The Stacks project

Remark 48.12.5. Let $Y$ be a quasi-compact and quasi-separated scheme. Let $f : X \to Y$ be a proper, flat morphism of finite presentation. Let $a$ be the adjoint of Lemma 48.3.1 for $f$. In this situation, $\omega _{X/Y}^\bullet = a(\mathcal{O}_ Y)$ is sometimes called the relative dualizing complex. By Lemma 48.12.3 there is a functorial isomorphism $a(K) = Lf^*K \otimes _{\mathcal{O}_ X}^\mathbf {L} \omega _{X/Y}^\bullet $ for $K \in D_\mathit{QCoh}(\mathcal{O}_ Y)$. Moreover, the trace map

\[ \text{Tr}_{f, \mathcal{O}_ Y} : Rf_*\omega _{X/Y}^\bullet \to \mathcal{O}_ Y \]

of Section 48.7 induces the trace map for all $K$ in $D_\mathit{QCoh}(\mathcal{O}_ Y)$. More precisely the diagram

\[ \xymatrix{ Rf_*a(K) \ar[rrr]_{\text{Tr}_{f, K}} \ar@{=}[d] & & & K \ar@{=}[d] \\ Rf_*(Lf^*K \otimes _{\mathcal{O}_ X}^\mathbf {L} \omega _{X/Y}^\bullet ) \ar@{=}[r] & K \otimes _{\mathcal{O}_ Y}^\mathbf {L} Rf_*\omega _{X/Y}^\bullet \ar[rr]^-{\text{id}_ K \otimes \text{Tr}_{f, \mathcal{O}_ Y}} & & K } \]

where the equality on the lower right is Derived Categories of Schemes, Lemma 36.21.1. If $g : Y' \to Y$ is a morphism of quasi-compact and quasi-separated schemes and $X' = Y' \times _ Y X$, then by Lemma 48.12.4 we have $\omega _{X'/Y'}^\bullet = L(g')^*\omega _{X/Y}^\bullet $ where $g' : X' \to X$ is the projection and by Lemma 48.7.1 the trace map

\[ \text{Tr}_{f', \mathcal{O}_{Y'}} : Rf'_*\omega _{X'/Y'}^\bullet \to \mathcal{O}_{Y'} \]

for $f' : X' \to Y'$ is the base change of $\text{Tr}_{f, \mathcal{O}_ Y}$ via the base change isomorphism.


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Your email address will not be published.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 0B6S. Beware of the difference between the letter 'O' and the digit '0'.