The Stacks project

Lemma 38.43.12. In Situation 38.43.1. Let $b : X' \to X$, $D' \subset X'$, and $Q$ be as in Lemma 38.43.7. Let $\rho = (\rho _ i)_{i \in \mathbf{Z}}$ be integers. Let $W(\rho ) \subset X$ be the maximal open subscheme where $H^ i(M)$ is locally free of rank $\rho _ i$ for all $i$. Let $i : T \to D'$ be as in Lemma 38.43.11. Then there exists an open and closed subscheme $T(\rho ) \subset T$ containing $D' \cap b^{-1}(W(\rho ))$ scheme theoretically such that $H^ i(Li^*Q|_{T(\rho )})$ is locally free of rank $\rho _ i$ for all $i$.