Derived equivalences between triangular matrix algebras

ABSTRACT

In this paper, we study derived equivalences between triangular matrix algebras using certain classical recollements. We show that special properties of these recollements actually characterize triangular matrix algebras and describe methods to construct tilting modules and tilting complexes inducing derived equivalences between them.

KEYWORDS:

• Derived equivalences
• tilting objects
• triangular matrix algebras

MATHEMATICS SUBJECT CLASSIFICATION:

• 16E35
• 16G10

Acknowledgement

The author is supported by the National Natural Science Foundation of China 11541002, the Construct Program of the Key Discipline in Hunan Province, and the Start-Up Funds of Hunan Normal University 830122-0037. He also would like to thank the anonymous referee for carefully checking the manuscript and providing many valuable suggestions to clarify a few ambiguities and improve the paper.

Notes

¹Note that different from Theorem 4.5 in [14], we do not need to assume that A, B, and C have finite global dimensions.

²Here $T_1\oplus T_2$ generates D(Λ) if and only for $X\in D\left(\Lambda \right),{Hom}_{D\left(\Lambda \right)}(T_1\oplus T_2,X[n\left]\right)=0$ for all n∈ℤ implies X = 0. But in general $D\left(\Lambda \right)\ne tria(T_1\oplus T_2)$, which is the smallest triangulated category containing $T_1\oplus T_2$.