Seminarberichte Nr. 16

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Seminarberichte aus dem Fachbereich Mathematik der FernUniversität

16 – 1982

der Mathematik (Hrsg.)

Seminarbericht Nr. 16

Dieser Band der Seminarberichte enthält nur Manuskripte von Vorträgen, die auch während der Jahrestagung der Deutschen Mathematiker-Vereinigung 1982 vom 20. bis 24.

September 1982 in der Sektion "Kategorien und homologische

Algebra" gehalten wurden.

Remark on the Simplicial-Cosimplicial Tensor von R. Fritsch

Cartesian closed categories and analysis of smooth mavs

von A. Frölicher

Kategorien abelscher Galoiserweiterungen von C. Greither

How to use abelian group theory for the study of diagrams over posets

von M. Höppner

The injective hull and the CL-Compactification of a continuous poset -

von R.-E. Hoffmann

Eine Bemerkung über Mayer-Vietoris Folgen für Gruppoide

von K.H. Kamps

Kategorielle Darstellungstheorie von H. Kleisli

Wallman compactification of T4-relational algebras and Mal'cev monads

von A. Möbus

Würfelsätze in Kategorien mit Homotopiesystem von Th. Müller

A sup-preserving completion of ordered partial aZgebras

von A. Pasztor

Seite

1 -

5 -

17 -

19 -

31 -

93 -

- 103 -

- 111 -

- 133 -

- 14 7 -

Semitopologiaal funators preserving regular epimorphisms

von H.-E. Porst

Krull-Schmidt for arbitrary aategories von G. Richter

Grothendieak-Verdier aompletions of arbitrary funators

von W. Tholen

Seite

- 165 -

- 185 -

- 209 -

Rudolf Fritsch

ABSTRACT, We show that the existence of canonical repre- sentatives for the elements of the tensor product (coend) of a simplicial and a cosimplicial set depends only on the Eilenberg-Zilber property of the given cosimplicial set.

Thus the second condition, which is used in [5] for achie- ving this result, is superfluous.

AMS subject classification 1980: 18 G 30

Key words and phrases: simplicial set, cosimplicial set, tensorproduct, coend, Eilenberg-Zilber decomposition.

Let X: Äop • S be a simplicial set and Y: Ä • S a cosimpli- cial set. We consider X as a JN-graded set with Ä acting on the right and correspondingly Y is a JN-graded set wi th t>. ac- ting on the left. The Eilenberg-Zilber Lemma states that every x EX has a unique decomposition

( 1 ) X= X ⁺ X⁰

with x+ non-degenerate and

x

⁰ surjective. We assume that

X

has the dual property, i.e.

every y E Y has a unique decomposition

(2) +

Y = Y Yo

with y + in;ective and y⁰ interior

(That the proof of the Eilenberg-Zilber Lemma fails tobe dualizable depends on the fact, that any surjective map is uniquely determined by the set of its sec- tions; but different injective maps with the same one-element domain and the same codomain have the same set of retractions. Thus the Eilenber-Zilber pro- perty for cosimplicial sets is a real restriction; see [2;4.4] and [4] for a further discussion of this phenomenon).

Now take

(3) T - { (x,y) 1 x EX, y E Y, degree x. = degree y = n}

n

(4) T =

LJ

^{T i}

n E JN ⁿ

thus T also is a JN-graded set.

A pair (x, ^y} E T is called simi!ar to the pair (x' , y ') E T

if there is an operator a E ~ such that

(5) x=x'a., a.y=y'

Similarity generates an equivalence relation

~

^{on T} such that (x' a, y)

~

^{(x' , a y)}

for all suitable a.. The set of all equivalence classes is called the tensor product (coend [ 3

J)

of X and Y.

A pair (x, y) E T is called minimal, if x is nondegenerate and y is interior. Our aim is to prove

T

H E O R E M. Every equivalence c/ass in T contains exactly one minimal efement.

To this end we follow the lines of the proof in [1;2.1] formulated for "subdivision" functors; see also [S]. We define maps

( 6) ( 7) ( 8)

Then clearly

(M2, M

1 , M in t

(x,

y) = (xy , y + ^{0 )}

r

t.9.. (x,y) ^{= (x , x+ 0y}}

[S])by taking

(9) (x,y)

~

t t (x,y)

~

^tr~x,y)

~

t(x,y)

(10) t(x,y) ~ (x,y} ~ degree t(x,y) < degree (x,y}, (11) t (x,y) = (x,y) ~ y interior

r

(12) t

1 ^{(x,y) = (x,y) ~} x nondegenerate and finally

(13) t(x,y) = (x,y} ~ (x,y) minimal.

Since the set of degrees is bounded below, i t follows from (10), tha t for any pair (x, y) E T the sequence

(14) (tn(x,y) )nEJN

This proves the existence of a minimal pair in every class.

The key to the uniqueness lies in the

LEMMA. /f (x >Y) is similar to (x' >Y') then t (x~ ,y') is similar to r

t(x,y).

This comes out by taking the oFerator

( 1 5) + + + +

a' = (x' (ay) )o ( (ay )oyo)

where the exponent+ on an operator denotes its injective part while the exponent ^O stands for the surjective component.

Now, for our goal i t is enough to show that, if we start two sequences (14) with similar pairs in T, we end up with the same minimal pair. So let the pair (x,y) be similar to the pair (x' ,y') The lemma provides us with the inductive argument for showi~g:

For all n EIN the pair trtn (x >Y) is similar to the pair tn (x' >Y' ).

But analogously as before, we have for sufficiently large n t tn(x,y) = tn(x,y)

r

and both, as well tn(x,y) as tn(x' ,y'), are minimal pairs.

In view of the equations (5) we see that in this case the corres- ponding a is injective and surjective, thus a = id, i.e.

tn(x,y) = tn(x',y') which finishes the proof.

The reason for having written this note is that almost nobody seems to have read the german paper [1].

Bibliography

1. R. Fritsch, Zur Unterteilung semisimpfizialer Mengen I, Math.

z.

108 (1969), 329-367.

