Let $k$ be a perfect field of characteristic $p>0$. If $R$ is a perfect $k$-algebra, then the cotangent complex $L_{R/k}$ vanishes (the Frobenius is zero and induces an isomorphism on $L_{R/k}$ since $R$ is perfect). Is the converse true? Namely, if $L_{R/k}$ vanishes, then is $R$ perfect over $k$? I suspect not, but I can't find a counterexample. What happens if $k$ is a field of characteristic $0$?


1 Answer 1


see Bhargav Bhatt's example (suggested by Gabber) of an imperfect ring with trivial cotangent complex


  • $\begingroup$ Thanks. It still leaves open the case when k has characteristic 0, though. What happens then? $\endgroup$
    – skd
    Jan 2, 2018 at 1:46
  • $\begingroup$ @skd: Judging by Bhatt's note, it looks like it is not known. $\endgroup$
    – guest
    Jan 2, 2018 at 2:56

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