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UNIFORM LOCAL CONSTANCY OF ÉTALE COHOMOLOGY OF RIGID ANALYTIC VARIETIES
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 23 / Issue 2 / March 2024
- Published online by Cambridge University Press:
- 28 February 2023, pp. 839-896
- Print publication:
- March 2024
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- Article
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We prove some
$\ell $-independence results on local constancy of étale cohomology of rigid analytic varieties. As a result, we show that a closed subscheme of a proper scheme over an algebraically closed complete non-archimedean field has a small open neighbourhood in the analytic topology such that, for every prime number 
$\ell $ different from the residue characteristic, the closed subscheme and the open neighbourhood have the same étale cohomology with 
${\mathbb Z}/\ell {\mathbb Z}$-coefficients. The existence of such an open neighbourhood for each 
$\ell $ was proved by Huber. A key ingredient in the proof is a uniform refinement of a theorem of Orgogozo on the compatibility of the nearby cycles over general bases with base change.
