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  "body": "\n## Statement\n\nLet $f \\colon X \\to Y$ be a [[proper morphism|9537RKBZPP]] of [[Noetherian schemes|ND6XEY68P4]].\nLet $D^+_{qc}(X)$ be the bounded below [[derived category|5ST7NVJFSM]] of $\\mathcal{O}_X$-modules with [[quasi-coherent|KPKND0K7W6]] cohomology sheaves, and similarly for $Y$.\nLet $Rf_* \\colon D^+_{qc}(X) \\to D^+_{qc}(Y)$ be the right [[derived functor|NPMC3K6J2F]] of $f_*$.\nThen $Rf_*$ has a right adjoint $f^! \\colon D^+_{qc}(Y) \\to D^+_{qc}(X)$: for $\\mathcal{F} \\in D^+_{qc}(X)$ and $\\mathcal{G} \\in D^+_{qc}(Y)$ there is a natural isomorphism\n$$\\Hom_{D(Y)}(Rf_* \\mathcal{F}, \\mathcal{G}) \\cong \\Hom_{D(X)}(\\mathcal{F}, f^! \\mathcal{G}).$$\nIf moreover $f$ is smooth of relative dimension $n$, then $f^! \\mathcal{G} \\cong Lf^* \\mathcal{G} \\otimes \\omega_{X/Y}[n]$, where $\\omega_{X/Y} = \\bigwedge^n \\Omega_{X/Y}$ is the top exterior power of the [[sheaf of relative differentials|6VC6HQTFSA]].\n\n## Remarks\n\nGrothendieck's theory of duality was first written up in Hartshorne's book Residues and Duality.\nIn a later approach by Neeman, a right adjoint of $Rf_*$ on the unbounded category $D_{qc}(X)$ exists by Brown representability.\nThe main content is then the explicit form of $f^!$ and of the trace map $Rf_* f^! \\to \\mathrm{id}$.\n\nExample: let $Y = \\operatorname{Spec} k$ and $X$ a smooth projective variety of dimension $n$ over $k$.\nThen $f^! k = \\omega_X[n]$, and taking $\\mathcal{F}$ to be a vector bundle placed in degree $-i$ gives Serre duality $H^{n-i}(X, \\mathcal{E}^\\vee \\otimes \\omega_X) \\cong H^i(X, \\mathcal{E})^\\vee$.\n\nFor a finite morphism $f$, the right adjoint is $f^! \\mathcal{G} = R\\mathcal{H}om_{\\mathcal{O}_Y}(f_* \\mathcal{O}_X, \\mathcal{G})$, viewed as a complex of $\\mathcal{O}_X$-modules.\n\n## Explanations\n\n### L1 human\n\nGrothendieck duality is a relative version of Serre duality.\nInstead of one variety over a field, it treats a family of varieties over a base, and it works with complexes of sheaves instead of single sheaves.\nIt says that pushing sheaves forward along a proper map has a partner that pulls them back in a twisted way, and the twist is by the relative canonical bundle.\n\n### L2 human\n\nThe formula $f^! = Lf^* \\otimes \\omega_{X/Y}[n]$ for smooth $f$ and the description of $f^!$ for finite $f$ together compute $f^!$ for projective morphisms, by factoring through a closed immersion into projective space.\nThe complex $f^! \\mathcal{O}_Y$ is the relative dualizing complex; for flat $f$ of finite type it is concentrated in one degree exactly when the fibers of $f$ are Cohen–Macaulay.\nGrothendieck duality underlies the theory of dualizing complexes, local duality and the Grothendieck–Riemann–Roch formalism of pushforwards.\n",
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