Commutative algebra Homological methods Theorem

Direct summand conjecture

Builds on Regular local ring Noetherian ring Finitely generated module

Statement

Let $R$ be a regular local ring and let $R \subseteq S$ be an extension of commutative rings such that $S$ is a finitely generated $R$ -module. Then the inclusion $R \to S$ splits as a map of $R$ -modules: there is an $R$ -linear map $\rho \colon S \to R$ with $\rho(1) = 1$ .

Remarks

Suppose that $R$ contains the rational numbers and $S$ is a domain, and let $F$ and $L$ be the fraction fields of $R$ and $S$ . Then $\rho = \frac{1}{[L : F]} \operatorname{Tr}_{L/F}$ is a splitting. It maps $S$ into $R$ because $R$ is regular, hence integrally closed. Hochster proved the statement when $R$ contains a field, and André proved it in mixed characteristic, using perfectoid methods. It is one of Hochster's homological conjectures, and it kept the name "conjecture" after its proof. Regularity 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.

Going deeper

For a student

The statement is equivalent to the monomial conjecture, and it implies several other homological conjectures. A 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.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Tag	`CQJ1FPKKHT`
URI	`m2://CQJ1FPKKHT`
Kind	statement, theorem
Writing	W0 stub
MSC	13D22, 13H05, 13B21
Status	This copy of the site has no build data. Run `m2 status CQJ1FPKKHT`.
About	Regular local ring
Uses	Regular local ring, Noetherian ring, Finitely generated module
Changed	2 November 2026