The Stacks project

Lemma 57.12.4. Let $X$ be a Noetherian scheme. Let $x \in X$ be a closed point and denote $\mathcal{O}_ x$ the skyscraper sheaf at $x$ with value $\kappa (x)$. Let $K$ in $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$. Let $b \in \mathbf{Z}$. The following are equivalent

  1. $H^ i(K)_ x = 0$ for all $i > b$ and

  2. $\mathop{\mathrm{Hom}}\nolimits _ X(K, \mathcal{O}_ x[-i]) = 0$ for all $i > b$.

Proof. Consider the complex $K_ x$ in $D^ b_{\textit{Coh}}(\mathcal{O}_{X, x})$. There exist an integer $b_ x \in \mathbf{Z}$ such that $K_ x$ can be represented by a bounded above complex

\[ \ldots \to \mathcal{O}_{X, x}^{\oplus n_{b_ x - 2}} \to \mathcal{O}_{X, x}^{\oplus n_{b_ x - 1}} \to \mathcal{O}_{X, x}^{\oplus n_{b_ x}} \to 0 \to \ldots \]

with $\mathcal{O}_{X, x}^{\oplus n_ i}$ sitting in degree $i$ where all the transition maps are given by matrices whose coefficients are in $\mathfrak m_ x$. See More on Algebra, Lemma 15.75.5. The result follows easily from this (and the equivalent conditions hold if and only if $b \geq b_ x$). $\square$


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