Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T07:19:01.154Z Has data issue: false hasContentIssue false

Sheaf representations of strongly harmonic rings

Published online by Cambridge University Press:  14 November 2011

Harold Simmons
Affiliation:
Department of Mathematics, University of Aberdeen, Aberdeen AB9 2TY

Synopsis

We describe five different sheaf representations of a ring, all of which are full and four of which are faithful. We give a characterization of strongly harmonic rings, and show that for such rings, the four faithful representations agree.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1985

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Behrens, E. A.. Ring theory (New York: Academic Press, 1972).Google Scholar
2Bkouche, R.. Pureté, mollesse et puracompacité. C. R. Acad. Sci. Paris Sér. A 270 (1970), 16531655.Google Scholar
3Bkouche, R.. Couples spectraux et faisceaux associes. Applications aux anneaux de fonctions. Bull. Soc. Math. France 98 (1970), 253295.CrossRefGoogle Scholar
4Borceux, F., Simmons, H. and Van den Bossche, G.. A sheaf representation for modules with applications to Gelfand rings. Proc. London Math. Soc. 48 (1984), 230246.CrossRefGoogle Scholar
5Borceux, F. and Van den Bossche, G.. Recovering a frame from its sheaves of algebras. J. Pure Appl. Algebra 28 (1983), 141154.CrossRefGoogle Scholar
6Borceux, F. and Van den Bossche, G.. Algebra in a localic topos with applications to ring theory. Lecture Notes in Mathematics 1038 (Berlin: Springer, 1983).Google Scholar
7Dauns, J. and Hofmann, K. H.. The representation of biregular rings by sheaves. Math. Z. 92 (1966), 103123.CrossRefGoogle Scholar
8Hofmann, K. H.. Representation of algebras by continuous sections. Bull. Amer. Math. Soc. 78 (1972), 291373.CrossRefGoogle Scholar
9Keimel, K.. The representation of lattice-ordered groups and rings by sections of sheaves. In: Lectures on the applications of sheaves to ring theory. Lectures Notes in Mathematics 248, 198 (Berlin: Springer, 1971).Google Scholar
10Koh, K.. On functional representations of a ring without nilpotent elements. Canad. Math. Bull. 14 (1971), 349352.CrossRefGoogle Scholar
11Koh, K.. On a representation of strongly harmonic rings by sheaves. Pacific J. Math. 41 (1972), 459468.CrossRefGoogle Scholar
12Lambek, J.. On the representation of modules by sheaves of factor modules. Canad. Math. Bull. 14 (1971), 359368.CrossRefGoogle Scholar
13Mulvey, C. J.. Represéntations des produits sous-directs d'anneaux par espaces annelés C. R. Acad. Sci. Paris Sér. A 270 (1970), 564567.Google Scholar
14Mulvey, C. J.. A generalisation of Gelfand duality. J. Algebra 56 (1979), 499505.CrossRefGoogle Scholar
15Mulvey, C. J.. Representations of rings and modules. In: Applications of sheaves. Lecture Notes in Mathematics 753, 524585 (Berlin: Springer, 1979).Google Scholar
16Pierce, R. S.. Modules over commutative regular rings. Mem. Amer. Math. Soc. 70 (1967).Google Scholar
17Teleman, S.. Représentations par faisceaux des modules sur les anneaux harmoniques. C. R. Acad. Sci. Paris Ser. A 269 (1969), 753756.Google Scholar
18Teleman, S.. Theory of harmonic algebras with application to von Neumann algebras and cohomology of locally compact spaces (de Rham's theorem). Lecture Notes in Mathematics 248 99315 (Berlin: Springer, 1971).Google Scholar
19Tennison, B. R.. Sheaf theory (Edinburgh: Cambridge University Press, 1975).CrossRefGoogle Scholar