Mod-p Poincaré Duality for some non-proper spaces in p-adic analytic geometry.
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M7-411
p-adic analytic geometry, as started by J.Tate in 1962, forms a p-adic analogue of complex analytic geometry, where spaces are locally defined by convergent power series over extensions of Q_p. Adapting Grothendieck's constructions for schemes, R.Huber constructed a theory of étale cohomology in this setup, that is very well behaved (admits a six-functor formalism) for l-torsion coefficients, with l prime to p. When l = p, many constructions fail and the situation is more mysterious, but results of P.Scholze, and more recently of L.Mann and B.Zavyalov establish some good properties of cohomology (finiteness of cohomology, Poincaré duality) for smooth proper spaces, that do not hold in the non-proper setup. These results rely on perfectoid techniques, a tool that admits no analogue in algebraic geometry.
In this talk, after recollecting the necessary background on p-adic geometry and étale cohomology, I will present the key ideas behind the results above, before focusing on some of my work on extending Poincaré Duality to some non-proper spaces admitting a nicely controlled geometry, such as Drinfeld's symmetric spaces. No prior knowledge of p-adic analytic geometry is required !