2. R. Fritsch and D.M. Latch, Homotopy inverses for nerve, Math. Z. 177 (1981 147-179.

3. S. Mac Lane, Categories. For the Working Mathematician, Springer-Verlag Berlin-Heidelberg-New York 1972.

4. C. Ruiz and R. Ruiz, Remarks about the Eilenberg-Zilber type decompo- sition in cosimplicia/ sets, Rev. Colombiana Mat. ~ (1978), 61-82.

5. C. Ruiz and R. Ruiz, Characterization of the set-theoretical geometric realization in the noneuc!idean case, Proc • .Am. Math. Soc. 81 (1981), 321-324.

MATHEMATISCHES lNSTITUTJ LUDWIG-MAXIMILIANS-UNIVERSITÄTJ THERESIENSTRAßE 39J D - 8000 MüNCHENJ GERMANY

- 4.-

By Alfred FRÖLICHER

The purpose of this article is to give a survey of a new and simple approach to the analysis of smooth rnaps, to indicate its good categorical properties, and to show that i t includes classical calculus based on Banach spaces.

We shall equip for classical smooth manifolds V and W (in fact V,W can be more general) the function space c=(V,W) with a smooth structure. This is not possible i f one restricts

to Banach rnanifolds. Many attempts have been made to generalize the notions "differentiable" and "smooth" from maps between Banach spaces to maps between more general vector spaces (all vector spaces are supposed overIR). In infinite dimension, the differentiability of a map between vector spaces depends on some additional structure of the vector spaces, as e.g. a norm, a

topology, a pseudotopology, a bornology. Since in the traditional set-up the derivative of a map f : E

1 - E

2 involves the function

&pace L(E 1,E

2), thare were problems concerning the existence o:[

a function-space structure with the property that the composite.

of two twice differentiable functions is again twice differen- tiable. The best way to overcorne these difficulties_ was by using vector spaces over a cartesian closed category. So pseudotopolo- gies were used in [l],[4], compactly. generated topologies in [16]

bornologies in [ 3 ], are-wise determined spaces .in fl0]. We try to give a natural answer to the question: which is the

"good¹¹ category of vector spaces for which calculus should be developped, and we shall see that the explicit description of these vector spaces can be given in many different ways.

The author wishes to thank A. Krieg~ F.W. Lawvere, L.Nel and S.H. Schanuel for their very helpful suggestions and discus- sions.

Categories generated by sets of maps

In [ 5], any monoid M of maps of any fixed set B into itself was used to generate a category _%M, and a condition on M f o r ~ tobe cartesian closed was given. Instead of using a submonoid M of BB one could take any subset M of BB, but the category ~ depends only on the submonoid generated by M.

However, as pointed out by Lawvere, Schanuel and Zame [13] i t is useful to start with a set M of maps between two possibly different basic sets A and B, i. e. M c BA.

~

is also cal·led the Petermann category of M.

A M-structure on a set E i s a couple (C,F} where CcE ^A and F c BE such tha t for c : A - E resp. f : E - B one has c E C iff foc EM for all f E F and similarly f E F iff f ⁰c E M for all c E C. The obj ects of ~ are triples (E ,C, F) where E is a set and (C,F) a M-structure on E; the morphisms from

(E1

,c

₁,F

1) to (E 2

,c

2,F

2) are those maps ~ :E

1 - E

2 which sa- tisfy

~*

(C

1) c c

2 or equivalently

~*

(F

2) c F

1• All M-structures on a given set E forma complete lattice with respect to the order "finer" (for which one requires the identity map of E to be a morphism). Any set F

0 c BE generates a M-structure (C,F) for which C == {c : A - E; f ⁰ cE MV fE F

0}; i t is the coarsest satisfying F

O c F. Similarly for any gi ven c

0 ^c EA. In %M all limits and colimits exist, they are obtained explicitly by putting on the (co-)limit of the underlying sets the initial resp. final M-structure.

We shall use one example where A ~ B because i t is very useful for investigating the example M == C^{00 (} lR, m) (cf [13]) , namely A == JN, B ==

m,

^{M == 9.,«>} (the set of bounded sequences of lR) •

«J

9., -structures are related to Kolmogorov-bornologies (cf[8]).

Examples of . .t 00 -structures are obtained by taking as set E the underlying set of either a metric space or a locally convex space and by defining

C - {c: ^{JN -} E; c(JN) is bounded},

«J

F - {f : E - lR; f ⁰c E 9., Vc E C}.

One verifies that (C,F) is a i -structure on E. CO

For claculus, examples where M consists of certain (differentiable) maps lR --+ lR have to be investigated, e. g.

k ^{CO • '} k,l

C( lR, lR) ;C ( lR, lR) ;C ( lR, lR) ;Lip( lR, lR) ;C ( lR, lR) ;

B ( lR, lR} ; Bi ( lR, lR) • Here "Lip" refers to the Lipschi tz con- di tion with exponent 1 which has to' be satisfied locally; ck,l means k times differentiable and such that the k-th derivative

lies in Lip( lR, lR) ;B(resp B

1) means bounded (resp. locally

bounded) functions. B ( lR,IR) and B,.o( lR,IR) yield the same

%

^{as R,cx}

· e

-M

C7/ k

For k EJN, the category5L?M generated by M = C ( lR, JR) turns out tobe not very useful, since for n>l

( lRn, ck (

m.,

lRn) , ck ( lRn, lR ))

fails tobe an object. However, the above triple is an object for k

=

^eo and also if one replaces ck by ck,l; this is in fact essentially equivalent to the famous theorem of Boman [ ·2 ] saying that a real function on lRn is Cco (resp. Ck' 1) if i t is cco (resp. Ck'l) along each smooth curve oflRn. Using partitions of unity one deduces easily that the same holds iflRn is re- placed by a paracompact C⁰⁰-resp. Ck'1

-manifold. I t follows that these manifolds form full subcategories of the respective category ~ - In order to show the same for many infinite di- mensional manifolds one uses the generalization of Boman's theorem given in [ 6 ] ; we restrict to the smooth case.

Theorem 1. Fora map ~ : E

1 ^--+ E

2 between Frechet spaces the following conditions are equivalent:

1.

2.

3.

4.

~ is smooth,

~ * ( C^{00 (} lR, E l) ) c Cco ( lR, E 2) ,

~*(E_CO

2

^)cCco(E1_CO

^,m.),

_CO

C (E2, lR )⁰~°C ( lR,El) c C ( lR, lR) .

