# zbMATH — the first resource for mathematics

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)

##### MSC:
 14P99 Real algebraic and real-analytic geometry 14F20 Étale and other Grothendieck topologies and (co)homologies 14C20 Divisors, linear systems, invertible sheaves
##### Keywords:
purity; real spectrum; cohomology; fundamental classes; cycle maps