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  "body": "\n## Summary\n\nTight closure is an operation on ideals of Noetherian rings of prime characteristic $p$, defined with the [[Frobenius endomorphism|0T3WVTD675]].\nIt was introduced by Melvin Hochster and Craig Huneke.\nThe tight closure $I^*$ of an ideal $I$ contains $I$ and is contained in the integral closure of $I$.\nIn a regular ring, $I^* = I$.\nIt gives short proofs of results such as the [[Hochster–Roberts theorem|56107GMFYV]] and the Briançon–Skoda theorem.\nIt also leads to the classes of F-regular and F-rational rings.\n\n## Explanations\n\n### L1 human\n\nIn characteristic $p$, raising to the $p$-th power is a ring homomorphism, and one can raise elements to the powers $q = p, p^2, p^3, \\ldots$.\nAn element $x$ is in the tight closure of $I$ when its powers $x^q$ almost lie in the ideal generated by the $q$-th powers of elements of $I$.\n\"Almost\" means: up to one fixed multiplier $c$ that is not too degenerate.\nElements of $I$ are in it, and so are elements that differ from $I$ by something \"infinitely small\", such as a nilpotent element.\nIn a polynomial ring, nothing else is added: every ideal is tightly closed.\n",
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