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  "body": "\n## Definition\n\nLet $X$ be a smooth [[quasi-projective variety|EQSHJPE7H8]] over a field, with [[Chow ring|4HV4WTAX9P]] $CH^\\bullet(X)$ built from its [[Chow groups|EBEF6PEK4N]].\nFor an [[invertible sheaf|7JCF2TE3V5]] $L$ on $X$, let $c_1(L) \\in CH^1(X)$ be the class of a [[Cartier divisor|FHV05ZK99F]] $D$ with $L \\cong \\mathcal{O}_X(D)$.\nLet $E$ be a [[vector bundle|P4DZM975BV]] of rank $r$ on $X$.\nLet $\\pi \\colon \\mathbb{P}(E) \\to X$ be the associated projective bundle, $\\mathbb{P}(E) = \\operatorname{Proj} \\operatorname{Sym} E$, and let $\\xi = c_1(\\mathcal{O}_{\\mathbb{P}(E)}(1)) \\in CH^1(\\mathbb{P}(E))$, where $\\mathcal{O}_{\\mathbb{P}(E)}(1)$ is the [[twisting sheaf|V93Y0XYPZD]].\nThen $CH^\\bullet(\\mathbb{P}(E))$ is a free $CH^\\bullet(X)$-module, through $\\pi^*$, with basis $1, \\xi, \\ldots, \\xi^{r-1}$.\nThe *Chern classes* $c_i(E) \\in CH^i(X)$, for $0 \\le i \\le r$, are the unique classes with $c_0(E) = 1$ and\n$$\\sum_{i=0}^{r} (-1)^i \\, \\pi^* c_i(E) \\, \\xi^{r-i} = 0.$$\nOne sets $c_i(E) = 0$ for $i \u003e r$, and the *total Chern class* is $c(E) = \\sum_i c_i(E)$.\n\n## Remarks\n\nThis is Grothendieck's definition.\nThe Chern classes satisfy the Whitney formula $c(E) = c(E') \\, c(E'')$ for every exact sequence $0 \\to E' \\to E \\to E'' \\to 0$, and they are compatible with pullback.\nExample: for a line bundle $L$, $\\mathbb{P}(L) = X$ and $\\mathcal{O}_{\\mathbb{P}(L)}(1) = L$, so $\\xi = c_1(L)$, and the relation $\\xi - c_1(L) = 0$ agrees with the definition of $c_1(L)$ for invertible sheaves.\nOn $\\mathbb{P}^n$ with hyperplane class $h$, the Euler sequence gives $c(T_{\\mathbb{P}^n}) = (1 + h)^{n+1}$.\nNon-example: the skyscraper sheaf $k(P)$ at a point $P$ of a curve is coherent but not locally free.\nSo it is not a vector bundle, and the definition does not apply to it directly.\nOver $\\C$, the cycle class map sends $c_i(E)$ to the topological Chern class $c_i(E) \\in H^{2i}(X(\\C), \\Z)$.\n",
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