5
$\begingroup$

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 $0$?

Improve this question
$\endgroup$

1 Answer 1

Reset to default
3
$\begingroup$

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

http://www-personal.umich.edu/~bhattb/papers.html

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

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .