Relative Perversity

2021/09/14 by Hansen, David, Scholze, Peter · 3 citations
#14F08 #14F20 #18G80 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2109.06766

Abstract

We define and study a relative perverse t-structure associated with any finitely presented morphism of schemes f: X→ S, with relative perversity equivalent to perversity of the restrictions to all geometric fibres of f. The existence of this t-structure is closely related to perverse t-exactness properties of nearby cycles. This t-structure preserves universally locally acyclic sheaves, and one gets a resulting abelian category Perv^ULA(X/S) with many of the same properties familiar in the absolute setting (e.g., noetherian, artinian, compatible with Verdier duality). For S connected and geometrically unibranch with generic point η, the functor Perv^ULA(X/S)→ Perv(X_η) 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.

Cited by

• Six-Functor Formalisms
• Bézout's theorem for abelian varieties
• Non-archimedean topological mirror symmetry for SL_n and PGL_n Higgs bundles

Related

• Artin perverse sheaves
• Explicit models for perverse sheaves
• Perverse pullbacks
• Perverse sheaves on smooth toric varieties and stacks
• Perverse coherent sheaves (after Deligne)