Lambda-pure global dimension of Grothendieck categories and some applications

We study the $\lambda$-pure global dimension of a Grothendieck category $\mathcal{A}$, and provide two different applications about this dimension. We obtain that if the $\lambda$-pure global dimension $\text{plgldA}<\mathrm{\infty }$, then (1) The ordinary bounded derived category (where $\mathcal{A}$ has enough projective objects) and the bounded $\lambda$-pure one differ only by a homotopy category; (2) The $\lambda$-pure singularity category $\text{DlsgA}=0$. At last, we explore the reason why the general construction of classic Buchweitz-Happel Theorem is not feasible for $\lambda$-pure one.