The Stacks project

Lemma 38.43.14. Let $X$ be a scheme and let $D \subset X$ be an effective Cartier divisor. Let $M \in D(\mathcal{O}_ X)$ be a perfect object. Let $W \subset X$ be the maximal open over which the cohomology sheaves $H^ i(M)$ are locally free. There exists a proper morphism $b : X' \longrightarrow X$ and an object $Q$ in $D(\mathcal{O}_{X'})$ with the following properties

  1. $b : X' \to X$ is an isomorphism over $X \setminus D$,

  2. $b : X' \to X$ is an isomorphism over $W$,

  3. $D' = b^{-1}D$ is an effective Cartier divisor,

  4. $Q = L\eta _{\mathcal{I}'}Lb^*M$ where $\mathcal{I}'$ is the ideal sheaf of $D'$,

  5. $Q$ is a perfect object of $D(\mathcal{O}_{X'})$,

  6. there exists a closed immersion $i : T \to D'$ of finite presentation such that

    1. $D' \cap b^{-1}(W) \subset T$ scheme theoretically,

    2. $Li^*Q$ has finite locally free cohomology sheaves,

    3. for $t \in T$ with image $w \in W$ the rank of $H^ i(Li^*Q)_ t$ over $\mathcal{O}_{T, t}$ is equal to the rank of $H^ i(M)_ x$ over $\mathcal{O}_{X, x}$,

  7. for any affine chart $(U, A, f, M^\bullet )$ for $(X, D, M)$ the restriction of $b$ to $U$ is the blowing up of $U = \mathop{\mathrm{Spec}}(A)$ in the ideal $I = \prod I_ i(M^\bullet , f)$, and

  8. for any affine chart $(V, B, g, N^\bullet )$ for $(X', D', Lb^*N)$ such that $I_ i(N^\bullet , g)$ is principal, we have

    1. $Q|_ V$ corresponds to $\eta _ gN^\bullet $,

    2. $T \subset V \cap D'$ corresponds to the ideal $J(N^\bullet , g) = \sum J_ i(N^\bullet , g) \subset B/gB$ studied in More on Algebra, Lemma 15.96.9.

  9. If $M$ can be represented by a locally bounded complex of finite locally free $\mathcal{O}_ X$-modules, then $Q$ can be represented by a bounded complex of finite locally free $\mathcal{O}_{X'}$-modules.

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