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Purity theorems for real spectra and applications. (English) Zbl 0840.14035
Broglia, Fabrizio (ed.) et al., Real analytic and algebraic geometry. Proceedings of the international conference, Trento, Italy, September 21-25, 1992. Berlin: Walter de Gruyter. 229-250 (1995).
Summary: The following purity theorem is proved. Let \(A\) be an excellent regular ring and \({\mathfrak p}\) a prime ideal of \(A\) of height \(c\) such that \(A/{\mathfrak p}\) is regular. If \(F\) is a locally constant sheaf on the real spectrum \(\text{sper }A\) then the sheaf \(\underline {H}^i_{\text{sper }A/ {\mathfrak p}} (F)\) of cohomology with supports is for \(i=c\) locally isomorphic to the restriction \(F|_{\text{sper } A/{\mathfrak p}}\), and is 0 for \(i\neq c\). As an application we construct, on every excellent regular noetherian scheme \(X\), a Gersten type resolution of the sheaf \(\mathbb{Z}/2\) on the real spectrum of \(X\). This resolution is used for a new construction of fundamental classes and cycle maps with values in \(\mathbb{Z}/2\)-cohomology of the real spectrum. As another application we give a new proof to a well-known theorem about nullhomologous divisors.
For the entire collection see [Zbl 0812.00016].
Reviewer: Reviewer (Berlin)

14P99 Real algebraic and real-analytic geometry
14F20 √Čtale and other Grothendieck topologies and (co)homologies
14C20 Divisors, linear systems, invertible sheaves