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Rigid geometry, Lefschetz-Verdier trace formula and Deligne’s conjecture. (English) Zbl 0920.14005

The author shows among other results the Deligne conjecture \[ \text{Lef(Fr}^{n }\cdot b,K) =\sum_{D\in\Pi_{0}(\text{Fix Fr}^{n}\cdot b)} \text{ naive.loc}_{p}( \text{Fr}^{n}\cdot b,K)\tag{1} \] for \(U\) an open subset of a proper scheme \(X\), for \(b: V\to U\times_{k}U\) a correspondence (\(k\) the algebraic closure of a finite field) for \(K\) an element of the derived category \(D^{b}_{c}(U,\overline{\mathbb Q}_{l})\) of bounded complexes with constructible cohomology sheaves, for the corresponding Frobenius map Fr and if \(n\) is suitably large. Here Lef means the global trace, naive.loc means the naive local term (as an important property this term vanishes if the fiber of \(K\) is zero), and Fr means the geometric Frobenius over \(\mathbb{F}_q\). This result follows by upgrading in stages the Lefschetz-Verdier trace formula as found in Sémin. Géom. Algebr. 1965-66, SGA5, Lect. Notes Math. 589 [Exposé III, 73-137 (1977; Zbl 0355.14004) by A. Grothendieck, and Exposé III B, 138-203 (1977; Zbl 0354.14006) by L. Illusie]. For a proper statement of (1), it is necessary to use rigid geometry.

MSC:

14F20 Étale and other Grothendieck topologies and (co)homologies
14G20 Local ground fields in algebraic geometry
14G15 Finite ground fields in algebraic geometry
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