Is there a notion of “localization morphism” of schemes

algebraic-geometrycommutative-algebrareference-requestschemes

Let $A$ be a ring and $S\subseteq A$ a multiplicative system. Then the localization homomorphism of rings $\phi:A\to S^{-1}A$ induces a morphism between the spectra: $f=\mathrm{Spec}(\phi):\mathrm{Spec}(S^{-1}A)\to\mathrm{Spec}(A)$.

Is there a property of scheme morphisms $f:X\to Y$ that captures the
idea that $f$ locally looks like the Spec of a localization
homomorphism of rings? Does this have a name? Interesting properties/characterizations?

Notice that $S$ doesn't have to be of the form $\{f^n\}_{n\geq 0}$ for $f\in A$ or $A\smallsetminus \mathfrak{p}$ for a prime ideal $\mathfrak{p}\subset A$.

Here's an attempt. $f:X\to Y$ is a localization morphism if there is an affine open cover $\{V_i\}$ of $Y$ such that every $f^{-1}(V_i)$ has an affine open cover $\{U_j\}$ such that the morphism $f:U_j \to V_i$ corresponds to a homomorphism $\phi=f^{\sharp}:A\to B$, where $A=\mathcal{O}_Y(V_i)$ and $B=\mathcal{O}_X(U_j)$, and there is a multiplicative system $S\subseteq A$ and an isomorphism $\alpha:B\tilde{\to} S^{-1}A$ such that $\alpha\circ\phi=\lambda$, where $\lambda:A\to S^{-1}A$ is the canonical map to the localization.

Best Answer

The notion you have defined is called a localizing morphism in [Nayak, Definition 2.1], at least for morphisms between noetherian schemes. Nayak points out that this definition has some peculiar features. For example, if $Y$ is a noetherian scheme, then we can have the following [Nayak, (2.4)]:

  1. Let $X$ be the scheme obtained by gluing two open sets $U \subseteq Y$ and $V \subseteq Y$ along a nonempty open subset in $U \cap V$. Then, $X \to Y$ is a localizing morphism that is not separated in general.
  2. Let $\{U_i\}$ be a finite collection of open subsets of $Y$, and let $X = \coprod_i U_i$. Then, the natural map $X \to Y$ is a localizing morphism that is separated, but is not an open immersion in general.

This is why Nayak defines a localizing immersion between noetherian schemes as a localizing morphism that is set-theoretically injective, or equivalently a localizing morphism that is separated and maps generic points to generic points [Nayak, Lemma 2.6 and Definition 2.7].

You can read about some properties of localizing immersions in [Nayak, (2.8)], but one of the most important results is the following:

Theorem [Nayak, Theorem 3.6]. Let $f\colon X \to S$ be a separated morphism essentially of finite type between noetherian schemes. Then, $f$ factors as $$X \overset{k}{\longrightarrow} Y \overset{p}{\longrightarrow} S,$$ where $k$ is a localizing immersion and $p$ is separated and of finite type.

Nayak uses this result to obtain a version of Nagata's compactification theorem for separated morphisms essentially of finite type.

Theorem [Nayak, Theorem 4.1]. Let $f\colon X \to S$ be a separated morphism essentially of finite type between noetherian schemes. Then, $f$ factors as $$X \overset{k}{\longrightarrow} Y \overset{p}{\longrightarrow} S,$$ where $k$ is a localizing immersion and $p$ is proper.

Nayak also obtains versions of Zariski's main theorem [Nayak, Theorem 4.3], Chow's lemma [Nayak, Theorem 4.8], and Grothendieck duality [Nayak, Theorem 5.3] for separated morphisms essentially of finite type.

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