I am interested in Poincaré duality from the point of view of Grothendieck's 6-functor formalism. I am predominantly interested in the proof that Poincaré duality holds in étale cohomology from this perspective. Where is there an account of the most generic situation in which Poincaré duality of an étale cohomology arises?
$\begingroup$
$\endgroup$
8
-
$\begingroup$ Can you clarify (for us non english-speaking natives), what you mean by "the most generic situation" ? $\endgroup$– A.B.Commented Nov 13, 2021 at 23:13
-
2$\begingroup$ As I understand it, Poincaré duality in étale cohomology comes down to the existence of a suitable trace morphism $Tr_f: Rf_{!}f^{\ast}L(d)[2d]\to L$ for a smooth morphism $f:X\to Y$ pure of relative dimension $d$ and a torsion complex $L$ on $Y$. The trace morphism is constructed, by devissage, from the case of a smooth irreducible projective curve $X$ over an algebraically closed field, where it is taken to be the degree map $\deg: Pic(X)\to \mathbb{Z}$ modulo $n$. $\endgroup$– A.B.Commented Nov 14, 2021 at 13:50
-
2$\begingroup$ An easy to navigate reference for this is [Etale cohomology theory, 8.2, 8.3 and 8.5] by Lei Fu. $\endgroup$– A.B.Commented Nov 14, 2021 at 13:54
-
3$\begingroup$ Relevant paper $\endgroup$– Denis NardinCommented Nov 15, 2021 at 9:30
-
2$\begingroup$ If you really want the most general statement, look at Theorem 0.1.4 here (arxiv.org/pdf/1211.5948.pdf). $\endgroup$– David HansenCommented Nov 15, 2021 at 9:51
|
Show 3 more comments