Pre-weight structures, pseudo-identities and canonical derived equivalences
Xiao-Wu Chen
TL;DR
The paper extends derived Morita theory beyond flat algebras by introducing pre-weight structures and pseudo-identities, and defining canonical derived equivalences via tilting data and truncated dg endomorphism algebras. A central factorization theorem shows any derived equivalence splits uniquely as a pseudo-identity followed by a canonical derived equivalence, enabling a unified treatment of non-flat and flat cases. It proves that every derived equivalence from a left hereditary algebra is canonical, and that in the flat setting canonical and standard notions coincide, aligning with Rickard’s framework. The work also develops a robust toolkit (dg endomorphism algebras, towers, and semi-orthogonal decompositions) to analyze when endofunctors are identities and when derived functors arise from tilting data, with potential broad impact on representation theory and algebraic geometry.
Abstract
We introduce the notion of pre-weight structure on a triangulated category and study the corresponding pseudo-identities. We propose the notion of canonical derived equivalence between algebras that are not necessarily flat, which is associated to a tilting complex. In the flat situation, canonical derived equivalences coincide with standard derived equivalences in the sense of Rickard. We prove that any derived equivalence starting from a hereditary algebra is canonical. The key tool is a general factorization theorem: any derived equivalence is uniquely factorized as a pseudo-identity followed by a canonical derived equivalence.