... ?A(i")). Then MA C A. Hence eachelement of [ R "] is equivalent to some M\ and the smooth algebra (C°°( R)n , [ R n]) is complete.Clearly f C M (R n ) , [R" l ) is separaied and admits Y,l=] R • x* as a basic submodule. Considerthe function <p : R n —• R defined by

o if t1 = o= (tlt2...tn)

All the derivativesof <p are zero at ( 0 , 0 . . .0) . Therefore (C°°( R n ) , [ R n ] ) is not semi-analytic.

The uniqueness follows from Lemma 1.3.3.

D

Lemma 1.3.6 Let / g C ° ( R " ) and X £ R. Then there exist n functions p,- r- C" (R" ) ,i = 1,2,... ,n, such that / = /(X)e + Et - i (** - ^*e)9*-

In a similar way without using the above lemma one can see that the algebra of polynomialfunctions on an integral domain R and the algebra of entire functions on IK "(IK = R or IK = C)admit unique smooth structures. These smooth algebras will be denoted by (P(R"): \Rn\) and(C"( IK "), (IK ")), respectively. They are both separated and have basic submodules. The first isof polynomial type and the second is semi-analytic.

8

Now let Q C R n be ncn-empt). Assume that X £ £1. Define <p\ : R™ - {0} -» R as

follows

Let .A be the R-algebra generated by C O (R") U{pitX g f l } . Notice that

As in Proposition 1.3.5, one sees that A admits a smooth structure E which contains the

se t{Mj |> e f l ) . Assume that A E _4C. Then clearly \ = (rA(xv) H i 2 ) . . . rA U " ) ) is an

element of R n . LetX £ I i . Then 1 = ?A(e) = ?A(yjj • ^ ' ) = ?A ((£>>) rA(<,o~1) = r A ( ^ x ) x 0.

This is a contradiction. Therefore, every element of E is equivalent to some MA, A 6 £2, EO = Q

and ( A , E) is complete. The smooth algebra ( A £ ) is separated if and only if f! is dense in R".

It is called the smooth algebra associated with £1 and it will be denoted b y ( j 4 ( Q ) , £ ( £ 2 ) ) .

Proposition 1.3.7. Let JW be a closed submanifold of R ". Assume that A = C°°( M). Then

Ac - M. In other words C°°( M) is complete.

Proof. L e t ( , 4 ( A O , £ ( M ) ) denote the smooth algebra associated with M. Letp : A(M) —y A

be the restriction homomorphism. Then ip : (A(M), Z(M); M -* (A:AC,AC) is a surjective

C-homomorphism. By Lemma ].2.2.iii, ip : Ac —> ( .4(M)) C is injective. But A{M)C - M and

M C -4C. Furthermore, from Lemma 1,1,2 and the fact that A separates the points of M, it follows

that, ^\KI = idjj' Therefore ,4 c = M.

•From the above proposition and Whitney's embedding theorem we have the following

theorems.

Theorem 1.3.8. Let M be a finite dimensional smooth manifold. Then C°°( M) is complete.

Theoretn 1.3.9. Let M and N be two finite dimensional smooth manifolds. Assume that p :

C°°(W) —* C°°(M) is a homomorphism of R-algebras. Then there exists exactly one smooth

mapping <$>' M —* N such that tf>' = p-

Proof. The uniqueness of <j> is clear. By the above theoretn C°°( M) and C°°( JV) are complete.

Therefore, p_ is a map from M into JV. Since for each / £ C™(JV), ip'(f) = <p(f) e C°° (M) ,

¥> : M -+ iV is smooth.

D

Proposition 1.3.10. Let M be a finite dimensional smooth manifold and let w be a point whichis not in M. Assume that p : C°°( M U {w}) —» C*( M) denotes the restriction homomorphism.Then for each section of the homomorphism ip, say t), there exists a point TO £ M such that foreach/eC°°(M),T)(/(w)) =

Proof, Clearly the set of all / £ C°°( M) such that TJ( f)(u) = 0 is an ideal of type C forCX(M). Since C°°(M) is complete, M = J4C. Therefore there exists m £ M such Lhat Im = Iw.

The proof is complete.

