Lambda-pure global dimension of Grothendieck categories and some applications

Abstract

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