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We prove a universal coefficients theorem for the overconvergent cohomologymodules introduced by Ash and Stevens, and give several applications. Inparticular, we sketch a very simple construction of eigenvarieties usingoverconvergent cohomology and prove many instances of a conjecture of Urban onthe dimensions of these spaces. For example, when the underlying reductivegroup is an inner form of GL(2) over a quadratic imaginary extension of therationals, the cuspidal component of the eigenvariety is a rigid analyticcurve.

Universal coefficients for overconvergent cohomology and the geometry of eigenvarieties