G

1.4 Maxi mal smooth tri pies

Proposition 1.4.1. L e t ( 7 , < ) be a directed set. Assume that ( ( A , £ , , E , o ) ,p/i),jG/> * < J

is a direct system of smooth triples over R. where for each i < j , A, is a subalgebra of 4 and

Pji, the canonical injection of Ai into ^1^ is a domination. Furthermore, suppose that for some

v G / , (A*, I),; 2 ^ ) is C-complete. Then the i?-algebra ^ = U;erA" is the underlying H-algebra

of a unique C-complete smooth triple over R, such as ( A , £ ; £o) which satisfies the following

conditions:

i) For each i e / , p t : ( A , E , , ! « ) —» ( J4 , I ; ^31 is a domination. (Here p ; : A —• 4is the canonical injection).

ii) If for a l i i > v, ( A ^ ; £ t o ) is separated (resp. semi-analytic). t h e n ( A S ; £ o ) is

separated (resp. semi-analytic).

Proof. Let fi be an element of I and let A 6 Ep. We are going to prove that ,4 is a submodule of

type D for A- Let A = A • A and for i > / j . Let A< = A • A - Clearly we have A = Ujg/ A,. Since

.4 = U , e j A , for each y E A, there exists i G / such that y £ A - As A is an ideal of type D for

-4'. !/ = rAi(j/)e + rA,(s/), where TA,(y) £ A*. Since A( C A, rAi(j/) £ A. Hpnce 1/ e & + A .

But j / is an arbitrary element of A. Therefore A = Re + A. Suppose that for some k E IN,

E<where the sign ^ on E means that S is with finite support, xn = 1"' xj2 .. .Xp ,n= Y^\ "*> i £

A,yn 6 - 4 and ^j n g R. Since .4 = U,e/A^ and for i < ; , A is a subalgebra a' Aj, there exists

i 6 / , 1' > ^ , such that all yk £ A"- Thus in A'

= 0 = E yn^n6,4*nV+1

Since A is a sub-module of type D for A and generates the ideal A ,

= 0 =

Therefore 4 is a submodule of type D for A On the other hand let A and A' be two equivalent

submodules of type Dior A^. Since each A;is a domination, J4' • Aj = A • A = -

10

Hence A' • A = U,ci(A' • A,) = U^ = A. Therefore A and A' are equivalent. Let C £ ! „ be

compatible with B 6 £„. Then the™- exist J4 € EM equivalent to C and B ' £ ! „ equivalent to

B such that a i |fl< : fl' -> >1 is an isomorphism. As we have seen above <^(B') is equivalent to

Pfi(B) and/p^A) is equivalent to pM<C). By Lemma 1.2.5, ai|Pl,(B') : pM(B') - • ¥V(-4) is an

isomorphism. Thus ^ ( C ) is compatible with ip^iB). Therefore there exists a smooth triple over

E s u c h a s ( ^ 4 , £ ; £ o ) where£ D U;Ef£,andio is the subset of Io characterized by the following

property:

"An element A in IQ is in Eo if and only if A is generated by an element

that for some i £ I, A • A € £<o."

, such

Clearly each <pf : (A,Zio\%) —» (A,!•;%}) is a domination. Since (A,,, £„; £uo) is

C-complete (.4,X; £0) is also C-complete. The uniqueness of this smooth triple is trivial.

Assume that for all i > ;/, (.4,, £,;£«)) is separated. Lety 6 r \ c ^ A . Then there exists

x > v, such that y e Ai. Thus y = - n A G £ o ¥ ) ," ' (A) = n A c t , 0 A =

Therefore (,4, L; £0) is separated.

In the same way one can see that if for all i > u, (A, I , ; X,o) is semi-analytic, then

( .4 ,E ;£o) is semi-analytic.

n

Let T be a subcategory of the category R-ST. An object (_A, I ; io ) in F is called

maximal in F if every domination ip : ( . 4 , 1 ; So) —* ( J 4 ' , ^ ' ; ^d) which is in T is an isomorphism.

Proposition 1.4.2. The R -smooth algebra (C°°( R " ) , I R " j ) is not separatedly maximal (is not

maximal in the category of separated R-smooth algebras).

Sketch of Proof. Let X g Q and /J e R - Q. Define p i , 0 y : R -> R as follows

.. J l / t - X < f t g Q , ,., / 0 if

Let^bethesubalgebraof R R generated by C°°(R B ) U {^x|X 6 Q}U{^ lM ^ Q}- As we have

done in Proposition 1.3.5 one can check eosify that A admits a unique complete separated smooth

structure £ , such that the canonical injection (C°°(R) , [ R 1 ] ) —» (^4,2) is a domination which

is not an isomorphism.

