Relative perversity

We define and study a relative perverse t t -structure associated with any finitely presented morphism of schemes f : X → S f: X\to S , with relative perversity equivalent to perversity of the restrictions to all geometric fibres of f f . The existence of this t t -structure is closely related to perverse t t -exactness properties of nearby cycles. This t t -structure preserves universally locally acyclic sheaves, and one gets a resulting abelian category P e r v U L A ( X / S ) \mathrm {Perv}^{\mathrm {ULA}}(X/S) with many of the same properties familiar in the absolute setting (e.g., noetherian, artinian, compatible with Verdier duality). For S S connected and geometrically unibranch with generic point η \eta , the functor P e r v U L A ( X / S ) → P e r v ( X η ) \mathrm {Perv}^{\mathrm {ULA}}(X/S)\to \mathrm {Perv}(X_\eta ) is exact and fully faithful, and its essential image is stable under passage to subquotients. This yields a notion of “good reduction” for perverse sheaves.