Assume you have a variety $X$ defined over $\mathbb Z$ (or if you prefer, a discrete valuation ring), and let $F$ be a locally free tilting sheaf on $X$, that is, a locally free sheaf such that $\mathrm{Ext}^i(F,F)=0$ for all $i>0$. Assume that this sheaf is a generator of the derived category of coherent sheaves after base change to $\mathbb{Q}$ (or basically equivalently, to $\mathbb{F}_p$ for all sufficient large primes); does it follow that it is a generator over $\mathbb{Z}$ or modulo all primes?

Assume you have a smooth quasi-projective scheme $X$ (you can actually assume $X$ is projective over an affine scheme of finite type) defined over $\mathbb Z$ (or if you prefer, a discrete valuation ring), and let $F$ be a locally free tilting sheaf on $X$, that is, a locally free sheaf such that $\mathrm{Ext}^i(F,F)=0$ for all $i>0$. Assume that this sheaf is a generator of the derived category of coherent sheaves after base change to $\mathbb{Q}$ (or basically equivalently, to $\mathbb{F}_p$ for all sufficient large primes); does it follow that it is a generator over $\mathbb{Z}$ or modulo all primes?