2015/09/15 by Linquan Ma, Ma, Linquan
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Rings, Modules, and Algebras #math.AC #math.AG
paper · pdf · doi:10.48550/arxiv.1509.04400
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openalex publication_date 2015/09/15 · arxiv created 2015/12/04 · arxiv updated 2015/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 TorAi(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. 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.