Lemma 88.30.1. Let $A \to B$ be a ring homomorphism of Noetherian rings inducing an isomorphism on $I$-adic completions for some ideal $I \subset A$ (for example if $B$ is the $I$-adic completion of $A$). Then base change defines an equivalence of categories between the category (88.30.0.1) for $(A, I)$ with the category (88.30.0.1) for $(B, IB)$.
88.30 Application to modifications
Let $A$ be a Noetherian ring and let $I \subset A$ be an ideal. We set $X = \mathop{\mathrm{Spec}}(A)$ and $U = X \setminus V(I)$. In this section we will consider the category
88.30.0.1
\begin{equation} \label{restricted-equation-modification} \left\{ f : X' \longrightarrow X \quad \middle | \quad \begin{matrix} X'\text{ is an algebraic space}
\\ f\text{ is locally of finite type}
\\ f^{-1}(U) \to U\text{ is an isomorphism}
\end{matrix} \right\} \end{equation}
A morphism from $X'/X$ to $X''/X$ will be a morphism of algebraic spaces $X' \to X''$ over $X$.
Let $A \to B$ be a homomorphism of Noetherian rings and let $J \subset B$ be an ideal such that $J = \sqrt{I B}$. Then base change along the morphism $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ gives a functor from the category (88.30.0.1) for $A$ to the category (88.30.0.1) for $B$.
Proof. Set $X = \mathop{\mathrm{Spec}}(A)$ and $T = V(I)$. Set $X_1 = \mathop{\mathrm{Spec}}(B)$ and $T_1 = V(IB)$. By Theorem 88.27.4 (in fact we only need the affine case treated in Lemma 88.27.3) the category (88.30.0.1) for $X$ and $T$ is equivalent to the the category of rig-étale morphisms $W \to X_{/T}$ of locally Noetherian formal algebraic spaces. Similarly, the the category (88.30.0.1) for $X_1$ and $T_1$ is equivalent to the category of rig-étale morphisms $W_1 \to X_{1, /T_1}$ of locally Noetherian formal algebraic spaces. Since $X_{/T} = \text{Spf}(A^\wedge )$ and $X_{1, /T_1} = \text{Spf}(B^\wedge )$ (Formal Spaces, Lemma 87.14.6) we see that these categories are equivalent by our assumption that $A^\wedge \to B^\wedge $ is an isomorphism. We omit the verification that this equivalence is given by base change. $\square$
Lemma 88.30.2. Notation and assumptions as in Lemma 88.30.1. Let $f : X' \to \mathop{\mathrm{Spec}}(A)$ correspond to $g : Y' \to \mathop{\mathrm{Spec}}(B)$ via the equivalence. Then $f$ is quasi-compact, quasi-separated, separated, proper, finite, and add more here if and only if $g$ is so.
Proof. You can deduce this for the statements quasi-compact, quasi-separated, separated, and proper by using Lemmas 88.28.1 88.28.2, 88.28.3, 88.28.2, and 88.28.4 to translate the corresponding property into a property of the formal completion and using the argument of the proof of Lemma 88.30.1. However, there is a direct argument using fpqc descent as follows. First, you can reduce to proving the lemma for $A \to A^\wedge $ and $B \to B^\wedge $ since $A^\wedge \to B^\wedge $ is an isomorphism. Then note that $\{ U \to \mathop{\mathrm{Spec}}(A), \mathop{\mathrm{Spec}}(A^\wedge ) \to \mathop{\mathrm{Spec}}(A)\} $ is an fpqc covering with $U = \mathop{\mathrm{Spec}}(A) \setminus V(I)$ as before. The base change of $f$ by $U \to \mathop{\mathrm{Spec}}(A)$ is $\text{id}_ U$ by definition of our category (88.30.0.1). Let $P$ be a property of morphisms of algebraic spaces which is fpqc local on the base (Descent on Spaces, Definition 74.10.1) such that $P$ holds for identity morphisms. Then we see that $P$ holds for $f$ if and only if $P$ holds for $g$. This applies to $P$ equal to quasi-compact, quasi-separated, separated, proper, and finite by Descent on Spaces, Lemmas 74.11.1, 74.11.2, 74.11.18, 74.11.19, and 74.11.23. $\square$
Lemma 88.30.3. Let $A \to B$ be a local map of local Noetherian rings such that
$A \to B$ is flat,
$\mathfrak m_ B = \mathfrak m_ A B$, and
$\kappa (\mathfrak m_ A) = \kappa (\mathfrak m_ B)$
Then the base change functor from the category (88.30.0.1) for $(A, \mathfrak m_ A)$ to the category (88.30.0.1) for $(B, \mathfrak m_ B)$ is an equivalence.
Proof. The conditions signify that $A \to B$ induces an isomorphism on completions, see More on Algebra, Lemma 15.43.9. Hence this lemma is a special case of Lemma 88.30.1. $\square$
Lemma 88.30.4. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring. Let $f : X \to S$ be an object of (88.30.0.1). Then there exists a $U$-admissible blowup $S' \to S$ which dominates $X$.
Proof. Special case of More on Morphisms of Spaces, Lemma 76.39.5. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (1)
Comment #9775 by Keerthi Madapusi on