The vanishing of 𝑇𝑜𝑟₁^{𝑅}(𝑅⁺,𝑘) implies that 𝑅 is regular

Let ( R , m , k ) (R,m,k) be an excellent local ring of positive prime characteristic. We show that if Tor 1 R ⁡ ( R + , k ) = 0 \operatorname {Tor}_1^R(R^+,k) = 0 , then R R is regular. This improves a result of Schoutens, in which the additional hypothesis that R R was an isolated singularity was required for the proof.