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  "body": "\n## Summary\n\nAn elliptic complex is a sequence of [[differential operators between vector bundles|CGBX1N9V8H]] whose consecutive composites vanish.\nIts [[principal symbols|S8GC34D5ZG]] must form an exact sequence at each nonzero covector.\nIt generalises an [[elliptic operator|MT9P2X85V8]], which is an elliptic complex of length two.\nOn a compact manifold its cohomology groups are finite-dimensional, and its index is their alternating sum of dimensions.\nThe [[de Rham complex|GX74C07T0G]] and the Dolbeault complex are the basic examples.\n\n## Explanations\n\n### L1 human\n\nSome important operators come in chains, where applying two consecutive operators always gives zero, as $d \\circ d = 0$ for the exterior derivative.\nThen one studies the cohomology: what the next operator kills, modulo what the previous one produces.\nAn elliptic complex is a chain of this kind that is elliptic in the sense that, at top order and in every nonzero frequency direction, the chain is exact.\nThis makes the cohomology finite-dimensional on a closed manifold.\nFor the de Rham complex, this cohomology is the de Rham cohomology, and its alternating sum of dimensions is the Euler characteristic.\n",
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