2 Added another example of L-trivial maps

The question is already in the title. Less succinctly, let's call a map $f:X \to Y$ of schemes $L$-trivial if its cotangent complex is quasi-isomorphic to $0$. Such maps have striking deformation-theoretic consequences; for example, any deformation of $Y$ can be followed uniquely by a deformation of $X$.

My primary (and probably naive) question is:

Is there a classification of $L$-trivial maps?

I am sure this question has been asked before, but I did not find any literature that deals with it. The two three examples of $L$-trivial maps I am familiar with are:

• Etale morphisms (and these are the only examples under finiteness constraints).
• Any map between perfect $\mathbb{F}_p$-schemes.
• The inclusion of the closed point in the spectrum of a valuation ring with divisible value group, or similar "divisible" constructions. For example, $\mathrm{Spec}(\mathbb{C}) \hookrightarrow \mathrm{Spec}(\mathbb{C}[ t^{\mathbb{Q}_{\geq 0}}])$ is $L$-trivial.

[ Edit: I learnt the last one in conversation after positing the first version of this question. ]

More examples can be obtained by taking filtered colimits of the above examples, but those are only slightly different. Hence, a second question is: are there other fundamentally different examples of $L$-trivial maps?

Perhaps a classification is unreasonable to expect, so I am also happy to learn more about $L$-trivial maps in other geometric categories, like algebraic stacks, or derived/spectral schemes/stacks, or (complex/rigid) analytic spaces, etc.. In particular, I am especially curious to know if $L$-trivial maps can be better understood using derived algebraic geometry.

1

When does the cotangent complex vanish?

The question is already in the title. Less succinctly, let's call a map $f:X \to Y$ of schemes $L$-trivial if its cotangent complex is quasi-isomorphic to $0$. Such maps have striking deformation-theoretic consequences; for example, any deformation of $Y$ can be followed uniquely by a deformation of $X$.

My primary (and probably naive) question is:

Is there a classification of $L$-trivial maps?

I am sure this question has been asked before, but I did not find any literature that deals with it. The two examples of $L$-trivial maps I am familiar with are:

• Etale morphisms (and these are the only examples under finiteness constraints).
• Any map between perfect $\mathbb{F}_p$-schemes.

More examples can be obtained by taking filtered colimits of the above examples, but those are only slightly different. Hence, a second question is: are there other fundamentally different examples of $L$-trivial maps?

Perhaps a classification is unreasonable to expect, so I am also happy to learn more about $L$-trivial maps in other geometric categories, like algebraic stacks, or derived/spectral schemes/stacks, or (complex/rigid) analytic spaces, etc.. In particular, I am especially curious to know if $L$-trivial maps can be better understood using derived algebraic geometry.