Duality in the étale cohomology of one-dimensional proper schemes and generalizations
On an equidimensional scheme X we consider the complex of étale sheaves in degrees 0 and 1 given by: \({\mathcal G}:\oplus _{y\in X^{(0)}}j_{y*}{\mathbb{G}}_{m/y}\to ^{\partial}\oplus _{x\in X^{(1)}}i_{x*}{\mathbb{Z}}\) and we set \({\mathcal G}(n)=R \underline{Hom}_ X({\mathbb{Z}}/n,{\mathcal G})\in D^ +(X,{\mathbb{Z}}/n)\). For regular X we have quasi-isomorphisms \({\mathbb{G}}_ m\to ^{\sim}{\mathcal G}\) and \(\mu _ n\to ^{\sim}{\mathcal G}(n)\). Results: (a) Let X be a one-dimensional proper \({\mathbb{Z}}\)-scheme. Then for any constructible sheaf F on X the Yoneda-pairing \(H^ i(X,F)\times {\mathbb{E}}xt_ X^{3-i}(F,{\mathcal G})\to {\mathbb{H}}^ 3(X,{\mathcal G})\to ^{tr}{\mathbb{Q}}/{\mathbb{Z}}\) is a perfect pairing of finite groups, \(i\in {\mathbb{Z}}\). If X is irreducible the trace map is an isomorphism. (The 2- primary components may have to be excluded.) This extends the case where X is regular, i.e. Artin-Verdier duality for number and function fields. [cf. \textit{B. Mazur}, Ann. Sci. Ec. Norm. Supér., IV. Sér. 6(1973), 521-552 (1974; Zbl 0282.14004) and the author, Math. Z. 188, 91-100 (1984; Zbl 0585.14014)]. (b) There is also a local version of (a) where X is the spectrum of a henselian one-dimensional noetherian local ring with finite residue field. (c) Assume Y is separated, smooth of relative dimension \(d\) over an open subscheme of X as in (a). Then for constructible F on Y with \(nF=0\), n invertible on Y there is a perfect Yoneda duality on Y in the category of \({\mathbb{Z}}/n\)-sheaves with dualizing complex \({\mathcal G}(n)\otimes \mu _ n^{\otimes d}\). (d) If \(\mu_ n\) is replaced by \({\mathcal G}(n)\) Poincaré duality remains valid for singular proper curves over algebraically closed fields. Remarks: The fibres of a regular arithmetical surface can be singular. Therefore (a) should be useful for establishing arithmetical duality on such surfaces. Moreover (a), (b) suggest that higher dimensional arithmetical duality should not be confined to regular schemes.
scientific article; zbMATH DE number 3981330
scientific article