Birational spaces (Q330146)

Studying resolutions of singularities for varieties of characteristic zero, Zariski extensively used the Riemann-Zariski space \(RZ_K(k)\) of a finitely generated field extension \(k\subset K\), i.e. the space of all valuations on \(K/k\), which can also be obtained as the projective limit of all projective k-models of \(K\). This paper describes the category of birational spaces where the relative Riemann-Zariski spaces introduced by \textit{M. Temkin} in [Isr. J. Math. 185, 1--42 (2011; Zbl 1273.14007)] are naturally included.NEWLINENEWLINEIn Section \(2\) the author constructs the topological space \(\text{Spa}(B,A)\), where \(A\subset B\) are commutative rings with identity, of all \(A\)-valuations of \(B\) such that \(v(a)\leq 1\) for all \(a\in A\). He then studies the subspace \(\text{Val}(B,A)\) of unbounded valuations, its topological properties and proves it to be a locally ringed space with two sheaves of rings \(\mathcal{O}_{\text{Val}(B,A)}\subset \mathcal{M}_{\text{Val}(B,A)}\).NEWLINENEWLINEA Birational Space is defined in section \(3\) as a pair-ringed space \((\mathscr{X},\mathcal{M}_\mathscr{X},\mathcal{O}_\mathscr{X})\) that for every point in \(\mathscr{X}\) has an open neighbourhood with the induced subspace isomorphic to an affinoid birational space, i.e. \(\left(\text{Val}(B,A),\mathcal{M}_{\text{Val}(B,A)},\mathcal{O}_{\text{Val}(B,A)}\right)\) for some pair of rings of rings \(A\subset B\). Furthermore, it is proven that the functor \(\text{bir}\) that takes a pair of rings \(A\subset B\) to \(\text{Val}(B,A)\) gives rise to an anti-equivalence from the localisation of the category of pairs of rings with respect to relative normalisation to the category of affinoid birational spaces. This functor is then extended to to a functor from the category of affine and schematically dominant morphisms between quasi-compact and quasi-separated schemes to the category of birational spaces.NEWLINENEWLINESection \(4\) is dedicated to an introduction and study of relative blow ups so that in Section \(5\) the main theorem is proved: the functor \(\text{bir}\) provides an equivalence of categories between the localisation of the category of pairs of quasi-compact and quasi-separated schemes with an affine dominating morphism between them, with respect to simple relative blow ups and relative normalisations, and the category of quasi-compact and quasi-separated birational spaces.