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

87.30.0.1
\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 (87.30.0.1) for $A$ to the category (87.30.0.1) 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 (87.30.0.1) for $(A, I)$ with the category (87.30.0.1) 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 (87.30.0.1) 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 (87.30.0.1) 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 (87.30.0.1). 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 (87.30.0.1) for $(A, \mathfrak m_ A)$ to the category (87.30.0.1) 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 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 (87.30.0.1). Then there exists a $U$-admissible blowup $S' \to S$ which dominates $X$.

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


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