Tags: ? Refs: dualizable objects in a category

dualizing sheaf

• Definition: a sheaf \(\omega\in {\mathsf{Coh}}(X)\) is dualizing iff \begin{align*} {\mathcal{F}}\in {\mathsf{Sh}}(X)^{{\mathrm{lf}}} \implies H^{n-i}\left(X; {\mathcal{F}} {}^{ \vee }\otimes \omega\right) \simeq H^{i}(X; {\mathcal{F}}) {}^{ \vee } \end{align*}
  • If \(X\in{\mathsf{sm}}\mathop{\mathrm{Proj}}{\mathsf{Var}}\), then \(\omega\) is a representative of the canonical sheaf.