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  "body": "\n## Statement\n\nLet $R$ be a [[regular local ring|63XXTF6B5M]] and let $R \\subseteq S$ be an extension of commutative rings such that $S$ is a [[finitely generated|05QY1VEBYD]] $R$-module.\nThen the inclusion $R \\to S$ splits as a map of $R$-modules: there is an $R$-linear map $\\rho \\colon S \\to R$ with $\\rho(1) = 1$.\n\n## Remarks\n\nSuppose that $R$ contains the rational numbers and $S$ is a domain, and let $F$ and $L$ be the fraction fields of $R$ and $S$.\nThen $\\rho = \\frac{1}{[L : F]} \\operatorname{Tr}_{L/F}$ is a splitting.\nIt maps $S$ into $R$ because $R$ is regular, hence integrally closed.\nHochster proved the statement when $R$ contains a field, and André proved it in mixed characteristic, using perfectoid methods.\nIt is one of [[Hochster's homological conjectures|1TNS604ZST]], and it kept the name \"conjecture\" after its proof.\nRegularity cannot be dropped: for $R = K[\\![t^2, t^3]\\!] \\subseteq S = K[\\![t]\\!]$, a splitting $\\rho$ would give $t^3 = \\rho(t^3) = t^2 \\rho(t)$, so $\\rho(t) = t \\notin R$, a contradiction.\n\n## Explanations\n\n### L1 human\n\nIf a nice ring sits inside a finite extension ring, one can project the big ring back onto the small one in a linear way.\nFor rings containing the rationals this is easy, by averaging with a trace.\nFor rings like the $p$-adic integers it was a famous open problem for decades.\n\n### L2 human\n\nThe statement is equivalent to the monomial conjecture, and it implies several other homological conjectures.\nA ring that is a direct summand of a ring is often forced to inherit good properties; for example, a direct summand of a normal domain is normal.\n",
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  "msc": [
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  "name": "Direct summand conjecture",
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