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  "body": "\n## Definition\n\nLet $X$ be a [[Noetherian scheme|ND6XEY68P4]], and let $\\operatorname{Coh}(X)$ be the abelian category of [[coherent sheaves|56RXEJ8Z6S]] on $X$.\nA [[chain complex|WQ82C0M3QB]] $F^\\bullet$ of coherent sheaves is *bounded* if $F^i = 0$ for all but finitely many $i$.\nConsider the category of bounded complexes in $\\operatorname{Coh}(X)$ with chain maps.\nThe *bounded derived category of coherent sheaves* of $X$ is the category $D^b(X) = D^b(\\operatorname{Coh}(X))$ obtained from it by formally inverting all [[quasi-isomorphisms|C9JEC2WQKV]].\nIt is the bounded [[derived category|5ST7NVJFSM]] of $\\operatorname{Coh}(X)$.\n\n## Remarks\n\n$D^b(X)$ is a [[triangulated category|T2V7GJJ2EN]] with shift $F^\\bullet \\mapsto F^\\bullet[1]$.\nIt is equivalent to the full subcategory of the derived category of all $\\mathcal{O}_X$-modules whose objects are the complexes with coherent cohomology sheaves, nonzero in finitely many degrees.\nExample: for $X = \\operatorname{Spec} k$ with $k$ a field, coherent sheaves are finite-dimensional vector spaces, and every object of $D^b(X)$ is isomorphic to $\\bigoplus_i H^i(F^\\bullet)[-i]$, its cohomology placed in each degree.\nNon-example: on $X = \\mathbb{A}^1_k = \\operatorname{Spec} k[x]$, the complex $k[x] \\xrightarrow{x} k[x]$ in degrees $-1, 0$ maps to $k[x]/(x)$ in degree $0$ by a quasi-isomorphism.\nIn the homotopy category of complexes, before quasi-isomorphisms are inverted, these two complexes are not isomorphic, since there is no nonzero map of modules $k[x]/(x) \\to k[x]$.\nIn $D^b(X)$ they are isomorphic.\nTheorems on $D^b$: Beilinson showed that $D^b(\\mathbb{P}^n)$ is generated by $\\mathcal{O}, \\mathcal{O}(1), \\ldots, \\mathcal{O}(n)$, and Mukai showed that an abelian variety and its dual have equivalent derived categories.\n",
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