The smooth projective locally free theorem is the special case

Under AC, if $X\mathrm{/}k$ is smooth projective of pure dimension $d$, regular local rings make it CM. The regular-immersion Koszul calculation Koszul sheaf Ext of a smooth regular immersion is concentrated in codimension and conormal adjunction Adjunction for a smooth closed subvariety identify the ambient sheaf Ext in the local CM packet with ${\bigwedge }^d{\mathrm{\Omega }}_{X\mathrm{/}k}^1$. Thus the normalized ${\omega }_X$ is the canonical line bundle. For finite locally free $E$, $\mathcal{R}\text{\hspace{0.17em}⁣}Hom(E,{\omega }_X)=E^{\vee }\otimes {\omega }_X$: locally a finite free module has exact Hom, so the positive sheaf Ext terms vanish, and taking derived global sections gives ${\mathrm{Ext}}_X^{d-i}(E,{\omega }_X)=H^{d-i}(X,E^{\vee }\otimes {\omega }_X)$.

To compare traces, fix the normalized identification with the canonical line bundle. For an embedding of codimension $c$, Local-to-global Ext collapse for a regular immersion uses ${\sigma }_c=(-1{)}^{c(c+1)\mathrm{/}2}$ times the Koszul/Hodge determinant identification, rather than the unmodified determinant map. Use that same identification to transport the coherent theorem's dualizing sheaf and trace. Its global adjunction is the one-row Ext comparison, and its counit is precomposition with ${\mathcal{O}}_P\to i_{\ast }{\mathcal{O}}_X$; hence the transported trace is the published Gysin trace. The cup/evaluation compatibility and embedding independence proved in Embedding compatibility of smooth-projective Gysin traces then identify the pairing with Serre duality for locally free sheaves on a smooth projective variety. For singular projective $X$ that is Cohen–Macaulay and pure of dimension $d$, the coherent Ext statement remains valid and the dualizing sheaf may fail to be invertible. Without the Cohen–Macaulay hypothesis, duality generally requires the normalized dualizing complex $D_X$, rather than a shift of a single sheaf.