The vanishing conjecture for maps of Tor and derived splinters

We say an excellent local domain $(S,n)$ satisfies the vanishing conditions for maps of Tor, if for every $A\to R\to S$ with $A$ regular and $A\to R$ module-finite torsion-free extension, and every $A$-module $M$, the map $Tor^A_i(M, R)\to Tor_i^A(M, S)$ vanishes for every $i\geq 1$. Hochster-Huneke's conjecture (theorem in equal characteristic) thus states that regular rings satisfy such vanishing conditions. The main theorem of this paper shows that, in equal characteristic, rings that satisfy the vanishing conditions for maps of Tor are exactly derived splinters in the sense of Bhatt. In particular, rational singularities in characteristic $0$ satisfy the vanishing conditions. This greatly generalizes Hochster-Huneke's result and Boutot's theorem. Moreover, this leads to a new (and surprising) characterization of rational singularities in terms of splittings in module-finite extensions.