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We prove a finiteness theorem and a comparison theorem in the theory of\'etale cohomology of rigid analytic varieties. By a result of Huber, for aquasi-compact separated morphism of rigid analytic varieties with target beingof dimension $\le1$, the compactly supported higher direct image preservesquasi-constructibility. Though the analogous statement for morphisms withhigher dimensional target fails in general, we prove that, in the algebraizablecase, it holds after replacing the target with a modification. We deduce itfrom a known finiteness result in the theory of nearby cycles over generalbases and a new comparison result, which gives an identification of thecompactly supported higher direct image sheaves, up to modification of thetarget, in terms of nearby cycles over general bases.

\'Etale cohomology of algebraizable rigid analytic varieties via nearby cycles over general bases