Relative Riemann-Zariski spaces

By M. Temkin. 2008

56 citations recorded by the source corpus.

In this paper we study relative Riemann-Zariski spaces associated to a morphism of schemes and generalizing the classical Riemann-Zariski space of a field. We prove that similarly to the classical RZ spaces, the relative ones can be described either as projective limits of schemes in the category of locally ringed spaces or as certain spaces of valuations. We apply these spaces to prove the following two new results: a strong version of stable modification theorem for relative curves; a decomposition theorem which asserts that any separated morphism between quasi-compact and quasiseparated schemes factors as a composition of an affine morphism and a proper morphism. In particular, we obtain a new proof of Nagata’s compactification theorem.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/59583f10c7a04b714a10613d1981c819e6cce75d

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Coherent Tannaka duality and algebraicity of Hom-stacks (2014, background): “a valuation ring with fraction ﬁeld K. Finally let W = W1 ×K OS,s1. The local ring W is a semivaluation ring in the terminology of Temkin and a local ring of the relative Riemann–Zariski space RZU(S) [Tem11]. Note that A →V factors via W. Moreover, Spec(W) →Spec(S) restricts to Spec(OS,s1) →U over U”
Stable modification of relative curves (2007, methodology): “The case of an arbitrary qcqs scheme S will be deduced in a subsequent work [ Tem3 ] by use of relative Riemann-Zariski spaces RZU(S), see appendix B.2.”
Noetherian approximation of algebraic spaces and stacks (2009, background): “C], B. Conrad [Con07, Thm. 4.3, App. A] and M. Temkin [ Tem08 , Thm.”