Lambda-pure global dimension of Grothendieck categories and some applications

TL;DR

This paper investigates the $mbda$-pure global dimension of Grothendieck categories, revealing its implications on derived and singularity categories, and discusses limitations of classic theorems in this context.

Contribution

It introduces the concept of $mbda$-pure global dimension in Grothendieck categories and explores its impact on derived categories and singularity categories, highlighting new theoretical insights.

Findings

Finite $mbda$-pure global dimension implies equivalence of bounded derived categories

Finite $mbda$-pure global dimension results in vanishing $mbda$-pure singularity category

Classic Buchweitz-Happel Theorem cannot be directly extended to $mbda$-pure setting

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.