The Stacks project

Lemma 54.8.2. Let $(A, \mathfrak m, \kappa )$ be a local normal Nagata domain of dimension $2$. Let $a \in A$ be nonzero. There exists an integer $N$ such that for every modification $f : X \to \mathop{\mathrm{Spec}}(A)$ with $X$ normal the $A$-module

\[ M_{X, a} = \mathop{\mathrm{Coker}}(A \longrightarrow H^0(Z, \mathcal{O}_ Z)) \]

where $Z \subset X$ is cut out by $a$ has length bounded by $N$.

Proof. By the short exact sequence $ 0 \to \mathcal{O}_ X \xrightarrow {a} \mathcal{O}_ X \to \mathcal{O}_ Z \to 0 $ we see that
\begin{equation} \label{resolve-equation-a-torsion} M_{X, a} = H^1(X, \mathcal{O}_ X)[a] \end{equation}

Here $N[a] = \{ n \in N \mid an = 0\} $ for an $A$-module $N$. Thus if $a$ divides $b$, then $M_{X, a} \subset M_{X, b}$. Suppose that for some $c \in A$ the modules $M_{X, c}$ have bounded length. Then for every $X$ we have an exact sequence

\[ 0 \to M_{X, c} \to M_{X, c^2} \to M_{X, c} \]

where the second arrow is given by multiplication by $c$. Hence we see that $M_{X, c^2}$ has bounded length as well. Thus it suffices to find a $c \in A$ for which the lemma is true such that $a$ divides $c^ n$ for some $n > 0$. By More on Algebra, Lemma 15.125.6 we may assume $A/(a)$ is a reduced ring.

Assume that $A/(a)$ is reduced. Let $A/(a) \subset B$ be the normalization of $A/(a)$ in its quotient ring. Because $A$ is Nagata, we see that $\mathop{\mathrm{Coker}}(A \to B)$ is finite. We claim the length of this finite module is a bound. To see this, consider $f : X \to \mathop{\mathrm{Spec}}(A)$ as in the lemma and let $Z' \subset Z$ be the scheme theoretic closure of $Z \cap f^{-1}(U)$. Then $Z' \to \mathop{\mathrm{Spec}}(A/(a))$ is finite for example by Varieties, Lemma 33.17.2. Hence $Z' = \mathop{\mathrm{Spec}}(B')$ with $A/(a) \subset B' \subset B$. On the other hand, we claim the map

\[ H^0(Z, \mathcal{O}_ Z) \to H^0(Z', \mathcal{O}_{Z'}) \]

is injective. Namely, if $s \in H^0(Z, \mathcal{O}_ Z)$ is in the kernel, then the restriction of $s$ to $f^{-1}(U) \cap Z$ is zero. Hence the image of $s$ in $H^1(X, \mathcal{O}_ X)$ vanishes in $H^1(f^{-1}(U), \mathcal{O}_ X)$. By Lemma 54.7.5 we see that $s$ comes from an element $\tilde s$ of $A$. But by assumption $\tilde s$ maps to zero in $B'$ which implies that $s = 0$. Putting everything together we see that $M_{X, a}$ is a subquotient of $B'/A$, namely not every element of $B'$ extends to a global section of $\mathcal{O}_ Z$, but in any case the length of $M_{X, a}$ is bounded by the length of $B/A$. $\square$

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