Abstract
AbstractWe prove a relative Lefschetz–Verdier theorem for locally acyclic objects over a Noetherian base scheme. This is done by studying duals and traces in the symmetric monoidal$2$-category of cohomological correspondences. We show that local acyclicity is equivalent to dualisability and deduce that duality preserves local acyclicity. As another application of the category of cohomological correspondences, we show that the nearby cycle functor over a Henselian valuation ring preserves duals, generalising a theorem of Gabber.
Highlights
The notions of dual and trace in symmetric monoidal categories were introduced by Dold and Puppe [DP]. They have been extended to higher categories and have found important applications in algebraic geometry and other contexts
We show that under the assumption L ∈ Dcft (X, Λ), dualisability is equivalent to local acyclicity (Theorem 2.16)
By studying specialisation of cohomological correspondences, we generalise Gabber’s theorem that Ψ preserves duals and a fixed point theorem of Vidal to Henselian valuation rings (Corollaries 3.8 and 3.13)
Summary
The notions of dual and trace in symmetric monoidal categories were introduced by Dold and Puppe [DP]. They have been extended to higher categories and have found important applications in algebraic geometry and other contexts (see [BZN] by Ben-Zvi and Nadler and the references therein). One of our main results is the following relative Lefschetz–Verdier theorem. Let. GX be a commutative diagram of schemes separated of finite type over S, with p and D → D ×Y Y proper. Dcft (X, Λ) ⊆ D (X, Λ) denotes the full subcategory spanned by objects of finite tor-dimension and of constructible cohomology sheaves, and u, v is the relative Lefschetz–Verdier pairing.
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