Quillen defines a morphism $f:X→Y$ to be "of finite tor dimension" if $\mathcal{O}_X$ is of finite tor dimension as a module over $f^{-1}(\mathcal{O}_Y)$. The question is if there are sufficient conditions for $f$ to be of finite tor dimension, which are not too difficult to check.
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$\begingroup$ It suffices for $X$ to be regular. Are you specifically interested in the non-regular case? $\endgroup$– Steven LandsburgMay 5, 2014 at 0:36
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$\begingroup$ @StevenLandsburg: don't you mean Y? (another tautologically sufficient condition is for f to be flat) $\endgroup$– bananastackMay 5, 2014 at 0:56
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$\begingroup$ Yes, I meant $Y$. $\endgroup$– Steven LandsburgMay 5, 2014 at 0:59
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1$\begingroup$ The other straightforward case is when $X$ is a closed subscheme of $Y$, with the ideal of $X$ locally generated by a regular sequence, but I expect you already know this too. $\endgroup$– Steven LandsburgMay 5, 2014 at 1:13
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1$\begingroup$ I am specifically interested in case Y is not regular. Is there a good condition in case Y is integral and f is proper surjective? $\endgroup$– Shuji SaitoMay 5, 2014 at 8:26
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