Geodesic
Homology and cohomology groups of commutativeBanach algebras and analytic polydiscs
Glasgow mathematical journal, Tome 42 (2000) no. 1, pp. 15-24

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In this paper we deduce the existence of analyticstructure in a neighbourhood of a maximal ideal M in thespectrum of a commutative Banach algebra, A, from homologicalassumptions. We assume properties of certain of the cohomology groupsH^n(A,A/M), rather than the stronger conditions on thehomological dimension of the maximal ideal the first author has considered inprevious papers. The conclusion is correspondingly weaker: in the previous workone deduces the existence of a Gel'fand neighbourhood with analytic structure,here we deduce only the existence of a metric neighbourhood with analyticstructure. The main method is to consider products of certain co-cycles to deducefacts about the symmetric second cohomology, which is known to be related to thedeformation theory of algebras.1991 Mathematics Subject Classification. 46J20, 46M20. 
Pugach, L.I.; White, M.C. Homology and cohomology groups of commutativeBanach algebras and analytic polydiscs. Glasgow mathematical journal, Tome 42 (2000) no. 1, pp. 15-24. doi: 10.1017/S0017089500010028
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