The Stacks project

Lemma 88.8.4. The cotangent complex $L_{B/A}$ is zero in each of the following cases:

  1. $A \to B$ and $B \otimes _ A B \to B$ are flat, i.e., $A \to B$ is weakly étale (More on Algebra, Definition 15.93.1),

  2. $A \to B$ is a flat epimorphism of rings,

  3. $B = S^{-1}A$ for some multiplicative subset $S \subset A$,

  4. $A \to B$ is unramified and flat,

  5. $A \to B$ is étale,

  6. $A \to B$ is a filtered colimit of ring maps for which the cotangent complex vanishes,

  7. $B$ is a henselization of a local ring of $A$,

  8. $B$ is a strict henselization of a local ring of $A$, and

  9. add more here.

Proof. In case (1) we may apply Lemma 88.8.2 to the surjective flat ring map $B \otimes _ A B \to B$ to conclude that $L_{B \otimes _ A B/B} = 0$ and then we use Lemma 88.8.3 to conclude. The cases (2) – (5) are each special cases of (1). Part (6) follows from Lemma 88.3.4. Parts (7) and (8) follows from the fact that (strict) henselizations are filtered colimits of étale ring extensions of $A$, see Algebra, Lemmas 10.150.7 and 10.150.13. $\square$


Comments (1)

Comment #4517 by Manuel Hoff on

In the proof, should be replaced by .


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