•Theorem 1.4.3. Every C-complete separated .ft-smooth triple is dominated by a maximal one.

Proof. Let (.4,X; Io) be a C-complete separated iJ-smooth triple. Assume that £2 is the set

consisting of all separated H-smooth triples which dominate (A,^; £0). Since (_4,£; So) e

Q , 1} is not empty. Now we order Q by domination. By Proposition 1.4.1, each chain has an

11

upper bound in £2. By Zorn's lemma £2 has a maximal element (A\ £ ' ; £Q) which dominates

( , 4 , S ; £ o ) . By Lemma l.2A(A',Z';to) is C-complete. D

1.5 Smooth Structures on Rings

In the following x : R —> R denotes the identity mapping.

Let R be a ring. A subset ft of R is called absorbing with respect to X e R, if for each

a in R there exists 0 £ £2, /3 ^ 0 such that X + 0a £ £2. A subset £2 of R is called absorbing if

it is absorbing with respect to each of its elements.

Lemma 1.5.1. Let R be an integral domain which is not a field. Assume that £2 is an absorbing

subset of R. Suppose that X e ft, IJ € R and r) j 0 . Let y>,i7, : £2 —» f£ be defined as follows

I if t = X

Then there exists no function \j> : il —* R such that ipx,v can be written as

Proof. Assume that there exists $ : £2 -* R such that tpx^ can be written as above. Since ft is

absorbing there exists /3e R,f3^0, such that X + T]2 j3 £ £2. Hence

O = P v , U + r)2/3) =Tje + Tj2/3-V>(X + T;2/3) = r)[e + vP • ( > + V2P)\

Since R is an integral domain, the equality t)[e + TJ/3 • ifi(\ + T]2/3)] = 0 implies that t)[ -Pi>{\ +

rj2/3)] = e. Thus, 17 is a unit. After dividing the two sides of (*) by TJ we have px.^/v = e + {z -

Xe) -ifr/i}. Now the definition of px,* implies that for all a £ R, a =f 0 a •V'(a +X)/r ; = 1 Since

R is not a field, this is a contradiction.

•From new on R denotes an integral domain which is not a field.

Definition 1.5.2. Let £2 be an absorbing subset of R. A smooth structure on ft is an fl-smooth

triple (A, T.; £0) with the following properties:

i) The algebra .4 is a subalgebra of Ra and So = £2

ii) The module R • x is a basic submodule for ( A , £ ; £2).

iii) The smooth triple (,A, L; £2) is separated.

iv ) ( ,4 ,£ ;£2) is separatedly maximal.

Theorem 1.5.3. Let i i be as above. Assume that ft is an absorbing subset of R. Then, there

exists a unique smooth structure (.An, I n ; ft) on ft which satisfies the following condition:

12

If £1' is any other absorbing subset of R which •; contained in Q., then the restriction of

each element of An ti O.' is an element of

Proof. Let (P(R), \R\) be the ft-smooth algebra of polynomial functions on R. The smooth

triple ( P ( R), \R\; Q) satisfies all the conditions i-iii. It is clear that each smooth structure on £2

must dominate (P(R),]R\; £1). ByTheorem 1.4.3 there exists a separately maximal H-smooth

triple (A,T.;C1) which dominates (P(R), | H | ; I i ) . Clearly ( . 4 , 1 ; £2) satisfies the conditions i-

iv. We are going to prove that (A, £ ; £1) is the unique fl-smooth triple which satisfies shese condi-

tions. Suppose that (A', £ ' ; £i) is another R~ smooth triple which satisfies all the above conduions.

Let A V A' denote the subalgebra of Ra generated by A U A'. For A e R let M\ = fi(z - Ae)

and/> = (,4 V ,A')(:r - Xe). Assume that JI G - 4 V . 4 ' . Then, there exist a:, £.A,!/i C .4',

i = 1, 2,..., n, such that y can be written as y = JCJL] Zi • y<. Clearly ii is a subset of Lo n I(J.

