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  "body": "\n## Summary\n\nLet $M$ be a module over a commutative Noetherian ring and let $I$ be an ideal.\nThe local cohomology modules $H^i_I(M)$ are the right derived functors of the functor of sections supported on $I$.\nIt was introduced by Grothendieck.\nOver a Noetherian local ring, local cohomology at the maximal ideal detects the [[depth|M2RADBFW99]] and the [[Krull dimension|MVGBS3J1P7]] of a finitely generated module.\nIt is the subject of [[Grothendieck's local duality|VA3DY2YX83]].\n\n## Explanations\n\n### L1 human\n\nTake the elements of a module that are killed by some power of an ideal $I$.\nGeometrically, these are the sections that live only near the closed set cut out by $I$.\nTaking these elements is not an exact operation, and local cohomology records the correction terms.\nFor $R = K[x]$ and $I = (x)$, no nonzero polynomial is killed by a power of $x$, so $H^0_I(R) = 0$.\nThe first local cohomology is $H^1_I(R) = K[x, x^{-1}]/K[x]$, spanned by the classes of $x^{-1}, x^{-2}, \\ldots$.\n",
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