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
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