The Stacks project

Lemma 48.2.8. Let $X$ be a locally Noetherian scheme. Let $\omega _ X^\bullet $ be a dualizing complex on $X$ with associated dimension function $\delta $. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Set $\mathcal{E}^ i = \mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^{-i}_{\mathcal{O}_ X}(\mathcal{F}, \omega _ X^\bullet )$. Then $\mathcal{E}^ i$ is a coherent $\mathcal{O}_ X$-module and for $x \in X$ we have

  1. $\mathcal{E}^ i_ x$ is nonzero only for $\delta (x) \leq i \leq \delta (x) + \dim (\text{Supp}(\mathcal{F}_ x))$,

  2. $\dim (\text{Supp}(\mathcal{E}^{i + \delta (x)}_ x)) \leq i$,

  3. $\text{depth}(\mathcal{F}_ x)$ is the smallest integer $i \geq 0$ such that $\mathcal{E}_ x^{i + \delta (x)} \not= 0$, and

  4. we have $x \in \text{Supp}(\bigoplus _{j \leq i} \mathcal{E}^ j) \Leftrightarrow \text{depth}_{\mathcal{O}_{X, x}}(\mathcal{F}_ x) + \delta (x) \leq i$.

Proof. Lemma 48.2.5 tells us that $\mathcal{E}^ i$ is coherent. Choosing an affine neighbourhood of $x$ and using Derived Categories of Schemes, Lemma 36.10.8 and More on Algebra, Lemma 15.99.2 part (3) we have

\[ \mathcal{E}^ i_ x = \mathop{\mathcal{E}\! \mathit{xt}}\nolimits ^{-i}_{\mathcal{O}_ X}(\mathcal{F}, \omega _ X^\bullet )_ x = \mathop{\mathrm{Ext}}\nolimits ^{-i}_{\mathcal{O}_{X, x}}(\mathcal{F}_ x, \omega _{X, x}^\bullet ) = \mathop{\mathrm{Ext}}\nolimits ^{\delta (x) - i}_{\mathcal{O}_{X, x}}(\mathcal{F}_ x, \omega _{X, x}^\bullet [-\delta (x)]) \]

By construction of $\delta $ in Lemma 48.2.7 this reduces parts (1), (2), and (3) to Dualizing Complexes, Lemma 47.16.5. Part (4) is a formal consequence of (3) and (1). $\square$

Comments (0)

There are also:

  • 2 comment(s) on Section 48.2: Dualizing complexes on schemes

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Your email address will not be published.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