## 87.2 Modifications

Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring. We set $S = \mathop{\mathrm{Spec}}(A)$ and $U = S \setminus \{ \mathfrak m\}$. In this section we will consider the category

87.2.0.1
$$\label{spaces-resolve-equation-modification} \left\{ f : X \longrightarrow S \quad \middle | \quad \begin{matrix} X\text{ is an algebraic space} \\ f\text{ is a proper morphism} \\ f^{-1}(U) \to U\text{ is an isomorphism} \end{matrix} \right\}$$

A morphism from $X/S$ to $X'/S$ will be a morphism of algebraic spaces $X \to X'$ compatible with the structure morphisms over $S$. In Restricted Power Series, Section 86.11 we have seen that this category only depends on the completion of $A$ and we have proven some elementary properties of objects in this category. In this section we specifically study cases where $\dim (A) \leq 2$ or where the dimension of the closed fibre is at most $1$.

Lemma 87.2.1. Let $(A, \mathfrak m, \kappa )$ be a $2$-dimensional Noetherian local domain such that $U = \mathop{\mathrm{Spec}}(A) \setminus \{ \mathfrak m\}$ is a normal scheme. Then any modification $f : X \to \mathop{\mathrm{Spec}}(A)$ is a morphism as in (87.2.0.1).

Proof. Let $f : X \to S$ be a modification. We have to show that $f^{-1}(U) \to U$ is an isomorphism. Since every closed point $u$ of $U$ has codimension $1$, this follows from Spaces over Fields, Lemma 70.3.3. $\square$

Lemma 87.2.2. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local ring. Let $g : X \to Y$ be a morphism in the category (87.2.0.1). If the induced morphism $X_\kappa \to Y_\kappa$ of special fibres is a closed immersion, then $g$ is a closed immersion.

Proof. This is a special case of More on Morphisms of Spaces, Lemma 74.49.3. $\square$

Lemma 87.2.3. Let $(A, \mathfrak m, \kappa )$ be a Noetherian local domain of dimension $\geq 1$. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a morphism of algebraic spaces. Assume at least one of the following conditions is satisfied

1. $f$ is a modification (Spaces over Fields, Definition 70.8.1),

2. $f$ is an alteration (Spaces over Fields, Definition 70.8.3),

3. $f$ is locally of finite type, quasi-separated, $X$ is integral, and there is exactly one point of $|X|$ mapping to the generic point of $\mathop{\mathrm{Spec}}(A)$,

4. $f$ is locally of finite type, $X$ is decent, and the points of $|X|$ mapping to the generic point of $\mathop{\mathrm{Spec}}(A)$ are the generic points of irreducible components of $|X|$,

5. add more here.

Then $\dim (X_\kappa ) \leq \dim (A) - 1$.

Proof. Cases (1), (2), and (3) are special cases of (4). Choose an affine scheme $U = \mathop{\mathrm{Spec}}(B)$ and an étale morphism $U \to X$. The ring map $A \to B$ is of finite type. We have to show that $\dim (U_\kappa ) \leq \dim (A) - 1$. Since $X$ is decent, the generic points of irreducible components of $U$ are the points lying over generic points of irreducible components of $|X|$, see Decent Spaces, Lemma 66.20.1. Hence the fibre of $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ over $(0)$ is the (finite) set of minimal primes $\mathfrak q_1, \ldots , \mathfrak q_ r$ of $B$. Thus $A_ f \to B_ f$ is finite for some nonzero $f \in A$ (Algebra, Lemma 10.121.10). We conclude $\kappa (\mathfrak q_ i)$ is a finite extension of the fraction field of $A$. Let $\mathfrak q \subset B$ be a prime lying over $\mathfrak m$. Then

$\dim (B_\mathfrak q) = \max \dim ((B/\mathfrak q_ i)_{\mathfrak q}) \leq \dim (A)$

the inequality by the dimension formula for $A \subset B/\mathfrak q_ i$, see Algebra, Lemma 10.112.1. However, the dimension of $B_\mathfrak q/\mathfrak m B_\mathfrak q$ (which is the local ring of $U_\kappa$ at the corresponding point) is at least one less because the minimal primes $\mathfrak q_ i$ are not in $V(\mathfrak m)$. We conclude by Properties, Lemma 28.10.2. $\square$

Lemma 87.2.4. If $(A, \mathfrak m, \kappa )$ is a complete Noetherian local domain of dimension $2$, then every modification of $\mathop{\mathrm{Spec}}(A)$ is projective over $A$.

Proof. By More on Morphisms of Spaces, Lemma 74.43.5 it suffices to show that the special fibre of any modification $X$ of $\mathop{\mathrm{Spec}}(A)$ has dimension $\leq 1$. This follows from Lemma 87.2.3. $\square$

## 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).

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 0BH8. Beware of the difference between the letter 'O' and the digit '0'.