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

Aberbach, Ian

doi:10.1090/S0002-9939-04-07491-X

The vanishing of $\operatorname {Tor}_1^R(R^+,k)$ implies that $R$ is regular
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by Ian M. Aberbach PDF

Proc. Amer. Math. Soc. 133 (2005), 27-29 Request permission

Abstract:

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

References

I. M. Aberbach and F. Enescu, The structure of $F$-pure rings, preprint, 2003.
Melvin Hochster, Cyclic purity versus purity in excellent Noetherian rings, Trans. Amer. Math. Soc. 231 (1977), no. 2, 463–488. MR 463152, DOI 10.1090/S0002-9947-1977-0463152-5
Melvin Hochster and Craig Huneke, Infinite integral extensions and big Cohen-Macaulay algebras, Ann. of Math. (2) 135 (1992), no. 1, 53–89. MR 1147957, DOI 10.2307/2946563
Melvin Hochster and Craig Huneke, $F$-regularity, test elements, and smooth base change, Trans. Amer. Math. Soc. 346 (1994), no. 1, 1–62. MR 1273534, DOI 10.1090/S0002-9947-1994-1273534-X
Craig Huneke, Tight closure and its applications, CBMS Regional Conference Series in Mathematics, vol. 88, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996. With an appendix by Melvin Hochster. MR 1377268, DOI 10.1016/0167-4889(95)00136-0
Ernst Kunz, Characterizations of regular local rings of characteristic $p$, Amer. J. Math. 91 (1969), 772–784. MR 252389, DOI 10.2307/2373351
Ernst Kunz, On Noetherian rings of characteristic $p$, Amer. J. Math. 98 (1976), no. 4, 999–1013. MR 432625, DOI 10.2307/2374038
H. Schoutens, On the vanishing of Tor for the absolute integral closure, J. Algebra 275 (2004), 567–574.