The Stacks project

Lemma 48.18.3 (Makeshift base change). In Situation 48.16.1 let

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

be a cartesian diagram of $\textit{FTS}_ S$. Let $E \in D^+_\mathit{QCoh}(\mathcal{O}_ Y)$. If $f$ is flat, then $L(g')^*f^!E$ and $(f')^!Lg^*E$ restrict to isomorphic objects of $D(\mathcal{O}_{U'})$ for $U' \subset X'$ affine open mapping into affine opens of $Y$, $Y'$, and $X$.

Proof. By our assumptions we immediately reduce to the case where $X$, $Y$, $Y'$, and $X'$ are affine. Say $Y = \mathop{\mathrm{Spec}}(R)$, $Y' = \mathop{\mathrm{Spec}}(R')$, $X = \mathop{\mathrm{Spec}}(A)$, and $X' = \mathop{\mathrm{Spec}}(A')$. Then $A' = A \otimes _ R R'$. Let $E$ correspond to $K \in D^+(R)$. Denoting $\varphi : R \to A$ and $\varphi ' : R' \to A'$ the given maps we see from Remark 48.17.4 that $L(g')^*f^!E$ and $(f')^!Lg^*E$ correspond to $\varphi ^!(K) \otimes _ A^\mathbf {L} A'$ and $(\varphi ')^!(K \otimes _ R^\mathbf {L} R')$ where $\varphi ^!$ and $(\varphi ')^!$ are the functors from Dualizing Complexes, Section 47.24. The result follows from Dualizing Complexes, Lemma 47.24.6. $\square$

Comments (6)

Comment #4669 by Bogdan on

Can't we use, and to pin down an isomorphism ?

Comment #4676 by on

Yes, we can. This takes a lot more work though and it is very hard to use the extra information (about canonicalness of the identifications) obtained in this manner. But maybe we should update the text with a comment saying what you said.

Comment #4800 by on

Thanks again. I have added a bit of discussion. See here.

Comment #5101 by Bogdan on

I think one has to assume that since otherwise is not well-defined.

Comment #5102 by Bogdan on

I think one has to assume that since otherwise is not well-defined.

Comment #5103 by on

Argh! It seems you are correct. This just is a terrible lemma. Luckily it seems we only use it once!

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Your email address will not be published.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 0BZY. Beware of the difference between the letter 'O' and the digit '0'.