The D-standard and K-standard categories

TL;DR

This paper introduces the concepts of D-standard and K-standard categories, establishing their equivalence with derived autoequivalence properties in finite dimensional algebras, and provides new examples of D-standard categories.

Contribution

It defines D-standard and K-standard categories, proves their equivalence with derived autoequivalence properties, and offers new examples of D-standard module categories.

Findings

Module category is D-standard iff all derived autoequivalences are standard.

K-standard subcategory implies D-standard module category.

New examples of D-standard module categories are provided.

Abstract

We introduce the notions of a $\mathbf{D}$-standard abelian category and a $\mathbf{K}$-standard additive category. We prove that for a finite dimensional algebra $A$, its module category is $\mathbf{D}$-standard if and only if any derived autoequivalence on $A$ is standard, that is, given by a two-sided tilting complex. We prove that if the subcategory of projective $A$-modules is $\mathbf{K}$-standard, then the module category is $\mathbf{D}$-standard. We provide new examples of $\mathbf{D}$-standard module categories.