Let >, £ Zo n XJ. Then, JW\ is a submodule of type D for (.4.X; H ) . It is also a submodule of

type D ft r ( . 4 ' , £ ' ; i l ) . Therefore there exists some z< £ .4,1/i £ A' i = 1,2, — A. such that

,•+

e Re+ M\(A\/A'). Now assume that there exist some /j £ R, z £ A V A' such that fi(x ~ Ae)n

= ( i - A e J ^ ' z . Then ( i - \e)n[/ie - (x - \e)z\ = 0 . This equality and the fact that Ris an

integral domains imply that

li if t = A0 if iff X

By Lemma 1.5.1, M = 0 and i = 0. Thus Mx is a submodule of type D (or A V A'. Let

A, A' € £(i n Z(J. Then clearly M\ and Mx> are compatible with each other. Therefore there exists

a smooth structure £i on A V J4 ' which includes the set {Mj|A e l i f l £,(}. Let us denote the

injection .4 -^> AM A' (resp. .4' —> .A V ^ ' ) by > (resp. by tp'). Assume that A G (A V.A')C. Then

V~ l (A) £ .Ac anJ ^ ' ~ ! ( A ) t ^l'c. Since (A,Z) and (^4' ,! ' ) are C-comp!ete, ^ " ' ( A ) f Io

and p ' " 1 (A) 6 S,j. Thus there exis ts , fi' £ R such that ( s - ^ e ) E p " ' ( A ) and (x — fi'e) €

(f)'~'(A). Therefore i — fie' = tp(x - fie) G A and (x — fi'e') = <p'(x — fi,'e) e A. Since

A is an idea of type C, fi = / / and A G E l o . Let A' G £<), A" G S,j be generated by Mi. A

simple calculation shows that for each Rz 6 S such that .4z = A' and for each Rz' £ S ' such

that ,4V £ A", there exist invertible elements y 6 .A and JI' 6 j t ' such that z = {x — Xe) • y and

z' = ( i — Xe)j/'. Therefore

va : ( . A . I i Q ) -»(^4 v ^ l ' , X , ; £ i ) and p ' : ( ^ ; , S , £ i ) - » ( ^ v . A , E i ; O )

are dominations. Clearly ( ^ V A'^t ,£i) is separated. Since (A, E ;£ l ) and ( ^ t ' , I ' ; £2} are sep-

aratcdly maximal, (v4,i : ;£l) = ( ^ v ^4', S i , £2) = C-4',E'; £2). The unique smooth structure on

£2 will be denoted by (An, En; £2).

13

Ixt £1' be an absorbing subset of R which is included in £1. Assume that y and * are

two elements of _4n. Let X 6 £ i ' . Th^n there exist j , z £ . 4 a , such that y = y{\)e + ( i — Xe)y

and 2 = z{\)e + ( i - Ae)i. Suppose that y\a> = z\a>- Then the above equalities imply that

{x - Xe)(j/|n' - i |nO = 0 . Since R is an integral domain

if t = X _ ,

But £i' is an absorbing subset of R. Thus, Lemma 1.5.1, implies that y\n, = S|si.. Therefore the

restriction of elements of An to £1' is the underlying H-algebra of a separated .R-smooth triple

which admits Rx as a basic submodule. By the unicity of the smooth structure this /i-smooth triple

is included in (An ,£« ' , Q')-

•Let Q. be as above. Each y £ An is called a smooth/unction on Cl, Let A G £1, k G IN .

Then y a An can be written uniquely as

y = - Xe)

where a, e JJ and « £ An • The element a* of i i is called the fc-th derivative of y at A and it will

be denoted by 0 ( X ) . The function 0 : £i - , R defined by ^ : X i-> 0 f X ) is called the fc-th

der/rative of y. Clearly we have ^ ^ &

Important Remark 1.5.4. Let £i be as above. Suppose that 5 C Ra • The above theorem

and its proof show that to prove that 5 is included in An. it is sufficien! to construct an H-algebra

A C Ra containing SU{x} and prove thai for each y £ A and each A £ O , there exists j/> e A

such that y = y(\)e + ( i - Ae)so,.

Theorem 1.5.5. Let £i be an absorbing subset of R. Assume that ip : IN —> Q is bijective.

Suppose that fi : IN —> IN , i ~ 1,2 . . . , is a sequence of unbounded increasing functions. Let

(a<),-e[N be a sequence of elements of An. Then the function H : i i —< R given by

is a smooth function on il.

