The Stacks project

Lemma 38.28.6. In Situation 38.28.1 let $K$ be as in Lemma 38.28.2. Denote $X_0 \subset X$ the closed subset consisting of points lying over the closed subset $\mathop{\mathrm{Spec}}(A_1) = \mathop{\mathrm{Spec}}(A_2) = \ldots $ of $\mathop{\mathrm{Spec}}(A)$. There exists an open $W \subset X$ containing $X_0$ such that

  1. $H^ i(K)|_ W$ is zero unless $i = 0$,

  2. $\mathcal{F} = H^0(K)|_ W$ is of finite presentation, and

  3. $\mathcal{F}_ n = \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{O}_{X_ n}$.

Proof. Fix $n \geq 1$. By construction there is a canonical map $K \to \mathcal{F}_ n$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$ and hence a canonical map $H^0(K) \to \mathcal{F}_ n$ of quasi-coherent sheaves. This explains the meaning of part (3).

Let $x \in X_0$ be a point. We will find an open neighbourhood $W$ of $x$ such that (1), (2), and (3) are true. Since $X_0$ is quasi-compact this will prove the lemma. Let $U \subset X$ be an affine open neighbourhood of $x$. Say $U = \mathop{\mathrm{Spec}}(B)$. Choose a surjection $P \to B$ with $P$ smooth over $A$. By Lemma 38.28.4 and the definition of relative pseudo-coherence there exists a bounded above complex $F^\bullet $ of finite free $P$-modules representing $Ri_*K$ where $i : U \to \mathop{\mathrm{Spec}}(P)$ is the closed immersion induced by the presentation. Let $M_ n$ be the $B$-module corresponding to $\mathcal{F}_ n|_ U$. By Lemma 38.28.5

\[ H^ i(F^\bullet \otimes _ A A_ n) = \left\{ \begin{matrix} 0 & \text{if} & i \not= 0 \\ M_ n & \text{if} & i = 0 \end{matrix} \right. \]

Let $i$ be the maximal index such that $F^ i$ is nonzero. If $i \leq 0$, then (1), (2), and (3) are true. If not, then $i > 0$ and we see that the rank of the map

\[ F^{i - 1} \to F^ i \]

in the point $x$ is maximal. Hence in an open neighbourhood of $x$ inside $\mathop{\mathrm{Spec}}(P)$ the rank is maximal. Thus after replacing $P$ by a principal localization we may assume that the displayed map is surjective. Since $F^ i$ is finite free we may choose a splitting $F^{i - 1} = F' \oplus F^ i$. Then we may replace $F^\bullet $ by the complex

\[ \ldots \to F^{i - 2} \to F' \to 0 \to \ldots \]

and we win by induction on $i$. $\square$

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Your email address will not be published.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 0CTH. Beware of the difference between the letter 'O' and the digit '0'.