  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.
