On Serre-duality for coherent sheaves on rigid-analytic spaces
The main purpose of this article is to prove a Serre-duality theorem for coherent sheaves in the rigid-analytic context. Let \(X\) be a smooth proper rigid-analytic variety over \(K\) of dimension \(n\). Let \(\omega_X=\wedge^n\varOmega_{X/K}^1\) be the sheaf of holomorphic \(n-\)forms on \(X\). Then there exists a trace map \(t:H^n(X, \omega_X)\to K\) with the following property: If \(\mathcal F\) is a coherent sheaf on \(X,\) then the morphism \(t\) induces by composition with the canonical map a perfect pairing \[ \text{Hom}(\mathcal F,\omega_X)\times H^n(X,\mathcal F)\to H^n(X,\omega_X)\to K \] of \(K-\)vector spaces. Especially there is a canonical isomorphism \[ H^n(X,\mathcal F)^\lor\widetilde{\longrightarrow}\text{Hom}(\mathcal F, \omega_X). \] The proof is given by using results from algebraic local cohomology.
scientific article; zbMATH DE number 1038308