Geodesic
Homological dimension of cyclic Banach modules and homological characterization of metrizable compacta
Matematičeskie zametki, Tome 17 (1975) no. 2, pp. 301-305.

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It is proved that if the relative homological dimension of the $A$ module $A/I$ is less than two, then the spectrum (which is not necessarily complementable) of the closed ideal $I$ of the Banach algebra $A$ is paracompact. As an application in the realm of relative homological theory a criterion for metrizability of compacta is obtained. 
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     author = {Yu. V. Selivanov},
     title = {Homological dimension of cyclic {Banach} modules and homological characterization of metrizable compacta},
     journal = {Matemati\v{c}eskie zametki},
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     publisher = {mathdoc},
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     number = {2},
     year = {1975},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1975_17_2_a12/}
}
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Yu. V. Selivanov. Homological dimension of cyclic Banach modules and homological characterization of metrizable compacta. Matematičeskie zametki, Tome 17 (1975) no. 2, pp. 301-305. http://geodesic.mathdoc.fr/item/MZM_1975_17_2_a12/