Though there are many notions of differentiabi.li ty of a map between Frechet spaces, they almost all yield the same infinitely differentiable maps; this is the meaning of "smooth"

in (1), which in particular is the usual one in case of Banach spaces. In (3), E

2

notes the topological dual of E

2. As corollary one obtains :

00 00

Theorem 2. (V

,c (

lR, V), C (V, lR )) is an object of the category

(2? 00 00

2.. generated by C ( lR, lR) for a) V any Frechet space

b) V any paracompact C⁰⁰-manifold over a Frechet space E which has enough smooth functions (i.e. to any 0-neighborhood U there exists f E C⁰⁰ (E, lR) with f (0)

=

¹ and f (x)

=

^{0 for}

xt

^U; e.g. if ^E is nuclear).

It follows that these Frechet manifolds forma full subcategory of <jf^00•

Cartesian closedness

We assume that the given set Mc B A contains all constant maps

A - B. Then the one point set has a unique M-structure yielding an object s which is final i n ~ , hence Srr-~Id. S yields also a representation of the forgetful functor V from ~ into Set : V= ~ ( S , - ) .

Let us suppose now that the category%=

%~

is cartesian closed. Then there exists a functor H:

;jfx%

⁰P - .Jfsuch that

(1) 2[(X,H (Y, Z)) ~ }f(XTTY, Z).

Hence for X=S one has VH(Y,Z) ~ %(Y,Z) and i t is easy to show that by modifying H if necessary one can obtain VH(Y,Z) = %(Y,Z) and also the bijections in (1) tobe of the form f -

f

^where

f(x,y) - f(x)(y).

The basic sets A resp. B have natural M-structures of the form (CA,M) resp. (M,FB) where CA and FB are determined by

- 8 -

M(cf. the axioms of a M-structure). We note A resp. B the objects of%so obtained and we remark that for any object X :::

(Ex, ex,

F

x>

one has

ex :::

%CA, X) and F X :.:

%

^(X,

B) .

^The

(modified) functor H yields the obj,ect H (A,B); its underlying set is M, and we denote its M-structure by (r,~). Since

r :.:

jf°(A,H (A,B)) ;; Jt'(ATIA,B) we have (2)

(3)

-

^'

r - {y A - ^M; yo(cr,-r)EM Vcr,-rECA};

~ - . {c,p : M - B; c,p⁰ y E M Vy E r}.

Hence cartesian closedness of ~ implies the following conditioi on M (r and ~ being determined by (2) resp. (3)) :

(4) y : A - M

j=>

^yEr

c,poy EM Vc,p E ~

One easily shows that the converse holds; hence

Theorem 3. ~ 1 is cartesian closed if and only if M satisfies The verification of (4) is not difficult for M:.: C(JR, lR in this case ~~ is isomorphic to the cartesian closed category of imbeddable are-wise determined spaces. (4) also holds for M:.: i ; the verification is almest trivial. 00

The case M:.: C⁰⁰(JR, JR) is more difficult and justifies some further comment. By Boman's theorem one gets in this case

00

~

^ex, 2

r - {y : lR - C ( JR, lR ); y E C ( lR , lR )}.

Hence ~ - { c,p : C ( ⁰⁰ lR, lR ) - lR; c,p⁰ y E C ( ⁰⁰ lR, lR ) Vy E r}.

One knows (cf [ 15]) that ~

1. ::: {c,p E ~; c,p linear} consists exactly

in

of the distributions of compact support on JR. If one wants to verify (4), i t is useful to show that one has even

(4') y : A - M \ _=>

y

E

r

c,poy E M Vc,p E ~lin

The elegant proof due to Lawvere, Schanuel and Zame proceeds as follows : one first shows that the strenger property (4') holds for M = i , using the uniform boundedness principle. Using this (X)

(X)

one gets the result for M = C (

:m, :m)

by means of the following 1 enuna (cf • [ 7 ] and [ 13 ] ) .

Lemma. A function f : m.² - +

:m

is SJTIOOth iff all its partial difference quotients are ~oo-morphisms (i.e. are bounded on bounded sets).

Smooth spaces.

A srnooth space is an object of the category

w-»

generated

-

(X)

by C (

:m, :m) .

As an example of the consequences of the cartesian closedness of

w~

we rnention actions of smooth groups. A smooth group Gis a srnooth space with a compatible group structure

-

(i.e. such that the group operations are smooth). For any smooth space X, the set Diff(X) of diffeomorphisms (i.e. ~=-isornorphism

-

^.

of X becornes a smooth group if equipped with the srnooth structure initial with respect to the two maps i , j : Diff(X) - H(X,X)

where i(f) =fand j(f) = f-1

• An action of a srnooth group Gon a smooth space Xis an action GTIX-+ X which is smooth. Cartesian closedness of

W

⁰⁰ irnplies the

-

Proposition. There is bijection between the actions of Gon X and the smooth homomorphisms G-+ Diff(X). This bijection is in fact a diffeomorphism with respect to the natural smooth structur of the respective function spaces.

If Xis in particular a Banach manifold, one can show that Diff(X) becomes a subspace of H(X,X). This means that one can prove that the inversion map j is smooth with respect to the subspace structure by using an appropriate inverse function

theorem.

For an arbitrary smooth space X= (E,C,F) one can intro- duce tangent and cotangent spaces. Let p E E and Cp =={c E C;c (0) =p}.

One defines on C resp. F equivalence relations ~resp.~ as

p p

follows:

1~

:e

- -

^(f0c1) • (0) - (f1 °c) • (0) -

(f⁰c2). (0) (f2^{°c) •} (0)

Vf E F;

Vc E C •

p T X : = C /~ resp. TPx: = F/~ are called the tangent resp.

p p p

cotangent space of X at p. Frorn cartesian closedness i t follows that Fand hence also TPx are in a natural way smooth vector spaces. T X can be irnbedded in a (smooth) vector space (e.g.

p

into the dual of TPX, or into a vector space of derivations of F); but i t is not always a vector space as the following

e.xarnple shows : X consists of two intersecting lines ofIR2 and carries the subspace structure, p is the intersection point.

One can consider various subcategories of ~ ^00, e.g. by imposing the condition that for each p, T Xis a vector space,

p

and that Xis locally isornorphic to T X. Here, locally rneans with p . respect to the topology final with respect to the smooth curves.

