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Homological algebra with locally compact Abelian groups. (English) Zbl 1123.22002

In this paper some homological algebra methods for locally compact abelian (LCA) groups are developed: The category LCAb of LCA groups is not abelian. Here it is proved that the categories of LCA groups and, more generally, of abelian Hausdorff groups are quasiabelian. The authors observe that every LCA group has a canonical filtration of length three and so it generalizes the canonical torsion subgroup of a discrete abelian group and the dual canonical subgroup of a compact abelian group. A smallness condition, called finite rank, for LCA groups is introduced. Further, topological analogues of the fact that divisible abelian groups are injective in \(Ab\) are considered and corresponding statements in LCAb are given. Topological analogues of the fact that every abelian group can be resolved by divisible ones are also presented. Finally a derived Hom-functor is constructed: A bifunctor from the bounded derived category of LCA groups to the bounded derived category of abelian Hausdorff groups is obtained.

MSC:

22B05 General properties and structure of LCA groups
18G10 Resolutions; derived functors (category-theoretic aspects)
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References:

[1] Armacost, D. L., The Structure of Locally Compact Abelian Groups (1981), Marcel Dekker Inc.: Marcel Dekker Inc. New York · Zbl 0509.22003
[2] Balmer, P., The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math., 588, 149-168 (2005) · Zbl 1080.18007
[3] Bourbaki, N., Eléments de mathématique. Livre V: Espaces vectoriels topologiques (1953), Herman & Cie: Herman & Cie Paris · Zbl 0050.10703
[4] Fuchs, L., Abelian Groups (1958), Hungarian Academy of Sciences: Hungarian Academy of Sciences Budapest · Zbl 0090.02003
[5] Fulp, R. O.; Griffith, P. A., Extensions of locally compact abelian groups. I, II, Trans. Amer. Math. Soc., 154, 341-363 (1971) · Zbl 0216.34302
[6] Gelfand, S. I.; Manin, Yu. I., Methods of Homological Algebra (1996), Springer: Springer Berlin · Zbl 0855.18001
[7] Hewitt, E.; Ross, K. A., Abstract Harmonic Analysis, vol. I (1979), Springer: Springer Berlin · Zbl 0115.10603
[8] Iversen, B., Cohomology of Sheaves (1986), Springer: Springer Berlin · Zbl 1272.55001
[9] MacLane, S., Categories for the Working Mathematician (1971), Springer: Springer New York
[10] Morris, S. A., Pontryagin Duality and the Structure of Locally Compact Abelian Groups (1977), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0446.22006
[11] Moskowitz, M., Homological algebra in locally compact abelian groups, Trans. Amer. Math. Soc., 127, 361-404 (1967) · Zbl 0149.26302
[12] Neeman, A., Triangulated Categories, Ann. of Math. Stud., vol. 48 (2001), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ · Zbl 0974.18008
[13] Schneiders, J.-P., Quasi-abelian categories and sheaves, Mém. Soc. Math. Fr. (N.S.), 76, 1-134 (1999)
[14] Toën, B.; Vaquié, M., Under Spec Z, arXiv: · Zbl 1177.14022
[15] Toën, B.; Vezzosi, G., Homotopical Algebraic Geometry II: Geometric stacks and applications, Mem. Amer. Math. Soc., in press · Zbl 1145.14003
[16] Verdier, J.-L., Catégories dérivées, (SGA 4 1/2: Cohomologie étale. SGA 4 1/2: Cohomologie étale, Lecture Notes in Math., vol. 569 (1977), Springer: Springer Berlin), 262-311
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