Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lambda-pure global dimension of Grothendieck categories and some applications

Xi Wang, Hailou Yao, Lei Shen

Abstract

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

Lambda-pure global dimension of Grothendieck categories and some applications

Abstract

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

Paper Structure

This paper contains 4 sections, 21 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. $\lambda$-pure projective dimension
  3. The relations between derived categories and $\lambda$-pure derived categories
  4. $\lambda$-pure singularity categories

Key Result

Proposition 2.3

For any object $X$ in $\cal A$, assume the complex \xymatrix@C=0.5cm{ {\bm{P}}: &\cdots \ar[r] & P_1 \ar[r]^{d_1} & P_0 \ar[r]^{d_0} & X \ar[r] & 0 }is a $\lambda$-pure acyclic complex with $P_i$$(i=0,1,\cdots)$$\lambda$-pure projective. Then we have that $X$ is $\lambda$-pure projective if and on

Theorems & Definitions (45)

  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • proof
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Remark 2.7
  • ...and 35 more