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  "body": "\n## Statement\n\nLet $k$ be a field and let $f \\colon X \\to Y$ be a [[projective morphism|Z5G47MQ99B]] of [[smooth|EXKGG9X6J0]] quasi-projective varieties over $k$.\nLet $K_0(X)$ be the [[Grothendieck group|5FDMY7PDYY]] of [[coherent sheaves|56RXEJ8Z6S]] on $X$, and let $f_! \\colon K_0(X) \\to K_0(Y)$ be the map $[\\mathcal{F}] \\mapsto \\sum_{i \\ge 0} (-1)^i [R^i f_* \\mathcal{F}]$.\nLet $A^*(X)_\\Q$ be the rational [[Chow ring|EBEF6PEK4N]], with proper pushforward $f_* \\colon A^*(X)_\\Q \\to A^*(Y)_\\Q$.\nLet $\\operatorname{ch} \\colon K_0(X) \\to A^*(X)_\\Q$ be the Chern character, built from the [[Chern classes|PAD37GKDPD]], and let $\\operatorname{td}$ be the [[Todd class|2YDB5089FR]] of a tangent bundle.\nThen for every $x \\in K_0(X)$,\n$$\\operatorname{ch}(f_! x) \\, \\operatorname{td}(T_Y) = f_*\\big(\\operatorname{ch}(x) \\, \\operatorname{td}(T_X)\\big) \\quad \\text{in } A^*(Y)_\\Q.$$\n\n## Remarks\n\nThe idea of the proof: factor $f$ as a closed immersion $X \\to \\mathbb{P}^N_Y$ followed by the projection $\\mathbb{P}^N_Y \\to Y$.\nFor the projection, both sides are computed from the explicit structure of $K_0$ and of the Chow ring of a projective bundle.\nFor a closed immersion, one deforms to the normal bundle and reduces to the zero section of a vector bundle, where the Koszul complex gives the formula.\n\nExample: for $Y = \\operatorname{Spec} k$, $A^*(Y)_\\Q = \\Q$, $\\operatorname{td}(T_Y) = 1$ and $\\operatorname{ch}(f_! [\\mathcal{F}]) = \\chi(X, \\mathcal{F})$.\nSo the theorem becomes the Hirzebruch–Riemann–Roch formula $\\chi(X, \\mathcal{F}) = \\int_X \\operatorname{ch}(\\mathcal{F}) \\operatorname{td}(T_X)$.\n\nWithout the Todd classes the formula is false: the Chern character alone does not commute with pushforward, as already the case $f \\colon \\mathbb{P}^1_k \\to \\operatorname{Spec} k$ with $\\mathcal{F} = \\mathcal{O}$ shows, where $\\chi = 1$ but $f_* \\operatorname{ch}(\\mathcal{O}) = 0$.\n\n## Explanations\n\n### L1 human\n\nGrothendieck–Riemann–Roch is a version of Riemann–Roch for families.\nIt does not compute one number, the Euler characteristic of a sheaf on one variety.\nInstead it computes how the cohomology of a sheaf varies over a base, as a class on the base.\nThe correction factor is the Todd class, which measures the geometry of the varieties involved.\n\n### L2 human\n\nThe theorem says that the Chern character, corrected by Todd classes, is compatible with pushforward: a commutative square between K-theory and the Chow ring.\nIt is used to compute Chern classes of Hodge bundles and other natural bundles on moduli spaces, such as Mumford's computation on the moduli space of curves.\nThe case of a point recovers the Hirzebruch–Riemann–Roch theorem, and the topological analogue is the Atiyah–Singer family index theorem.\n",
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