The vanishing conjecture for maps of Tor and derived splinters

Abstract

We say an excellent local domain $(S,\mathfrak{n})$ 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 ${\mathrm{Tor}}_i^A(M,R)\to {\mathrm{Tor}}_i^A(M,S)$ vanishes for every $i\ge 1$. Hochster-Huneke's conjecture (theorem in equal characteristic) thus states that regular rings satisfy such vanishing conditions [HH95]. 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 [Bha12]. In particular, rational singularities in characteristic 0 satisfy the vanishing conditions. This greatly generalizes Hochster–Huneke’s result [HH95] and Boutot’s theorem [Bou87]. Moreover, our result leads to a new (and surprising) characterization of rational singularities in terms of splittings in module-finite extensions.