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  "body": "\n## Statement\n\nLet $k$ be a field and let $X$ be a [[projective variety|A4H4CZ2V35]] over $k$ of dimension $n$.\nAssume that $X$ is [[smooth|EXKGG9X6J0]] and geometrically connected, with [[canonical bundle|W05VYY4HBR]] $\\omega_X$.\nThen there is a trace isomorphism $H^n(X, \\omega_X) \\cong k$.\nLet $\\mathcal{E}$ be a [[vector bundle|P4DZM975BV]] on $X$ with dual $\\mathcal{E}^\\vee$, and let $0 \\le i \\le n$.\nThen the cup product followed by the trace,\n$$H^i(X, \\mathcal{E}) \\times H^{n-i}(X, \\mathcal{E}^\\vee \\otimes \\omega_X) \\to H^n(X, \\omega_X) \\cong k,$$\nis a perfect pairing of finite-dimensional vector spaces.\nSo $H^i(X, \\mathcal{E}) \\cong H^{n-i}(X, \\mathcal{E}^\\vee \\otimes \\omega_X)^\\vee$, where $H^i$ is [[sheaf cohomology|PQ6Q143ARJ]].\n\n## Remarks\n\nThe idea of the proof: first treat $\\mathbb{P}^N_k$, where the cohomology of the sheaves $\\mathcal{O}(d)$ is computed explicitly by Čech cohomology and the pairing is visible on monomials.\nFor $X \\subseteq \\mathbb{P}^N_k$ of codimension $r$, the dualizing sheaf of $X$ is $\\mathcal{E}xt^r(\\mathcal{O}_X, \\omega_{\\mathbb{P}^N})$, and for smooth $X$ this sheaf is $\\omega_X$.\n\nExample: for a smooth projective curve $X$ of genus $g$ and $\\mathcal{E} = \\mathcal{O}_X$, duality gives $h^1(X, \\mathcal{O}_X) = h^0(X, \\omega_X) = g$.\nCombined with Riemann–Roch it gives $\\deg \\omega_X = 2g - 2$.\n\nProperness is necessary: for $X = \\mathbb{A}^1_k$, $H^0(X, \\mathcal{O}_X) = k[t]$ is infinite-dimensional while $H^1(X, \\omega_X) = 0$ by affine vanishing.\nMore generally the statement holds for projective Cohen–Macaulay schemes of pure dimension $n$, with $\\omega_X$ replaced by the dualizing sheaf.\n\n## Explanations\n\n### L1 human\n\nSerre duality says that the cohomology of a vector bundle in degree $i$ and the cohomology of a twisted dual bundle in the complementary degree $n - i$ carry the same information.\nThe twist is by the canonical bundle.\nIt halves the work in many computations and links the top cohomology to global sections, which are easier to understand.\n\n### L2 human\n\nSerre duality is the algebraic analogue of Poincaré duality for compact oriented manifolds.\nIt is used constantly with the Riemann–Roch theorem: on a curve, $h^1(L) = h^0(K - L)$, which turns Riemann–Roch into a statement about global sections only.\nIts general form, valid for proper morphisms and complexes of sheaves, is Grothendieck duality.\n",
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