The Stacks project

38.44 Blowing up complexes, III

In this section we give an “algebra version” of the version of Macpherson's graph construction given in [Section 18.1, F].

Let $X$ be a scheme. Let $\mathcal{E}^\bullet $ be a bounded complex of finite locally free $\mathcal{O}_ X$-modules. Let $U \subset X$ be the maximal open subscheme such that $\mathcal{E}^\bullet |_ U$ has finite locally free cohomology sheaves.

Consider the projection morphism $p : \mathbf{P}^1_ X \to X$. The complement of the open subscheme $\mathbf{A}^1_ X \subset \mathbf{P}^1_ X$ is the image $(\mathbf{P}^1_ X)_\infty $ of the section $\infty : X \to \mathbf{P}^1_ X$ of $p$ or equivalently the inverse image of the divisor at $\infty $ in $\mathbf{P}^1_\mathbf {Z}$. Thus $(\mathbf{P}^1_ X)_\infty \subset \mathbf{P}^1_ X$ is an effective Cartier divisor. Let

\[ b : W \longrightarrow \mathbf{P}^1_ X \]

be the blowing up constructed in Lemma 38.43.3 starting with the effective Cartier divisor $(\mathbf{P}^1_ X)_\infty \subset \mathbf{P}^1_ X$ and the bounded complex $p^*\mathcal{E}^\bullet $ of finite locally free modules. We also denote

\[ \mathcal{Q}^\bullet = \eta _\mathcal {I} b^*p^*\mathcal{E}^\bullet \]

the complex considered in Lemma 38.43.3 where $\eta _\mathcal {I}$ is the operator of Section 38.42 associated to the ideal sheaf $\mathcal{I}$ of the effective Cartier divisor $W_\infty = b^{-1}(\mathbf{P}^1_ X)_\infty $ on $W$.

Lemma 38.44.1. The construction above has the following properties:

  1. $b$ is an isomorphism over $\mathbf{P}^1_ U \cup \mathbf{A}^1_ X$,

  2. the restriction of $\mathcal{Q}^\bullet $ to $\mathbf{A}^1_ X$ is equal to the pullback of $\mathcal{E}^\bullet $,

  3. there exists a closed immersion $T \to W_\infty $ of finite presentation such that $\infty (U) \subset T$ scheme theoretically and such that $\mathcal{Q}^\bullet |_ T$ has finite locally free cohomology sheaves.

Proof. This follows immediately from the results in Section 38.43, especially Lemma 38.43.7. $\square$

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