Let $\mathcal{E}$ be the class of finite-dimensional algebras isomorphic to endomorphism algebras of silting complexes over hereditary abelian categories. It is proved that the class $\mathcal{E}$ is closed under taking idempotent quotients, idempotent subalgebras and $τ$-reduction. We also show that the proper class consisting of shod algebras is also closed under these operations. In addition, several classic classes of algebras -- including laura, glued, weakly shod algebras -- are proved to be closed under idempotent quotients, thereby generalizing a known result originally established for specific idempotents.
For a given algebra A, the study of its idempotent subalgebra eAe and the idempotent quotient A/AeA associated with an idempotent e ∈ A is a central theme in representation theory (see, for instance, [CPS, AC, Xi, Xu, CK]). A primary motivation for focusing on these two constructions originates from the theory of recollements of derived categories. It is well known that under suitable conditions, the bounded derived category D b (A) can be "glued" from D b (eAe) and D b (A/AeA) via a recollement [CPS]. This framework provides a powerful mechanism for homological reduction, allowing one to decompose the homological properties of A-such as the finiteness of its global dimension-into those of eAe and A/AeA [AKLY].
From this perspective, establishing the closure properties of a specific class of algebras under these operations is of fundamental importance. It transforms the class into a selfcontained system where global conjectures and structural problems can be tackled through recursive arguments. For example, the fact that the class of 2-Calabi-Yau tilted algebras arising from hereditary categories is closed under idempotent quotients was a key step in proving the connectedness of cluster-tilting graphs for hereditary categories [FG]. Similarly, the proof of the connectedness of τ -tilting graphs for gentle algebras is based on the fact that the class of gentle algebras is closed under τ -reduction [FGLZ].
In the representation theory of finite-dimensional algebras, the study of endomorphism algebras of canonical objects-such as tilting modules, support τ -tilting modules, and (2-term) silting complexes-is a central theme. These constructions define several fundamental classes of algebras, including tilted, quasi-tilted, silted, and quasi-silted algebras. A landmark result by Happel, Reiten, and Smalø [HRS] established that almost hereditary algebras are precisely the endomorphism algebras of tilting objects in hereditary abelian categories. This categorical characterization was further extended by Buan and Zhou [BZa, BZb], who demonstrated that shod algebras can be realized as endomorphism algebras of 2-term silting complexes over hereditary abelian categories. As a further generalization, endomorphism algebras of silting complexes over a finite-dimensional hereditary algebras were investigated in [ALLT] from the view point of structure of categories.
The purpose of this work is to investigate the class E of finite-dimensional algebras realized as endomorphism algebras of basic silting complexes over hereditary abelian categories, specifically focusing on their behavior under the operations of taking idempotent subalgebras and idempotent quotients. Employing the machinery of silting reduction in triangulated categories, we develop an approach to analyze the stability of this class of algebras. Our main results, which are interconnected and proved using this method, establish the following closure properties.
Theorem 1.1 (Theorem 3.3, Theorem 3.5). Let A ∈ E and e ∈ A be an idempotent element.
(1) A/AeA ∈ E. Moreover, if A is quasi-silted, then so is A/AeA.
(2) eAe ∈ E. Moreover, if A is quasi-silted (resp. quasi-tilted), then so is eAe.
We remark that the final assertion of (2) was previously established in [AC]. However, our proof is entirely different from the one provided there.
Note that the idempotent quotient can be viewed as a special case of the τ -reduction introduced by Jasso [J]. It is therefore natural to consider whether the class E is closed under the more general operation of τ -reduction.
Theorem 1.2 (Theorem 3.9). Let A ∈ E and Z be a τ -rigid A-module.
(1) The τ -reduction of A with respect to Z belongs to E.
(2) If A is quasi-silted, then the τ -reduction of A with respect to Z is also quasi-silted.
The proof of Theorem 1.1, which establishes the closure property under idempotent quotients, naturally motivates a further investigation into the classical classes of algebras studied by Assem and Coelho [AC]. Specifically, we are led to consider whether these well-known classes-namely laura, glued, weakly shod, and shod algebras-are also closed under quotients by arbitrary idempotents. Utilizing an independent analysis with techniques distinct from silting reduction, we provide an affirmative answer to this question.
Theorem 1.3 (Theorem 4.1). Let A be a finite-dimensional k-algebra and e ∈ A be an idempotent element.
(1) If A is a laura algebra, then so is A/AeA.
(2) If A is a right (or left) glued algebra, then so is A/AeA.
(3) If A is a weakly shod algebra, then so is A/AeA.
(4) If A is a shod algebra, then so is A/AeA. This result significantly generalizes the work of Zito [Z], who proved a similar statement for specific idempotents.
The paper is organized as follows. We provide the necessary preliminaries and the main method of silting reduction in Section 2. In Section 3, we present the proof of Theorems 1.1-1.2. Section 4 is devoted to the proof of Theorem 1.3. In Sect
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