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 $\mathcal{E}^{i + \delta (x)} \not= 0$, 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). $\square$


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