# Lefschetz standard conjecture under specialization/generization

Let $$S$$ be a smooth connected noetherian scheme (not necessarily over a field) with residue fields that are all of finite type over their prime field.

Let $$f: \mathcal{X}\to S$$ be a smooth projective morphism.

Let $$s'\to s$$ be a specialization of points in $$S$$, ie. $$s\in\overline{\{s'\}}$$.

(1) If the Lefschetz standard conjecture is true for $$\mathcal{X}_s$$, is it true for $$\mathcal{X}_{s'}$$?

(2) If the Lefschetz standard conjecture is true for $$\mathcal{X}_{s'}$$, is it true for $$\mathcal{X}_{s}$$?

By Lefschetz standard conjecture, I mean "Conjecture B" in Grothendieck's paper on the standard conjectures.

Grothendieck does say Conjecture B is stable under specialization. Maybe this is easy to see. Any reference, please?

If your Weil cohomology theory is étale cohomology, then (2) is easy. Indeed, we have specialisation isomorphisms (for $\ell \neq \operatorname{char} \kappa(s)$) [SGA4$_3$, Exp. XVI, Cor. 2.2] $$H^i_{\text{ét}}(\mathcal X_{s'},\mathbb Q_\ell) \stackrel\sim\to H^i_{\text{ét}}(\mathcal X_s, \mathbb Q_\ell)$$ as well as specialisation maps [Ful98, Ex. 20.3.5] $$\operatorname{CH}^i(\mathcal X_{s'}) \to \operatorname{CH}^i(\mathcal X_s)$$ that are compatible with the cycle class maps $\operatorname{CH}^i \to H^{2i}$. Thus, if you have a cycle on $\mathcal X_{s'}\times \mathcal X_{s'}$ inducing the Lefschetz operator on $\mathcal X_{s'}$, then its specialisation induces the Lefschetz operator on $\mathcal X_s$. The same argument works whenever the Weil cohomology theory has some sort of specialisation isomorphism.
[SGA4$_3$] Artin, M.; Grothendieck, A.; Verdier, J.L.; et al, Seminaire de géométrie algébrique du Bois-Marie 1963--1964. Théorie des topos et cohomologie étale des schémas (SGA 4). Vol. 3: Exp. IX--XIX, Lecture Notes in Mathematics 305. Berlin-Heidelberg-New York: Springer-Verlag (1973). ZBL0245.00002.