I worry that I'm misinterpreting the mathematical content of the following sentences from Deligne's "Weil I" paper.
I include each sentence in French, followed by my translation of it. Please let me know if I should fix anything (or if it's all good). In particular, let me know if "relevable" has a better English translation (in the context of algebraic geometry) than "liftable". Thanks in advance for any help!
- Dans un article faisant suite à celui-ci, je donnerai divers raffinements des résultats intermédiaires, et des applications, parmi lesquelles le théorème de Lefschetz «difficile» (sur les cup-produits itérés par la classe de cohomologie d'une section hyperplane). (p. 273)
In an article following this one, I will give various refinements of intermediate results, and applications, among which is the "hard" Lefschetz theorem (on iterated cup-products with the cohomology class of a hyperplane section).
- Se basant sur des idées de Washnitzer, Lubkin a créé une variante de cette théorie, valable seulement pour $X$ propre, lisse et relevable en caractéristique zéro. (p. 274)
Based on the ideas of Washnitzer, Lubkin has created a variant of this theory, valid only for $X$ proper, smooth, and liftable to characteristic zero.
- On notera qu'une telle formule n'est raisonnable que parce que $dF = 0$, même à l'infini ($X$ n'est pas supposé propre). (p. 275)
We will note that such a formula is only reasonable because $dF = 0$, even at infinity ($X$ is not taken to be proper).
- Le vérifier pour $\dim(V) = 1$, et observer que les deux membres sont additifs en $V$ dans une suite exacte courte. (p. 276)
Check it for $\dim(V) = 1$, and observe that the two sides are summands of $V$ in a short exact sequence.