This topology has the good property that the cond~tion for a map to be a ~ ⁰⁰ -rnorphisrn is of local. character and that inclusions of open subsets of an object belong to initial morphisms.

Calculus for convenient vector spaces

It is natural to develop calculus for vector spdces over the cartesian closed category

<??,

i.e. for smooth vector spaces.

As in other cases (topological or compactly generated or

pseudotopological vector spaces), some further restrictions have tobe imposed in order to obtain theorems one wants to have. One restriction is in the direction of local convexity in order to have enough linear differentiable functions; the other is a completeness condition.

For any smooth vector space Ewe put E' = { i : E ^{- I R ;} i linear and smooth}

E' with its universal srnooth structure is called the dual of E.

The canonical map of E into the bidual is smooth.

Definition. A smooth vector space Eis called convenient if i) E' separates points;

ii) E' generates the smooth structure of E;

iii) E' yields a complete bornology on E.

Any Frechet space with its natural smooth structure is convenient. In fact, its dual coincides with its topological

dual, so i) holds. From theorem 1 follows ii). The bornology in i.

is formed by the subsets B of E for which t(B) is bounded for all 1 E E' . For a Frechet space these subsets are the saroe as the bounded subsets of Ein the usual sense. But for a locally convex space, bornological completeness (also called local completeness in [9]) is in general rnuch weaker than usual cornpleteness. This conf irms a remark of Hogbe who said (cf.• [ 8]) that nevertheless

"for a great many problems bornological cornpleteness turns out to be enough" .

The condi tions· i) , ii) , iii) can be expressed equi valently by

i') The canonical map E - E" is injectif;

ii') The smooth structure of Eis initial with respect to E - - E" ;

iii') The canonical map E - T

0E is surjectif.

The map in iii') associates to a E E the tangent vector at O represented by the curve Ar-- A•a.

Theorem 4. Fora %-morphism f : E1 - E

2 between convenient srnooth vector spaces one has:

1) For all a,hE E

1, df(a,h) 2) df(a,-) is linear;

- lirn

,A• Ü

f(a+>.h)-f(a)

A ^exists;

3) d f : E 1TIE

1 - E

2 is also a ~-~orphism and E 1TIE

1 is convenien 4} The map f - df is a (linear) <g7-morphism between the res-

pective function spaces.

-

- 12 -

i

t

The limit in (1) is in the streng sense of Mackey- convergence; hence also for the weak topology induced on

E2 by E

2.

According to (3) one obtains higher order differentials

n n-1

d f

=

d(d f) and one can prove for these the usual symmetry properties. Instead of working with df one can use

f' :

Et -

^L(E₁^,E₂) defined as f' (a) (h) = df (a,h) and f(n) = (f(n-l) '; L(E

1,E

2) is again convenient. (4) implies that the smooth structure of C (E00

1,E

2) is initial with respect to the maps d , n=O,l, .•. n

By showing that the conditions for convenient vector spaces are inherit~d by the function spaces C (E00

1,E

2) one obtains the first part of the following

Theorem 5 a) The category of convenient smooth vector spaces with the smooth maps as morphisms is cartesian closed.

b) The category of convenient smooth vector spaces with the linear smooth maps as morphisms has a tensor_product making i t a symmetric monoidal closed category.

In (11], A. Kriegl introduced certain locally convex spaces as the "right spaces for analysis in infinite dimension"

and showed many interesting and useful properties of them; cf.

also (12]. It is reassuring that these spaces can be identified with the convenient smooth vector spaces; cf. (1) and (4) in the following

Theorem 6. The following spaces can be identified with each other : 1)

2)

3) 4)

The convenient smooth vector spaces;

The vector spaces with compatible convenient i 00 -structure;

The vector spaces with compatible convenient Ck,l(lR, lR)- structure;

The separated locally convex spaces which are bornological and bornologically complete;

5) The separated convex bornological vector spaces which are topological and bornologically complete;

6) The dual pairs (E,E') for which the bornology of E determined by E' is complete and for whic~ E' satisfies one of the

following equivalent conditions : the M-structure (C,F) of E generated by E' has the property FL.

=

E' for

00 in

a) M - C ( JR, JR) , b) M - Ck ' l ( JR, JR ) , c) M - 9.^00•

In 2) and 3), convenient is defined as in the case of smooth vector spaces.

Using a general result of L. Nel (14] one can show that for any smooth space X the space FX of smooth functions X - J R is the dual of a smooth vector space LX; in fact X-+ LX belongs to a coadjoint to the forgetful functor from smooth vector spaces to smooth spaces. The dual of any smooth vector space being

convenient, FX is always convenient. Moreover, FX being a dual

FX

is a tri-dual and one therefore gets a retraction to the canonical map FX - F~. Hence FX is reflexive iff LX is

Mackey-dense in FX. For X a classical manifold, FX consists of the distributions of compact support on X and LX of the clernen- tary distributions (i.e. linear combinations of evaluations at points), andin this case A. Kriegl could show the respective density and hence the reflexivity of FX (serninar on smooth functions, University of Geneva, not published).

To appear in the Proceedinqs of the workshop on cateqories and foundations of continuum physics, Buffalo, May 17-21, 1982.

- 1.:1 -

R E F E R E N C E S

[l] A. Bastiani : "Applications differentiables et varietes differentiables de dimension infinie". Journal d'Analyse mathematique XIII, p. 1-114.

[2] J. Boman : "Differentiability of a function and of its

compositions with functions of one variable", Math. Scand.20, 1967, p. 249-268.

(3 ]

[4 ]

(5 ]

[6 ]

J.F. Colombeau: "Differentiation et Bornologie". These, Universite de Bordeaux I, 1973.

A. Frölicher and W. Bucher: "Calculus in Vector Spaces without Norm", Lecture Notes in-Math. 30, Springer 1966.

A. Frölicher "Categories cartesiennement fer.mees en- gendrees par des monoides", Cahiers de Top. et Geom. diff.

XXI/4, 1980, p. 367-375.

A. Frölicher : "Applications li~ses entre espaces et varietes de Frechet" C.R. Ac. Sei. Paris 293, 1981, p. 125-127.

(7] Haupt, Aumann, Pauc : "Differential- und Integralrechnung Bd II, Göschens Lehrbücherei Band 25, de Gruyter 1950.

