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  "body": "\n## Statement\n\nLet $X$ and $Y$ be [[complex analytic spaces|HXSZA44J5J]], let $f \\colon X \\to Y$ be a [[proper|15XEBGNGHC]] holomorphic map and let $\\mathcal{F}$ be a [[coherent analytic sheaf|DSJET02WPR]] on $X$.\nFor an integer $q \\ge 0$, let $R^q f_* \\mathcal{F}$ be the sheaf on $Y$ associated with the presheaf $V \\mapsto H^q(f^{-1}(V), \\mathcal{F})$ of [[sheaf cohomology|PQ6Q143ARJ]] groups.\nThen $R^q f_* \\mathcal{F}$ is a coherent analytic sheaf on $Y$ for every $q \\ge 0$.\n\n## Remarks\n\nWhen $Y$ is a single point, $R^q f_* \\mathcal{F}$ is the vector space $H^q(X, \\mathcal{F})$ and $X$ is compact, so the theorem contains the [[Cartan–Serre finiteness theorem|MPNGD4V271]].\nThe case where $f$ is finite and $q = 0$ is easier and is proved with the [[Weierstrass division theorem|FE83QQS17S]].\nThe general case is much harder.\nGrauert's original proof was long; a later proof by Kiehl and Verdier uses Čech cochains of Stein coverings and compact operators between Fréchet modules.\n\nExample: if $A \\subseteq X$ is an analytic subset with its reduced structure, then $f_* \\mathcal{O}_A$ is coherent on $Y$, and its support is $f(A)$.\nSince supports of coherent sheaves are analytic, this gives [[Remmert's proper mapping theorem|VA2JEWWB61]].\n\nProperness is necessary.\nConsider the inclusion $f \\colon \\C \\setminus \\set{0} \\to \\C$.\nThe stalk of $f_* \\mathcal{O}$ at $0$ is the ring of holomorphic functions on punctured discs around $0$, modulo agreement near $0$.\nIt is not finitely generated over $\\mathcal{O}_{\\C,0}$.\nIndeed it is nonzero and multiplication by $z$ is bijective on it, so Nakayama's lemma rules out finite generation.\n\n## Explanations\n\n### L1 human\n\nGiven a proper holomorphic map, one can push a coherent sheaf forward along it and take the cohomology along the fibres.\nThe theorem says that the result is again a coherent sheaf on the target.\nSo finiteness survives when one looks at a whole family of compact spaces at once, not only at one compact space.\n\n### L2 human\n\nThe theorem is the basic finiteness result of relative complex analytic geometry.\nIt is used to study families of compact complex spaces, for example to prove semicontinuity of the dimensions of cohomology groups of fibres.\nIt is the analytic analogue of the coherence of higher direct images of coherent sheaves under proper morphisms of schemes.\n",
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