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I have read a discussion (in a less common language) which discussed a homotopical interpretation of flatness, which went something like:

A map of commutative algebras is flat if pushing it out along any other morphism is quasi-isomorphic to the derived pushout when the algebras are embedded into dg-algebras. Hence flatness is just another name for quasi-fibrations.

Another bloke remarked:

Flatness is about giving the correct pullback, but it may also be obtained without flatness. Of course the correct condition is Tor-independence, i.e the tensor product $A\otimes_\Bbbk B$ is correct iff $\operatorname{Tor}^n_\Bbbk(A,B)=0$ for all $n$.

Where is this viewpoint written down and/or developed clearly? I'm not at a level where I can make out the picture from a few hints. Does 'derived' means 'homotopy' here? Why do dg-algebras pop up? What's this Tor-independence business all about?

Finally, the first bloke also wrote there's a homotopical characterization of étaleness, which I would also like to know more about.

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    $\begingroup$ Flatness is equivalent to vanishing of all higher Tor groups, which are the same as the homology groups of derived tensor products, and derived tensor products are homotopy pushouts of commutative dg-algebras. A homotopical characterization of étaleness or smoothness can be given using the cotangent complex. See any of the usual references on derived algebraic geometry, e.g. Lurie's thesis or HAG II by Toen-Vezzosi. This all goes back to Quillen, whose original notes on homotopical algebra and the cotangent complex are also very good. $\endgroup$
    – AAK
    Commented Jun 22, 2016 at 14:11

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The derived tensor product is defined as a homotopy pushout of $M\leftarrow R\rightarrow N$ in commutative DGAs. Now, if $R$ is a commutative ring and $M$ and $N$ are $R$-algebras, then the homology groups $H^i(M\otimes^\mathbf{L}_R N)=\operatorname{Tor}^R_i(M,N)$. Therefore $M\otimes_R^\mathbf{L} N$ and $M\otimes_R N$ are quasi-isomorphic for all $N$ iff $M$ is flat. Tor-independence is an isomorphic condition (Definition 55.1 in http://stacks.math.columbia.edu/download/more-algebra.pdf).

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  • $\begingroup$ The "iff" should just be "if". $\endgroup$
    – Qiaochu Yuan
    Commented Dec 26, 2016 at 19:58
  • $\begingroup$ or "iff $M$ or $N$ is flat" should be "for all $N$ iff $M$ is flat" $\endgroup$
    – Will Sawin
    Commented Dec 26, 2016 at 20:39

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