Geodesic
The vanishing conjecture for maps of Tor and derived splinters
Journal of the European Mathematical Society, Tome 20 (2018) no. 2, pp. 315-338

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We say an excellent local domain (S,n) satisfies the vanishing conditions for maps of Tor, if for every A→R→S with A regular and A→R module-finite torsion-free extension, and every A-module M, the map ToriA​(M,R)→ToriA​(M,S) vanishes for every i≥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.
DOI : 10.4171/jems/768
Classification : 13-XX, 14-XX
Keywords: The vanishing conjecture for maps of Tor, rational singularity, derived splinters 
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     title = {The vanishing conjecture for maps of {Tor} and derived splinters},
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     year = {2018},
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Linquan Ma. The vanishing conjecture for maps of Tor and derived splinters. Journal of the European Mathematical Society, Tome 20 (2018) no. 2, pp. 315-338. doi: 10.4171/jems/768

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