In some famous papers like Gaitsgory's "Construction of central elements in the affine Hecke algebra via nearby cycles" and Beilinson-Bernstein's "A proof of Jantzen conjectures", the authors used the nearby cycle functor $\Psi$ to schemes over an affine line (sometime even over an algebraic curve), and they always referred to Exposé XIII of SGA 7. However in loc. cit., I just find the definition of nearby cycle functor for schemes over strict traits (spectrum of strict henselian local rings). My question is what is the precise definition of nearby cycle functor over an affine line? Is there any authentic reference for this type of nearby cycle?
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4$\begingroup$ You just pull back to the étale local ring of the affine line at the point you want (often 0), which is a strict trait. $\endgroup$– Will SawinCommented Dec 12, 2023 at 12:23
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$\begingroup$ In this way, it seems that one can define nearby cycle functor over any base scheme (not necessarily a curve) around a distinguished point? $\endgroup$– Allen LeeCommented Dec 14, 2023 at 10:28
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$\begingroup$ You want a codimension $1$ point so the étale local ring will be a strict trait, but this indeed works in higher dimension. There are variants of nearby cycles that give more complicated invariants at higher codimensions points. $\endgroup$– Will SawinCommented Dec 14, 2023 at 13:39
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