The Stacks project

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 $Q$ to $\mathbf{A}^1_ X$ is equal to the pullback of $E$,

  3. there exists a closed immersion $i : T \to W_\infty $ of finite presentation such that $(W_\infty \to X)^{-1}U \subset T$ scheme theoretically and such that $Li^*Q$ has locally free cohomology sheaves,

  4. for $t \in T$ with image $u \in U$ we have that the rank $H^ i(Li^*Q)_ t$ over $\mathcal{O}_{T, t}$ is equal to the rank of $H^ i(M)_ u$ over $\mathcal{O}_{U, u}$,

  5. if $E$ can be represented by a locally bounded complex of finite locally free $\mathcal{O}_ X$-modules, then $Q$ can be represented by a locally bounded complex of finite locally free $\mathcal{O}_ W$-modules.

Proof. This follows immediately from the results in Section 38.43; for a statement collecting everything needed, see Lemma 38.43.14. $\square$

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