The Stacks project

Lemma 61.3.2. 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.

Lemma 61.3.3. 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.

Lemma 61.3.4. 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.

Lemma 61.3.5. Let $A \to B$ be a local isomorphism. Then

1. $A \to B$ is étale,

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

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

Lemma 61.3.6. 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 $A_{f_ i} \cong B_{g_ i}$.

Lemma 61.3.7. 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

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 $X$.

Lemma 61.3.8. Let $A$ be a ring. Set $X = \mathop{\mathrm{Spec}}(A)$. The functor

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.