The Stacks project

Lemma 50.23.11. Let $k$ be a field. Let $X$ be an irreducible smooth proper scheme over $k$ of dimension $d$. Let $Z \subset X$ be the reduced closed subscheme consisting of a single $k$-rational point $x$. Then the image of $1 \in k = H^0(Z, \mathcal{O}_ Z) = H^0(Z, \Omega ^0_{Z/k})$ by the map $H^0(Z, \Omega ^0_{Z/k}) \to H^ d(X, \Omega ^ d_{X/k})$ of Remark 50.23.7 is nonzero.

Proof. The map $\gamma ^0 : \mathcal{O}_ Z \to \mathcal{H}^ d_ Z(\Omega ^ d_{X/k}) = R\mathcal{H}_ Z(\Omega ^ d_{X/k})[d]$ is adjoint to a map

\[ g^0 : i_*\mathcal{O}_ Z \longrightarrow \Omega ^ d_{X/k}[d] \]

in $D(\mathcal{O}_ X)$. Recall that $\Omega ^ d_{X/k} = \omega _ X$ is a dualizing sheaf for $X/k$, see Duality for Schemes, Lemma 48.27.1. Hence the $k$-linear dual of the map in the statement of the lemma is the map

\[ H^0(X, \mathcal{O}_ X) \to \mathop{\mathrm{Ext}}\nolimits ^ d_ X(i_*\mathcal{O}_ Z, \omega _ X) \]

which sends $1$ to $g^0$. Thus it suffices to show that $g^0$ is nonzero. This we may do in any neighbourhood $U$ of the point $x$. Choose $U$ such that there exist $f_1, \ldots , f_ d \in \mathcal{O}_ X(U)$ vanishing only at $x$ and generating the maximal ideal $\mathfrak m_ x \subset \mathcal{O}_{X, x}$. We may assume assume $U = \mathop{\mathrm{Spec}}(R)$ is affine. Looking over the construction of $\gamma ^0$ we find that our extension is given by

\[ k \to (R \to \bigoplus \nolimits _{i_0} R_{f_{i_0}} \to \bigoplus \nolimits _{i_0 < i_1} R_{f_{i_0}f_{i_1}} \to \ldots \to R_{f_1\ldots f_ r})[d] \to R[d] \]

where $1$ maps to $1/f_1 \ldots f_ c$ under the first map. This is nonzero because $1/f_1 \ldots f_ c$ is a nonzero element of local cohomology group $H^ d_{(f_1, \ldots , f_ d)}(R)$ in this case, $\square$

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Your email address will not be published.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 0G8E. Beware of the difference between the letter 'O' and the digit '0'.