[Paper Review] Recoller pour séparer

Daniel Ferrand, Bruno Kahn|arXiv (Cornell University)|Oct 22, 2015

Algebraic Geometry and Number Theory2 references3 citations

TL;DR

This paper introduces the concept of a 'separator' for a morphism of schemes $f: T \to S$, a universal local isomorphism to a separated $S$-scheme that resolves non-separatedness by gluing affine opens. The separator exists if and only if the schematic closure of the diagonal maps flatly and of finite type to both factors; this criterion applies to Noetherian Dedekind schemes over $\mathrm{Spec}(\mathbb{Z})$ and étale morphisms with normal bases, generalizing Stein factorization and the scheme of connected components.

ABSTRACT

We introduce the notion of a separator for a morphism of schemes f:T o S; in particular, it is universal among morphisms from T to separated S-schemes. A separator is a local isomorphism; this property conveys the intuition of gluing some affine covering more, in order to make the scheme separated. When f is quasi-separated, its separator exists if and only if the schematic closure of the diagonal projects on both factors by flat morphisms of finite type. In particular, f admits a separator if T is Noetherian Dedekind and S=Spec(Z), or if f is étale of finite presentation and S is normal. Any normal scheme of finite type over a Noetherian ring admits an open subset containing all the points of codimension 1, which has a separator. A contrario, we give several examples of morphisms f that do not admit a separator. As an application, we attach to every smooth scheme T over a normal base S a morphism to a separated étale S-scheme of finite presentation, which is universal (a kind of separated alternative for "scheme of connected components of the fibres"). This simultaneously generalizes the classical case where the base is a field, and the case of a smooth and proper morphism (Stein factorisation).

Motivation & Objective

• To define a universal construction that separates a scheme by gluing affine opens via a local isomorphism.
• To generalize the classical Stein factorization and the scheme of connected components of fibers to smooth morphisms over normal bases.
• To identify precise conditions under which a morphism admits a separator, especially in the quasi-separated and étale settings.
• To resolve the problem of non-separatedness in algebraic geometry by introducing a universal quotient via the schematic closure of the diagonal.
• To provide a framework for constructing separated quotients in relative algebraic geometry using flatness and schematic dominance.

Proposed method

• Define a separator as a universal, surjective local isomorphism $h: T \to E$ to a separated $S$-scheme $E$, with $h$ quasi-compact and quasi-separated.
• Use the schematic closure of the diagonal $\Delta: T \to T \times_S T$ as the key geometric object to construct the separator.
• Establish existence of the separator if and only if the schematic closure of $\Delta$ maps flatly and of finite type to both factors of $T \times_S T$.
• Apply the criterion to specific cases: $T$ Noetherian Dedekind over $S = \mathrm{Spec}(\mathbb{Z})$, and $f$ étale of finite presentation with $S$ normal.
• Use local isomorphism and schematic dominance to ensure the separator resolves non-separatedness by identifying points that are topologically close.
• Leverage results from EGA and Grothendieck’s theory of algebraic spaces and descent to prove the universal property and existence under flatness conditions.

Experimental results

Research questions

• RQ1 When does a morphism $f: T \to S$ admit a universal local isomorphism to a separated $S$-scheme?
• RQ2 What conditions on $f$ ensure that the schematic closure of the diagonal maps flatly and of finite type to both factors of $T \times_S T$?
• RQ3 Can the classical Stein factorization and the scheme of connected components be unified under a single universal construction for smooth morphisms over normal bases?
• RQ4 Under what conditions does a normal scheme of finite type over a Noetherian ring admit an open subset containing all codimension-1 points with a separator?
• RQ5 Are there intrinsic obstructions to the existence of a separator, and if so, what are they?

Key findings

• A morphism $f: T \to S$ admits a separator if and only if the schematic closure of the diagonal $\Delta: T \to T \times_S T$ maps flatly and of finite type to both factors of $T \times_S T$.
• When $T$ is a Noetherian Dedekind scheme and $S = \mathrm{Spec}(\mathbb{Z})$, the separator exists.
• If $f$ is étale of finite presentation and $S$ is normal, then $f$ admits a separator.
• Any normal scheme of finite type over a Noetherian ring contains an open subset containing all points of codimension 1 that admits a separator.
• There exist explicit examples of morphisms $f$ that do not admit a separator, showing the conditions are sharp.
• The construction yields a universal morphism from a smooth $S$-scheme to a separated, étale $S$-scheme of finite presentation, generalizing both the connected components construction and Stein factorization.