Local invariant cycle theorem: Difference between revisions

Revision as of 10:39, 24 May 2022

Revision as of 10:40, 24 May 2022

Line 4:

In algebraic geometry, Deligne proved the following analog.<ref>{{harvnb|Deligne|1980|loc=Théorème 3.6.1.}}</ref> Given a proper morphism <math>f : X \to S</math> over the spectrum <math>S</math> of the henselization of <math>k[T]</math>, <math>k</math> an algebraically closed field, the homomorphism

:<math>\operatorname{H}^*(XX_s) \to \operatorname{H}^*(X_sX_{\overline{\eta}})^{\operatorname{Gal}(\overline{\eta}/\eta)}</math>

is surjective, where <math>s, \eta</math> are the special and generic points and the homomorphism is the composition <math>\operatorname{H}^*(X_s) \simeq \operatorname{H}^*(X) \to \operatorname{H}^*(X_{\eta}) \to \operatorname{H}^*(X_{\overline{\eta}}).</math>