The vanishing conjecture for maps of Tor and derived splinters
Linquan Ma
University of Utah, Salt Lake City, USA
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Abstract
We say an excellent local domain satisfies the vanishing conditions for maps of Tor, if for every with regular and module-finite torsion-free extension, and every -module , the map vanishes for every . 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.
Cite this article
Linquan Ma, The vanishing conjecture for maps of Tor and derived splinters. J. Eur. Math. Soc. 20 (2018), no. 2, pp. 315–338
DOI 10.4171/JEMS/768