Extensions of locally compact abelian groups. I

Authors:
Ronald O. Fulp and Phillip A. Griffith

Journal:
Trans. Amer. Math. Soc. **154** (1971), 341-356

MSC:
Primary 18.20; Secondary 22.00

Part II:
Trans. Amer. Math. Soc. (1971), 357-363

MathSciNet review:
0272870

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is concerned with the development of a (discrete) group-valued functor Ext defined on where is the category of locally compact abelian groups such that, for *A* and *B* groups in , Ext (*A, B*) is the group of all extensions of *B* by *A*. Topological versions of homological lemmas are proven to facilitate the proof of the existence of such a functor. Various properties of Ext are obtained which include the usual long exact sequence which connects Hom to Ext. Along the way some applications are obtained one of which yields a slight improvement of one of the Noether isomorphism theorems. Also the injectives and projectives of the category of locally compact abelian totally disconnected groups are obtained. They are found to be necessarily discrete and hence are the same as the injectives and projectives of the category of discrete abelian groups. Finally we obtain the structure of those connected groups *C* of which are direct summands of every *G* in which contains *C* as a component.

**[1]**Jean Braconnier,*Sur les groupes topologiques localement compacts*, J. Math. Pures Appl. (9)**27**(1948), 1–85 (French). MR**0025473****[2]**Phillip Griffith,*A solution to the splitting mixed group problem of Baer*, Trans. Amer. Math. Soc.**139**(1969), 261–269. MR**0238957**, 10.1090/S0002-9947-1969-0238957-1**[3]**Edwin Hewitt and Kenneth A. Ross,*Abstract harmonic analysis. Vol. I*, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 115, Springer-Verlag, Berlin-New York, 1979. Structure of topological groups, integration theory, group representations. MR**551496****[4]**G. Hochschild,*Group extensions of Lie groups*, Ann. of Math. (2)**54**(1951), 96–109. MR**0041858****[5]**G. Hochschild,*Group extensions of Lie groups. II*, Ann. of Math. (2)**54**(1951), 537–551. MR**0043789****[6]**K. H. Hofmann and Paul Mostert,*Splitting in topological groups*, Mem. Amer. Math. Soc. No.**43**(1963), 75. MR**0151544****[7]**Saunders Mac Lane,*Homology*, Die Grundlehren der math. Wissenschaften, Band 114, Academic Press, New York, and Springer-Verlag, Berlin, 1963. MR**28**#122.**[8]**Calvin C. Moore,*Extensions and low dimensional cohomology theory of locally compact groups. I, II*, Trans. Amer. Math. Soc.**113**(1964), 40–63. MR**0171880**, 10.1090/S0002-9947-1964-0171880-5**[9]**-,*Extensions and low dimensional cohomology theory of locally compact groups*. II, Trans. Amer. Math. Soc.**113**(1964), 64-86. MR**30**#2106.**[10]**Martin Moskowitz,*Homological algebra in locally compact abelian groups*, Trans. Amer. Math. Soc.**127**(1967), 361–404. MR**0215016**, 10.1090/S0002-9947-1967-0215016-3**[11]**L. S. Pontryagin,*Topological groups*, Translated from the second Russian edition by Arlen Brown, Gordon and Breach Science Publishers, Inc., New York-London-Paris, 1966. MR**0201557**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
18.20,
22.00

Retrieve articles in all journals with MSC: 18.20, 22.00

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1971-99931-0

Keywords:
Locally compact abelian groups,
topological group extensions,
homological-topological lemmas,
extension functor,
Noether theorem,
totally disconnected projectives,
totally disconnected injectives

Article copyright:
© Copyright 1971
American Mathematical Society