The Stacks project

60.3 Local isomorphisms

We start with a definition.

Definition 60.3.1. Let $\varphi : A \to B$ be a ring map.

  1. We say $A \to B$ is a local isomorphism if for every prime $\mathfrak q \subset B$ there exists a $g \in B$, $g \not\in \mathfrak q$ such that $A \to B_ g$ induces an open immersion $\mathop{\mathrm{Spec}}(B_ g) \to \mathop{\mathrm{Spec}}(A)$.

  2. We say $A \to B$ identifies local rings if for every prime $\mathfrak q \subset B$ the canonical map $A_{\varphi ^{-1}(\mathfrak q)} \to B_\mathfrak q$ is an isomorphism.

We list some elementary properties.

Lemma 60.3.2. Let $A \to B$ and $A \to A'$ be ring maps. Let $B' = B \otimes _ A A'$ be the base change of $B$.

  1. If $A \to B$ is a local isomorphism, then $A' \to B'$ is a local isomorphism.

  2. If $A \to B$ identifies local rings, then $A' \to B'$ identifies local rings.

Proof. Omitted. $\square$

Lemma 60.3.3. Let $A \to B$ and $B \to C$ be ring maps.

  1. If $A \to B$ and $B \to C$ are local isomorphisms, then $A \to C$ is a local isomorphism.

  2. If $A \to B$ and $B \to C$ identify local rings, then $A \to C$ identifies local rings.

Proof. Omitted. $\square$

Lemma 60.3.4. Let $A$ be a ring. Let $B \to C$ be an $A$-algebra homomorphism.

  1. If $A \to B$ and $A \to C$ are local isomorphisms, then $B \to C$ is a local isomorphism.

  2. If $A \to B$ and $A \to C$ identify local rings, then $B \to C$ identifies local rings.

Proof. Omitted. $\square$

Lemma 60.3.5. Let $A \to B$ be a local isomorphism. Then

  1. $A \to B$ is ├ętale,

  2. $A \to B$ identifies local rings,

  3. $A \to B$ is quasi-finite.

Proof. Omitted. $\square$

Lemma 60.3.6. Let $A \to B$ be a local isomorphism. Then there exist $n \geq 0$, $g_1, \ldots , g_ n \in B$, $f_1, \ldots , f_ n \in A$ such that $(g_1, \ldots , g_ n) = B$ and $A_{f_ i} \cong B_{g_ i}$.

Proof. Omitted. $\square$

Lemma 60.3.7. Let $p : (Y, \mathcal{O}_ Y) \to (X, \mathcal{O}_ X)$ and $q : (Z, \mathcal{O}_ Z) \to (X, \mathcal{O}_ X)$ be morphisms of locally ringed spaces. If $\mathcal{O}_ Y = p^{-1}\mathcal{O}_ X$, then

\[ \mathop{\mathrm{Mor}}\nolimits _{\text{LRS}/(X, \mathcal{O}_ X)}((Z, \mathcal{O}_ Z), (Y, \mathcal{O}_ Y)) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\textit{Top}/X}(Z, Y),\quad (f, f^\sharp ) \longmapsto f \]

is bijective. Here $\text{LRS}/(X, \mathcal{O}_ X)$ is the category of locally ringed spaces over $X$ and $\textit{Top}/X$ is the category of topological spaces over $X$.

Proof. This is immediate from the definitions. $\square$

Lemma 60.3.8. Let $A$ be a ring. Set $X = \mathop{\mathrm{Spec}}(A)$. The functor

\[ B \longmapsto \mathop{\mathrm{Spec}}(B) \]

from the category of $A$-algebras $B$ such that $A \to B$ identifies local rings to the category of topological spaces over $X$ is fully faithful.

Proof. This follows from Lemma 60.3.7 and the fact that if $A \to B$ identifies local rings, then the pullback of the structure sheaf of $\mathop{\mathrm{Spec}}(A)$ via $p : \mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ is equal to the structure sheaf of $\mathop{\mathrm{Spec}}(B)$. $\square$


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