The Stacks project

87.30 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
\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 ( for $A$ to the category ( for $B$.

Lemma 87.30.1. 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 ( for $(A, I)$ with the category ( for $(B, IB)$.

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 87.27.4 (in fact we only need the affine case treated in Lemma 87.27.3) the category ( for $X$ and $T$ is equivalent to the the category of rig-├ętale morphisms $W \to X_{/T}$ of locally Noetherian formal algebraic spaces. Similarly, the the category ( for $X_1$ and $T_1$ is equivalent to the category of rig-├ętale 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 86.14.6) 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 87.30.2. Notation and assumptions as in Lemma 87.30.1. 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 87.28.1 87.28.2, 87.28.3, 87.28.2, and 87.28.4 to translate the corresponding property into a property of the formal completion and using the argument of the proof of Lemma 87.30.1. 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 ( Let $P$ be a property of morphisms of algebraic spaces which is fpqc local on the base (Descent on Spaces, Definition 73.10.1) 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 73.11.1, 73.11.2, 73.11.18, 73.11.19, and 73.11.23. $\square$

Lemma 87.30.3. Let $A \to B$ be a local map of local Noetherian rings such that

  1. $A \to B$ is flat,

  2. $\mathfrak m_ B = \mathfrak m_ A B$, and

  3. $\kappa (\mathfrak m_ A) = \kappa (\mathfrak m_ B)$

Then the base change functor from the category ( for $(A, \mathfrak m_ A)$ to the category ( 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 87.30.1. $\square$

Lemma 87.30.4. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring. Let $f : X \to S$ be an object of ( Then there exists a $U$-admissible blowup $S' \to S$ which dominates $X$.

Proof. Special case of More on Morphisms of Spaces, Lemma 75.39.5. $\square$

Comments (0)

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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0AS1. Beware of the difference between the letter 'O' and the digit '0'.