Proof. Let .4 be the subalgebra of Rn generated by all the functions of the above form. Assume

that X G £1 and H = £™ 0 1 °< I l U < * - <P( k) e ) w f l ] is an clement of A. Then, there exists

n 6 ' N such that X = ip(n) and

*=0

where fk(i) = \ . {k^ \ f .^ f " , and h is such that fn(h) i 0 . Since the first part of the

above sum is a polynomial in x, for some fi E R and some P ( x ) e P( i?) it can be written as

14

E™J). Therefore H = y.e+ (x-\e)\P{x)

• — tp(k)e)^^]]. By the above remark H is a smooth function.

•Proposition 1.5.6. Let R be a local integral domain which is not a field. Suppose that M is the

maximal ideal of R. Assume that i/> : R —> R is a smooth function. Then for each A & M, the

function • is a smooth function.

Proof. Let A be the K-algebra generated by all the functions of the above form. Let <p = j-^L

and/i g R. Then ^ = u ^ + (^ - ^ ( M ) e ) [ n _ _ i _ J ] . Since V - i)(n)e is a smooth

function which is zero at p, there exists tj £ An such that i/> — i/>(/i)e = ( i — /ie)7j. Therefore

V ~+ (x ~ J- By Remark 1.5.4 the function <p is smooth.

D

Proposition 1.5.7 Let R be an integral domain which is not a field. Assume that Q. c R is an

absorbing subset. Let y be a smooth function on Q.. Then

1) For all n > 1, f £ : £2 -> R is smooth.

2) If - is defined on an absorbing subset of £1, it is also smooth and

3) If j/(£i) is included in an absorbing subset of R, say H' and z £ .4si\ then z o j i i s a

smooth function and ^ ^ = | o 5 x ^ .

Proof. 1) Let .4 be the R-algebra generated by all the smooth functions on Q and the derivatives

of all order of these functions. Let y £ A be a smooth function. Then for A £ Q, there exists a

smooth function z £ _4fl such that T/ = y(\}e+ (x-Xe). Thus for each JJ £ H, we have ^ ( / J ) =

z(fi) + ip — \)^(ji). Since z is smooth, there exists z £ -4<2 such that a = j( A)e + ( i - Ae)^.

Therefore ^ ( M ) = z(\) + (M - A)[Ufi) + ^ ( A * ) ] . Hence g = %(X) + (x - \e)(z + £ ) .

Now by induction on n one can see that for each n > 1, there exists yn £ A such that |£j£ =

£jf(X)e+ (x -Ae)y n . By Remark 1.5.4, ££ is a smooth function. The rest of the proposition can

be proved in the same way.

D

Let R[ [ T] 1 = R be the integral domain of formal power series with coefficients in R .

Assume that ( a n ^ i N , (On = E*^o ar*Tk) is a sequence in R. We say that the sequence (an

tends to a = E t t o akTk, if for each it £ IN, a^ tends to at in R with its natural topology.

Proposition 1.5.8. Let / : R

R.R be an entire function. Then / admits a smooth extension to

Proof. Let a = Y,Za a»- Define / ( a ) as follows

/ ( a ) = f(ao) + / ' ( a o ) ( a - a o ) + ^-4^-(a - a0)2 + • •. + f " (,

2! n!( a - a o ) n + • • •

By using the above definition of convergence one can see that / is an /J-valued function on R.

15

Let C""(R) denote the algebra of entire function on R , and let A = {f[f £ O'CR)}. Direct

computation usir.g the above mentioned definition of convergence shows that for a £ Kandy £ A

there exists z £ A such that y = y(a)e + ( i — «e ) j . By Remark 1.5.4 y is smooth.

•

An integral domain is called semi-analytic (resp. of polynomial type) if its smooth struc-

ture is semi-analytic (resp. of polynomial type). It is called fine, if there exists a smooth function

ip : R — {0 } —> R such that p doc~ not admit any smooth extension to R. An integrai domain R

is called wild if there exists a non-constant function i/> : H —> R such that ^ = 0. It is called fame

if there exists a smooth function TJ : i i —* ii satisfying the following conditions:

0 0 ^ 0 , n = 0 , l , 2 , . . .

ii) 0 ( 0 ) = 0 , n = 0 , l , 2 , . . .

Proposilion 1.5.9. A necf ssary and sufficient condition for R to be semi-analytic is that it be

neither wild nor tame.

