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**The vanishing conjecture for maps of Tor and derived splinters.**
*(English)*
Zbl 1391.13025

A theorem of Hochster and Huneke says that if \(A\) is an equal characteristic regular domain, \(R\) is a module-finite and torsion-free extension of \(A\) and \(R \rightarrow S\) is any homomorphism from \(R\) to a regular ring \(S\) then for every \(A\)-module \(M\) and every \(i \geq 1\), the map \(\mathrm{Tor}_i^A (M,R) \rightarrow \mathrm{Tor}_i^A(M,S)\) vanishes. They conjectured that it also holds in mixed characteristic. This has become known as the vanishing conjecture for vanishing of Tor, and it implies other well-known conjectures. Extending this, the author says that an excellent local domain \((S,\mathfrak n)\) satisfies the vanishing conditions for maps of Tor if, for every \(A \rightarrow R \rightarrow S\) with \(A\) regular and \(A \rightarrow R\) a module-finite torsion-free extension, and every \(A\)-module \(M\), the map \(\mathrm{Tor}_i^A (M,R) \rightarrow \mathrm{Tor}_i^A(M,S)\) vanishes for every \(i \geq 1\). The main result of this paper is that in equal characteristic, rings that satisfy the vanishing conditions for maps of Tor are exactly derived splinters in the sense of a paper of Bhatt. The author also shows that an equivalent condition is that for every regular local ring \(A\) with \(S = A/P\) and every module-finite torsion-free extension \(A \rightarrow B\) with \(Q \in \mathrm{Spec } B\) lying over \(P\), the map \(P \rightarrow Q\) splits as a map of \(A\)-modules. The author concludes with a corollary that characterizes rational singularities in terms of splittings in module-finite extensions.

Reviewer: Juan C. Migliore (Notre Dame)