(81 H. Hogbe-Nlend: "Bornologies and functional analysis", Mathematics Studies 26, North-Holland 1977.

(9] H. Jarchow: "Locally convex spaces", Teubner 1981.

(10] A. Kriegl: "Eine Theorie glatter Mannigfaltigkeiten und Vektorbünder; Dissertation, Wien 1980.

unendlich-dimensionalen", to appear in Monatshefte für Mathematik.

(12) A. Kriegl : "Eine kartesisch abgeschlossene Kategorie glatter Abbildungen zwischen beliebigen lokalkonvexen Vektorräumen", preprint 1982.

(13) F.W. Lawvere, S.H. Schanuel and W.R. Zame: "On C⁰⁰ Function Spaces^{11 ,} Preprint 1981.

(14) L. Nel: ¹¹Convenient topological Algebra and reflexive objects^{11 ,} iri Lecture Notes in Math. Vol. 719, p. 259 ff.

(15] N. van Que and G. Reyes : "Theorie des distributions et theoremes d'extension de Whitney", Expose 8, Geom. diff.

synth. fase. 2, Rapport de Recherches DMS 80-12, Universit1 de Montreal 1980.

(16] U. Seip: ¹¹A convenient Setting for Smooth Manifolds^11• J. of pure and appl. Algebra 21, 1981, p. 279-305.

- 16 -

Section de Mathematiques Universite de Geneve 2-4, rue du Lievre CH-1211 GENEVE 24

e

C. Greither Universität München

Dem Vortrag lag eine gemeinsame Arbeit mit R. Haggenmüller (m.m.38) zugrunde.

Wenn R c S kommutative Ringe sind und G c EndR-Alg (S) Z

endliche Gruppe, so heißt R c S §-galoissch, wenn S E: R-Mod e.e.projektiv ist und EndR(S) freier S-Linksmodul mit

Basis G ist.

Man erhält die Kategorie Gal(R,G) der G-Galoiserweiterungen S :> R . Die Morphismen sind die G-invarianten R-Algebren-

homorphismen. Alle Morphismen sind Isomorphismen. Gal(R,G) = Menge(!) der Isomorphieklassen von Gal(R,G) ist punktierte Menge (

*

= [ RG]) und für abelsches G nach HARRISON eine

abelsche Gruppe. Beispiel: Gal (JR, ?l /2:?l ) = { [ <r], [ JR x JR]}

=

^{?l /2:?l •}

Gal(-,-) ist ein Bifunktor. Wir fragen: Wann induziert R c R[ X] (=Polynomring in einer Variablen) einen Iso

Gal(R,G)-Gal(R[X],G)? Oder, anders gesagt: Wann ist jede G-Erweiterung von· R[X] erweitert von R?

Mit geometrischen Methoden sieht man zum Beispiel schnell, daß stets Gal(CC(X] ,G) =

*

= Gal(CC,G) ist. Dies motiviert

die

VERMUTUNG: k=k ,chark {' !GI =;> Gal(k[X],G)

= * .

1 1.

HAUPTSATZ. Sei G abelsch. Dann ist Gal(R,G)~Gal(R[X],G) ~ genau dann wenn n ^:= !GI in Rred kein Nullteiler ist.

Zum Beweis zeigt man zuerst allgemein Gal (R, G) ;; Gal (R d' G)

re

( CIPOLLA, KERSTEN) , reduziert auf G

=

^{2Z /p2Z} und verwendet u.a.

eine Mayer-Vietoris-Sequenz für den Funktor Gal(-,G) . (Si ehe m. m. 3 8 ) .

Weiter gilt:

SATZ. Falls R eine normalem-Algebra ist, so ist auch für nichtabelsches G : Gal(R,G);; Gal(R[X] ,G) .

C. Greither

Mathematisches Institut Universität München Fed. Rep. Germany

- 18 -

Michael Höppner

Universität-GH-Paderborn, 0-4790 Paderborn, Fed. Rep. Germany

Abstract The theory of abelian groups in exploited for the study of diagrams over a locally well-ordered poset. This method yields a complete list of indecomposables as well as some direct de- compositions into cyclics and a characterization of all

(L-)

pure-

injectives. Moreover, we show the existence of stacked bases for diagrams and give an analogue of a freeness criterion interesting in the theory of mixed groups.

Q.

Introduction

As was pointed out by Mitchell

[_1l], diagrams

over partially ordered sets with values in some category of modules may be

considered as generalized modules themselves. In this note we want to exemplify this thesis in another detail by exploiting the well- elaborated theory of

abelian groups

(e. g. as exposed by Kaplansky [11] or Fuchs [iJ) for the study of certain diagrams. Tabe precise, let I be a

linear locally well-ordered

set, i. e. for every

i EI let the cone of successors il

⁼

{j EI

I

i < j} be well-

ordered and let F-Mod be the category of vectorspaces over some

hereditary

and

ZocaZZy noetherian,

i.e. has global dimension one and possesses a generating set of noetherian objects. (See [1,f]

for a general description of such categories of diagrams.) As was already noted in [10] for the special case of diagrams over the ordered set of integers

71.,

a lot of group theoretic concepts may be recovered in this setting:

In the first section of this note we introduce the notions of divisible diagrams and of basic subdiagrams. This yields decompo- sitions into direct sums of cyclics for diagrams of locally bounded height and shows the indecomposable diagrams tobe constant an subintervalls of I. An analogue of Kulikov's theorem proves the pure global dimension of [I, F-Mod] tobe bounded by one. The second section gives a characterization of pure-injective diagrams in terms of a completeness property and shows that the reduced

~-pure-injective diagrams (i. e. diagrams with pure-injective copowers) are just the diagrams of locally bounded height. In the final section we give an analogue of a freeness criterion of Griffith [6] which is important for the study of mixed groups.

Moreover, we establish the existence of stacked bases for diagrams parallel to a theorem of Cohen and Gluck [4] which extends the well-known structure theorem for finitely generated abelian groups.