Proof. Let R be semi-analytic. Assume that p : R —> R is smooth. Then p g n ™ | i£, where

/o = AR • x. Therefore there exists n £ IN such that ip £ If. In other words there exists no

function \j> : R —> R such that <p can be written as i " • ip. Thus there exists £ < n such that

0The proof of the sufficiency is clear.

By above proposition each smooth function on a semi-analytic ring is uniquely deter-

mined by its derivatives at an element of R.

Let R be semi-analytic. Assume that y : R —> R is smooth. Without any ambiguity we

can write y in the following form

(*) !/ =

where n! an • ^ ( \). The relation (*) n, called the series representation ofy at \,

Proposition 1.5.7 implies the following

Proposition 1,5.10. The set of all the series representations of the smooth functions on a semi-

analytic integral domain R at X £ R, is a commutative H-algebra under componentwise additions

and Cauchy products. Moreover, this algebra is closed under termwise differentiation snd substi-

tution.

Proposition 1.5.11. Let z : TL - {0} - . 7L be defined as follows

z{t) = ( l - t

16

Then z is a smooth function and does not admit any smooth extension to TL.

Proof. By Theorem S.5.5, z is smooth. Assume that I : TL —> 71 is a smooth extension

of z. Then, there exists a smooth function y : TL —> TL such that z = z(0)e + xy. Since

7(2) = i ( 0 ) +21/(2) = - 3 , 5 ( 0 ) ^ 0 . Let

= ( 1 - - t 4 ) 2 ( 2 " _

1H-1

• 2 ( 1 + 2 - 2 2 2 . . . . - •

• ! ) " _ ( 2 ( n + l ) " ) n + l

, k2 "*). Clearly u n divides 5 —

+2(1 + 2 -2 2- 2 2) . . . ( l + 2 J ]* 2 "" ) • J}(fc2(n+ '

andw n = (1 + 2 • 2 8 ){1 + 2 • 2 5 4 • 35") • . . . - ( ] + Z l K - z

I n + i . Moreover, ? - in+i is a smooth extension of z - In+\. There exists yt £ As, such that

z - /n+1 = (z - /n+1 ) (0)e-r i • !/i- Therefore (z - J n + i ) ( u n ) = (z - J n + i ) (0) + wni/i(wB).

But {I - /n+i )(wn) = U - i ^ i )(wn) is divisible by wn. Thus(z - In+\ )(0) is divisible by u v

Furthermore, /„+] (0) is divisible by wn. Therefore for all n > 3, z{0) is divisible by un . This is

a contradiction.

a

In [2] we have proved that TL is a semi-analytic integral domain. The abovv proposition

shows that 2Z is also a fine integral domain. Some other properties of the smooch functions on TL

is contained in [2]. More about the subject of this paper will be given later.

1.6 Some Open Problems

1.1) Is there any .R-algebra admitting two different separated smooth structures'.'

1.2) Is there any separated ii-smooth algebra without any basic submodules?

1.3) Conjecture-Every separated 2£-smooth algebra is semi-analytic.

1.4) Is every smooth function on !E of the form given in Theorem 1.5.5 with u, £ P(7L)">.

1.5) Characterize all semi-analytic integral domains.

1.6) Is there any integral domain which is of polynomial type?

1.7) Characterize all wild integral domains.

1.8) Let (oB )flgiN be a sequence in 2Z. Under what conditions there exist a smooth func-

tion y :TL -+1L having ^ ™ 0 a , i " as its series representation around zero?

1.9) Characterize all integral domains R having the property that AR (the .R-algebra of

smooth functions on R) is an integral domain.

17

1.10) Can one embed R into a non-wild integral domain R which is not a field such that

each / e C°°(R) admits a smooth extension to Rt

1.11) Characterize all fine integral domains.

1.12) Is (C" ( R " ) , ( R ")) sem i-analyticly and separatedly maximal?

1.13) Are any two semi-analyticly and separatedly maximal semi-analytic subatgebras

of the R -algebra R R isomorphic?

Acknowledgments

The author would like to thank Professor Abdus Salam, the International Atomic Energy

Agency and UNESCO for hospitality at the International Centre for Theoretical Physics, Trieste.

REFERENCES

11 ] Hirsch, M.W., Differential Topology (Springer-Verlag, 1976).

[2] Shafei Deh Abad, A., "An introduction to the theory ofdifferentiable structures on infinUe

integral domains", 1. Sci. I.R. Iran, Volume 1, number 3 (1990).

18