The Stacks project

Lemma 37.71.2. Consider a cartesian diagram

\[ \xymatrix{ X \ar[r]_ i \ar[d]_ f & X' \ar[d]^{f'} \\ Y \ar[r]^ j & Y' } \]

of schemes. Assume $X' \to Y'$ is flat and locally of finite presentation and $Y \to Y'$ is a finite order thickening. Let $E' \in D(\mathcal{O}_{X'})$. If $E = Li^*(E')$ is $Y$-perfect, then $E'$ is $Y'$-perfect.

Proof. Recall that being $Y$-perfect for $E$ means $E$ is pseudo-coherent and locally has finite tor dimension as a complex of $f^{-1}\mathcal{O}_ Y$-modules (Derived Categories of Schemes, Definition 36.35.1). By Lemma 37.71.1 we find that $E'$ is pseudo-coherent. In particular, $E'$ is in $D_\mathit{QCoh}(\mathcal{O}_{X'})$, see Derived Categories of Schemes, Lemma 36.10.1. To prove that $E'$ locally has finite tor dimension we may work locally on $X'$. Hence we may assume $X'$, $S'$, $X$, $S$ are affine, say given by rings $A'$, $R'$, $A$, $R$. Then we reduce to the commutative algebra version by Derived Categories of Schemes, Lemma 36.35.3. The commutative algebra version in More on Algebra, Lemma 15.83.8. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0DJY. Beware of the difference between the letter 'O' and the digit '0'.