1. Direct sums of cyclic diagrams

Note, that a locally well-ordered set I has to contain a coun-

- 20 -

intervalls [in+1

0

As usual we take (SiF)i E 1 as a generating set of small projectives for [I,F-Mod], the diagrams SiF being constant with value F and identic morphisms on iI (the successors of i) and being zero else. By our assumtions we have {SjF

I

i < j} for the lattice of subdiagrams of SiF' therefore our diagrams should in fact behave like modules over a

(complete) discrete valuation ring.

Note, that a

cyclic

diagram is just a quotient SijF

=

SiF/SjF for some i < j. So the cyclics are even cyclically presented, hence pure-projective.

In accordance with abelian group theory we call a diagram D: I

^•

F-Mod

divisible (torsionfree)

iff d .. : D.

^•

D. is an

J 1 1 J

epimorphism (monomorphism) for every i < j. This gives rise to the usual torsion theories of (divisible, reduced) and (torsion,

torsionfree) diagrams. Note, that the divisible (torsionfree) diagrams are just the injectives (flats) by the injectivity (flat- ness) criterion in

[.1.Q].

Since every diagram D splits off a maximal divisible subdiagram div D whose structure is known by

[~], we may restrict ourselves to reduced diagrams when dealing with further structure theory.

Lemma 1.1.

Every non-divisible diagram

D: I

^•

F-Mod

has a cyclic direct swnmand.

Proof. Let dji: Di

^•

Dj be a non-epimorphism and suppose that j

diagram D' generated by {Ok

I

k

<:

i} is divisible in [iI,F-Mod]

and causes a direct decomposition D

=

0

¹ ~

0

¹¹

in [I,F-Mod].

II

Moreover, every non-trivial x E Dj generates a cyclic direct summand.

Because of lemma 1.1. we have the following complete list of indecomposable diagrams: 1) cyclic torsionfree diagramms SiF' 2) cyclic torsion diagrams S .. F, and 3) divisible diagrams 6.JF

lJ

which are constant with value F an an arbitrary left-open

•

intervall J of I (see the structure theorem for indecomposable injective diagrams in [~]). Hence, the indecomposables are just the constant diagrams an arbitrary subintervalls of I.

By transfinite induction we deduce from lemma 1 .1. that every diagram D has a

basic suhdiagram

0

^1,

i. e. 1) 0

¹

is a maximal direct sum of cyclic subdiagrams such that 2) 0

¹

is pure in D and 3) 0/0

¹

is divisible. As is known for abelian groups (see

[_§_]),

basic subdiagrams are unique up to isomorphisms only.

For the following proposition note, that for a diagram D and a non-trivial element

XE 0.

1

the

height

of

X

is defined by ht x

=

min { j

^<;

i

1 X E

im d .. }

lJ

or ht

X = 00

if the minimum does not exist. Moreover, D is called of

(ZocaZZy) bounded height

iff there is a lower bound for the heights of non-trivial elements of D ( every D.).

1

- 22 -

D is a direct swn of cyclics.

Proof. For the first case remember that there is an obvious notion of duality for diagrams D by Di

⁼

HomF(D;,F) and that D is pure-injective iff the canonical embedding D

^•

D** splits. So by assumptions the basic subdiagram

0¹

of

D

splits off, moreover

o;o•

⁼

div o

⁼

o.

For the second case just note that

0/0¹

has tobe divisible as well as locally of bounded height, hence trivial.

Especially, finitely generated diagrams are direct sums of cyclics. Therefore it is easy to show that a subdiagram

U

of an

arbitrary diagram D i s pure in

^{D iff d .. U. = U.} n d .. D.

lJ J 1 lJ J

every j..; i. The following is known as Kulikov

¹

s theorem in abelian group theory (see [~]).

Proposition 1.3.

A diagram

D

is a direct swn of cyclics iff there is a filtration O =

o

^{0 c}

o

^{1 c ··· c}

On

c ··· c D -with

for

Cl

D = u D ⁿ such that every

Dn

has bounded height taken in

D.

n E lJ

Proof. The necessity of such a filtration follows from the

existence of a countable initial subset in every locally well-

ordered set I. For sufficiency, we may assume the diagrams On to

be pure in D by considering 5n with

i.

Therefore every On is a direct summand of on+1, since the quotient

0

n+1

10

n is of bounded height hence a direct sum of cyclics and pure-projective. Moreover, O

⁼ m

on+ 1;on by a

n E :N

well-known induction argument.

Corollary 1.4. If

a diagram is a direct sum of cyclics then so is every subdiagram.

•

Especially, the pure global dimension of [I, F-Mod] is bounded by one. It is now an easy exercise to show that the pure global dimension is zero iff I is well-ordered. For further information about pure dimensions of diagrams over linear ordered sets we refer to [2].

2.

(L-)

pure-injective diagrams

By the splitting criterion mentioned above, we have already encountered diagrams of pointwise finite dimension as well as

diagrams of bounded height (see the remark on pure global dimension) as being pure-injective. Moreover, as is known for abelian groups (see [5]) every diagram O has a

pure-injective hull

P(O). The following is known as Sasiada's theorem in group theory (see [5]).

- 24 -

Proof. It suffices to show that P(D)/D is divisible in general.

Suppose the contrary then this quotient has a cyclic direct summand C which can be lifted to P(D) by purity such that D

n C = 0.

Hence P(D) would not be a pure-essential extension of D.

• ·

By considering the long exact sequence for Pext we obtain the following closure property.

Corollary 2.2.

Let

D

be are pure subdiagram of the pUPe-injeotive diagram

P

and iet D c

U

c

P

such that

U/D

⁼

div(P/D).

Then

U is pure-injective. _•

For any diagram D, we define the

d-adic topoZogy

by dinstinguishing the family

( d .. D.) . .

lJ J J ,( l

as a base of open neighbourhoods for every D;· This makes the D; topological

vectorspaces with continuous morphisms dij" It can be proved along the lines of abelian group theory that D is reduced and pure- injective iff D is hausdorff and complete in its d-adic topology.

Moreover, in the reduced case pure-injective hulls turn out tobe d-adic

corrrpZetions

(see

[..§_]).

As usual, we call a diagram D

L-pure-injective

if arbitrary direct sums D(X) are pure-injective.

For detailed information about r-pure-injective modules we refer

to

(15].

1) D is reduced and ~-pure-injective 2) D is locally of bounded height

3) D is discrete in its d-adic topology.

