Almost direct summands. (English) Zbl 1304.13042

Bhatt, Bhargav

Nagoya Math. J. 214, 195-204 (2014).

The direct summand conjecture of Hochster asserts that any module-finite extension \(R\to S\) of commutative rings with \(R\) regular is a direct summand as an \(R\)-module map. This conjecture is known for some special cases. The general mixed characteristic case remains wide open, and is a fundamental open problem in commutative algebra.
In the paper under review, the author uses Faltings’ theory of almost étale extensions in \(p\)-adic Hodge theory to prove some new cases of the direct summand conjecture. More precisely, the author shows that:
Let \(V\) be a complete \(p\)-adic discrete valuation ring whose residue field \(k\) satisfies \([k : k^p]<\infty\). Let \(R\) be a smooth \(V\)-algebra, and let \(f: R\to S\) be the normalization of \(R\) in a finite extension of its fraction field. Assume that there exists an ètale map \(V[T_1,\ldots, T_d]\to R\) such that \(f\otimes_R R[1/p\cdot T_1\cdots T_d]\) is unramified. Then \(f:R\to S\) is a direct summand as an \(R\)-module map.

Reviewer: Siamak Yassemi (Tehran)

MSC:

13H05	Regular local rings
13D22	Homological conjectures (intersection theorems) in commutative ring theory
14B05	Singularities in algebraic geometry

Keywords:

direct summand conjecture; almost étale extensions in \(p\)-adic Hodge theory

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