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\(n\)-dimensional local fields and adeles on \(n\)-dimensional schemes. (English) Zbl 1144.11078
Young, Nicholas (ed.) et al., Surveys in contemporary mathematics. Cambridge: Cambridge University Press (ISBN 978-0-521-70564-6/pbk). London Mathematical Society Lecture Note Series 347, 131-164 (2008).
This paper is concerned with the construction of various complexes which compute the Zariski cohomology of a quasicoherent sheaf \(\mathcal F\) on a scheme \(X\). Section 2 of the paper contains basic definitions, general classification theorems and several enlightening examples concerning higher-dimensional local fields. It is so well-written that I would recommend it as first reading for anyone interested in learning about these fields. In Section 3, adeles and adelic complexes are defined. In particular, it is shown there that the Zariski cohomology of a quasicoherent sheaf \(\mathcal F\) on a noetherian scheme \(X\) agrees with the cohomology of the associated adelic complex \(\mathcal A_{X}(\mathcal F)\). The cases \(\text{dim}\,X=1\) and \(2\) are discussed in detail.
Section 4 deals with the so-called restricted adelic complex \(\mathcal A(\mathcal F)\) associated to a quasicoherent sheaf \(\mathcal F\) on a scheme \(X\). The main difference with the adelic complexes is that restricted adelic complexes are associated with one fixed chain of irreducible subvarieties of a scheme \(X\) (as opposed to considering all nondegenerate simplices on \(X\)). Once again, the cases \(\text{dim}\,X=1\) and \(2\) are discussed in detail.
Further, the following result is obtained: let \(X\) be a projective algebraic scheme of dimension \(n\) over a field. Let \(Y_{0}=X\supset Y_{1}\supset \dots\supset Y_{n}\) be a chain of closed subschemes such that each \(Y_{i}\) is an ample divisor on \(Y_{i-1}\). Then, for any quasicoherent sheaf \(\mathcal F\) on \(X\) and any \(i\), we have \(H^{i}(X,\mathcal F)=H^{i}(\mathcal A(\mathcal F))\), where \(\mathcal A(\mathcal F)\) is the restricted adelic complex associated with the chain \(Y_{0}=X\supset Y_{1}\supset \dots\supset Y_{n}\). The last section of the paper discusses extensions of Weil’s reciprocity law from curves to surfaces.
For the entire collection see [Zbl 1128.00009].

11R56 Adèle rings and groups
14F99 (Co)homology theory in algebraic geometry