Proof. We restrict to

1) => 2).

Suppose there are non-trivial xn E Di such that ht xn+

₁ <

ht xn for every n E ~- Then

(x

₁

,x

_{2 , •••}

,xn,0,0, ••. )n E

~

would be a non-converging Cauchy- sequence in D. (~).

l

The following is a theorem of Baumslag and Blackburn

[1]

in abelian group theory.

Proposition 2.4.

Fora family

(Ds)s ES

of reduced pure-

•

injective diagrams the direct sum ^@

Ds

is pure-injective iff for every i

E I

there is a finite set

^s

^{E S}

S.

l such that ^@

D~

contains elements of uniformly bounded height only.

s E S,S.

l

Proof. Fora pure-injective direct sum

@

Os easy variants of

s E S

the proof of prop. 2.3. 1)

=>

2) assure that for every i EI

l

there are at most finitely many D~ which are not of bounded height and that the rest have tobe bounded in height uniformly. The

converse is clear.

Example 2.5. on

= rr

m < n

S

m,n

F: ?l. • F

-Mod is reduced and pure-

- 26 -

n

<

0

with different exceptional sets S;

=

{i, i+1, ..• , -1, O} for every i <

0.

3.

More group theory

In

[Jl]

Kaplansky listed the five theorem on abelian groups he considered most remarkable. We now give a short discussion of these theorems for diagrams.

Remember, that every projective diagram P: I

^•

F-Mod is

free

[1] ,

i. e.

P = (i) S.

P. wi th we 11-determi ned projecti ve modul es

i

EI

^1-1

P; and the diagrams S;P; defined analogous to S;F. Moreover, by a result of Mitchell [14]

P

is free iff

Pl

1 _ is free for every

l

i EI and I;

= {j I

j < i}. The following is known as a theorem of Griffith [6] in abelian group theory which solves a problem of Baer on

mixed groups.

Proposition 3.1. A torsionfree diagram F is free iff Ext(F,T)=O for every torsion diagram

T.

Proof. Let I~

=

{j

I j-<

i} and let E: 0

^•

X

^•

Y

^• F1

1 ~

^•

0

1

by any extension in [I~, F-Mod). Then

E

is the restriction of an

!

diagram X obtained from X by adding zeros. By assumption

E

splits, hence F1

₁

? is free ardeven

l

freeness.

is free by torsion-

•

There is a result of Cohen and Gluck

[_!]

extending the existence of

staaked bases

in free abelian groups of finite rank. The

analogue for diagrams is very easy because of the structure of projective diagrams. For ease of reference we call the canonical sequence O

•

S.F

•

S.F

•

S .. F

•

0 the

standard resolution

of the

J l l J

cycl ic 5;jF.

Proposition 3.2.

Let

C

be a not neaessarily finite direat sum of ayalia diagrams. Then any projeative resolution

E: 0

^•

Q

^•

P

^• f

C

^•

0

is isomorphia to the direat sum of standard resolutions and a trivial sequenae

O

^•

Q

^{1 •}

Q

^{1 •}

0

^•

0.

Proof. If C ^{= @} S. . F then we may chose an inverse image

t

E

T

¹

tJt

of C under f which is of the form

^ij

S. F and is a direct

t E T ¹

t

summand of P. The associated direct sum of standard resolutions proves tobe isomorphic to

E

up to a trivial complement by some diagram chaseing.

In contrast to abelian group theory Whitehead

¹

s problem for diagrams can be decided in the negative as was shown in [1.., lQ_].

We remark that the three problems above can be dealt with in the

- ')Q -

•

As for the remaining two theorems mentioned by Kaplansky, note that Specker's problem seems tobe rather artificial for diagrams, indeed the answer to this problem depends an the way of translating it into our setting. An analogue of Ulm

¹

s theorem has tobe stated carefully and is not yet obtained: note, that there are countably generated reduced torsion diagrams with trivial socle, hence the usual Ulm-Kaplansky invariants da not provide a complete set of invariants.

REFERENCES

[ 1] G. Baumslag and N. Blackburn: Direct summands of unrestricted direct sums of abelian groups. Archiv Math.

1Q_

(1959)

403 - 408

[ 2] H. Brune: Same left pure semisimple categories which are not right pure semisimple. Comm. Alg. 7 (1979) 1795 - 1803

[ 3] H. Brune: On projective representations of ordered sets.

to appear

[ 4] J. Cohen and H. Gluck: Stacked bases for modules over principal ideal rings. J. Alg. 1.! (1970)493 - 505

[ 5] L. Fuchs: Infinite Abelian Groups. Academic Press, New York, 1970

- 29 -

of Baer. Trans. Amer. Math. Soc. 139 (1969) 261 - 269

[ 7] M. Höppner: On the freeness of Whitehead diagrams. to appear [ 8] M. Höppner: A note on the structure of injective diagrams.

to appear

[ 9] M. Höppner und H. Lenzing: Projective diagrams over partially ordered sets are free. J. pure appl. Alg •

.f.Q_

(1981) 7 - 12 [10] M. Höppner und H. Lenzing: Diagrams over ordered sets: A

simple model of abelian group theory. Abelian Group Theory (ed. R. Göbel and E. Walker), Springer LNM 874 (1981) 417 - 430

[11] I. Kaplansky: Infinite Abelian Groups. University of Michigan Press, Ann Arbor, 1969

[12] I. Kaplansky: Five theorems on abelian groups. Topics in Algebra (ed. M. Newman), Springer LNM 697 (1978) 47 - 51 [13] B. Mitchell: Rings with several objects. Advances Math.

~

(1972) 1 - 161

[14] B. Mitchell: A remark on projectives in functor categories.

J. Alg. 69 (1981) 24 - 31

[15] W. Zimmermann: Rein injektive direkte Summen von Moduln.

