Let $ X $ be a projective scheme over a noetherian ring $ A $, $ \mathcal{F} $ a coherent sheaf and $ \mathcal{L} $ an ample line bundle, suppose $ H^1(X, \mathcal{F} \otimes \mathcal{L}^{\otimes m}) = 0 $, then is $\mathcal{F} \otimes \mathcal{L}^{\otimes m+1} $ globally generated?
At the end of page 58 in Scholze's Algebraic Geometry I notes, there is a footnote which says given a ring $R$ and $X = \mathbb{P}^n_R$, and $\mathcal{M}$ a quasi-coherent $\mathcal{O}_X$-module of finite type, $ \exists \, d \in \mathbb{Z}$ s.t. for every $m \geq d$ the $\mathcal{O}_X$-module $ \mathcal{M} \otimes_{\mathcal{O}_X} \mathcal{O}_X(m) $ is globally generated. I asked DeepSeek this question and its answer used ample line bundle and Serre's vanishing theorem to simplify the proof. The answer used a key lemma (the question above) whose proof given by DeepSeek seems to be wrong, that's why I want to raise this question here.
