[Paper Review] Hereditary triangulated categories

Xiao‐Wu Chen, Claus Michael Ringel|arXiv (Cornell University)|Jun 27, 2016

Algebraic structures and combinatorial models9 references5 citations

TL;DR

This paper provides two intrinsic characterizations of hereditary triangulated categories—defined as triangulated categories equivalent to the bounded derived category of a hereditary abelian category—using a full subcategory condition and the non-existence of certain paths. The key result establishes that a triangulated category is hereditary if and only if there is no path from $X[1]$ to $X$ for any indecomposable object $X$, with applications to piecewise hereditary algebras and derived equivalences.

ABSTRACT

We call a triangulated category \emph{hereditary} provided that it is equivalent to the bounded derived category of a hereditary abelian category, where the equivalence is required to commute with the translation functors. If the triangulated category is algebraical, we may replace the equivalence by a triangle equivalence. We give two intrinsic characterizations of hereditary triangulated categories using a certain full subcategory and the non-existence of certain paths. We apply them to piecewise hereditary algebras.

Motivation & Objective

• To provide intrinsic, category-theoretic characterizations of hereditary triangulated categories without relying on external equivalences.
• To resolve a gap in earlier work by showing that for algebraical triangulated categories, the equivalence to a derived category can be strengthened to a triangle equivalence.
• To establish a criterion for piecewise hereditary algebras based on the non-existence of paths from $X[1]$ to $X$ for indecomposable objects.
• To apply these characterizations to classify triangulated categories arising from hereditary abelian categories and to study directing objects in derived categories.

Proposed method

• Introduce and analyze hereditary $t$-structures in triangulated categories, using truncation functors and cohomological functors to define the heart as an abelian category.
• Prove that a triangulated category is hereditary if and only if it admits a hereditary $t$-structure with a hereditary heart.
• Use the existence of a realization functor for $t$-structures in algebraical triangulated categories to upgrade additive equivalences to triangle equivalences.
• Define and analyze paths in triangulated categories, particularly focusing on paths from $X[1]$ to $X$ for indecomposable objects.
• Establish that the non-existence of such paths characterizes hereditary triangulated categories, especially in the algebraical case.
• Apply the results to piecewise hereditary algebras by showing that the absence of paths from $X[1]$ to $X$ characterizes derived equivalence to a hereditary abelian category.

Experimental results

Research questions

• RQ1 Can hereditary triangulated categories be characterized intrinsically, without reference to an external derived category?
• RQ2 Under what conditions can an additive equivalence between a triangulated category and a derived category be upgraded to a triangle equivalence?
• RQ3 What is the role of path structures—specifically paths from $X[1]$ to $X$—in determining whether a triangulated category is hereditary?
• RQ4 How do these characterizations apply to piecewise hereditary algebras and their derived categories?
• RQ5 What is the relationship between directing objects in a derived category and the derived equivalence class of the underlying hereditary abelian category?

Key findings

• A triangulated category is hereditary if and only if it admits a hereditary $t$-structure, providing a structural characterization independent of external equivalences.
• For algebraical triangulated categories, any equivalence to a bounded derived category of a hereditary abelian category can be refined to a triangle equivalence.
• The non-existence of a path from $X[1]$ to $X$ for any indecomposable object $X$ characterizes hereditary triangulated categories.
• A triangulated category has a directing object if and only if it is triangle equivalent to the bounded derived category of a hereditary abelian category with a directing object.
• A finite-dimensional algebra $A$ is piecewise hereditary if and only if there is no path from $X[1]$ to $X$ for any indecomposable object $X$ in $\mathbf{D}^b(A\text{-mod})$, offering a new homological criterion.
• An algebra $A$ is derived equivalent to a finite-dimensional hereditary algebra if and only if its derived category contains a directing object.