Comm. Alg. l (1977) 1083 - 1117

- 30 -

by

Rudolf-E. Hoffmann

(Preliminary Communication) Introduction

In [sc

2] (2.12), D.S.Scott showed that the continuous lattices, invented by him in his study of a mathematical theory of computation [sc

1], are precisely - when they are made into topological spaces via the Scott topology - the injective T

0-spaces, i.e. the injective objects in the cate- gory

!o

^{of T}₀-spaces and continuous maps with regard to the classof all embeddings. Moreover, the sort of morphisms be- tween continuous lattices, Scott con~idered, are precisely the continuous maps with regard to the respective Scott to- pologies. These are fairly ~-Hausdorff topologies. (In- deed,the Scott topology induces the partial order of the lattice L via x

s

y iff x E cl {y} , the "specialization order" of the topology; hence L is Hausdorff in the Scott topology, iff L has cardinality one.) In topological alge- bra, compact Lawson semilattices (= compact Hausdorff topo- logical A-semilattices such that the A-preserving continuous maps into the unit interval, with its ordinary topology and the min-semilattice structure, separate the points) with a unit element 1 have attracted considerable interest. In

[HS], K.H. Hofmann and A.R. Stralka essentially proved that they are precisely the continuous lattices; their (compact Hausdorff) topology is uniquely determined by the lattice-

structure: i t is called the CL-topology or the Lawson to- pology of the continuous lattice - cf.[C] VI-3.4. (J.D.Law- son [L

1] showed that a semilattice admits at most one com- pact Hausdorff topology making i t into a topological semi- lattice.)

Interest in continuous posets is more recent than that in continuous lattices themselves. It was primarily initi- ated by theoretical computer scientists ([Sm], [Ma]). But soon continuous posets eguipped with the Scott topology were recoqnized as a significant class of topological spaces: They are the projective sober spaces (R.-E. Hoff- mann [H

6] 2.19), i.e. the retracts of the free.objects of a functor, the "specialization order" functor, which natu- rally arises in the study of sober spaces (and, more gene- rally, T

0-spaces). They are the prime spectra of the com- pletely distributive complete lattices (J.D. Lawson [L

2], R.-E.Hoffmann [H

9] 2.5), a fact which establishes a bi- jective correspondence between these two classes of struc- tures. In [H

7

3,

a characterization of continuous posets in terms of adjunctions (between partially ordered sets) has been given.

In [Ba

2] § 2, B. Banaschewski showed that in

'!'.o

^every

space X has a greatest essential extension X • 11.X

(extension = topological embedding) and he noted that the spaces X satisfying

(*) Whenever x E V~ Q(X), the lattice of open sets of X (ordered by inclusion), then there is an open neighborhood W of x in X such that

V n n{cl{z} lz E W} not=~

have an injective hull in T , i.e. 11.X is an injective T

0-

- - - 0

space ([Ba

2] cor. 2, p. 240), i.e. 11.X is - by D. Scott's result [sc

2] 2.12 - a continuous lattice in its Scott to- pology. In [H

6] 3.14, i t is established that the sober spaces satisfying (*) are precisely the cqntinuous posets

- 32 -

uous poset P, the injective hull (P,crp) ^• A(P,ap) has the form

• ( I ( P) , crI ( p) )

for some continuous lattice I(P), and the order-extension e: P • I(P)

is uniquely determined up to an isomorphism. This will be called the injective hull of the continuous poset P 1

) (The strenger assertion - in [Ba

2] cor. 2, p. 240 - that every space with an injective hull in T

-o

satisfies (*) is false, as K.H. Hofmann and M.W. Mislove [HM

2] have observed. In an appendix to this paper we give a corrected treatment explaining also its impact on the results in [H

6] and_[H 9] which are partly in need of reformulation.)

In section 1, we provide an intrinsic characterization of the injective hull of a continuous poset Pin order- theoretic and algebraic terms, the proof of which heavily relies on the universal properties of this concept. Fora continuous poset Panda complete lattice L, an order-em- bedding e: P ^• L is an (the) injective hull of P iff

(i) e[P] is join-dense in L, (ii) e: P ^• L preserves supre- ma of non-empty up-directed subsets, (iii) e: P ^• L pre- serves the way below relation, and (iv) e[P] generates L, i.e. there is no proper subset of L containing e[P] which is stable in L under arbitrary infima and under suprema of non-empty up-directed subsets. None of these conditions can be omitted.

In the following sections, we make a study of (the ana- logue of) the CL-topology ~ on a continuous poset. R.L.Wil- son ([Wi

1, Wi

2]) has dealt with continuous posets which are

1) This terminology sh.ould not be confused with the result of B.Banaschewski and G.Bruns ([BB]§4) that the MacNeille completion is the injective hull in the category Poset of partially ordered sets and isotone maps with regard to the class of all order-embeddings.

compact (Hausdorff) in their CL-topology

s•

Generally, however,

s

need not be compact, but i t is always completely regular Hausdorff (K.H.Hofmann and M.W.Mislove [HM 1 ],

p.243). Indeed, the CL-topology on a continuous poset P i s the trace of the (compact Hausdorff) CL-topology of the injective hull I(P) of P.

Every embedding of a space X into a compact Hausdorff space Y induces a natural Hausdorff compactification of X, viz. the closure of X in Y. We endow the closure C of a con- tinuous peset Pin its injective hull L with regard to the CL-topology ,L with the partial order inherited from L (but not with any topology). The order-extension

p • C

will be called the CL-cornpactification of P. Commenting on an earlier draft of this paper, K.H. Hofmann and M.W. Mis- love [HM

2] gave an exarnple to show that C need not be a continuous peset. Here we show by exarnples that aLIC need not be the intrinsic Scott topology of C with (1) C non-con- tinuous and (2) a continuous peset C with ascending chain condition (= a.c.c~.

A continuous 1,A-sernilattice is cornpact (Hausdorff) in its CL-topology iff i t is a cornplete lattice (hence a con- tinuous lattice). I t results that the CL-cornpactification of a continuous 1,A-semilattice coincides with the injective hull. This leads to another intrinsic characterization of this construction (which recently has led to an interesting application in [H

12

J).

In section 4, we show that the injective hull of an alge- braic peset P i s an algebraic lattice. We also provide a re- presentation of the CL-cornpactification of an algebraic peset.

In section 5, we show that the CL-cornpactification P ^• C of a continuous peset P i s equivalent to the underlying

order-embedding of a "Hausdorff cornpactification" of the lo- cally quasicornpact space (P,ap) (not an ordinary compactifi- cation) obtained by J.M.G. Fell ([Fe

1] § 2, [Fe

2]). Thus i t results frorn [H

10] 3.7.1 that the second factor in the

- 34 -