The Stacks project

Lemma 37.13.14. Consider a cartesian diagram of schemes

\[ \xymatrix{ X' \ar[r]_{g'} \ar[d] & X \ar[d] \\ Y' \ar[r] & Y } \]

If $X \to Y$ is flat, then the canonical map $(g')^*\mathop{N\! L}\nolimits _{X/Y} \to \mathop{N\! L}\nolimits _{X'/Y'}$ is a quasi-isomorphism. If in addition $\mathop{N\! L}\nolimits _{X/Y}$ has tor-amplitude in $[-1, 0]$ then $L(g')^*\mathop{N\! L}\nolimits _{X/Y} \to \mathop{N\! L}\nolimits _{X'/Y'}$ is a quasi-isomorphism too.

Proof. Translated into algebra this is More on Algebra, Lemma 15.78.3. To do the translation use Lemma 37.13.2 and Derived Categories of Schemes, Lemmas 36.3.5 and 36.9.4. $\square$


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