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Let Λ be a Noetherian algebra and T ∈ Λ - mod be an n -tilting module with Γ = End Λ (T). Then Λ satisfies Auslander–Reiten conjecture if and only if so does Γ. As a consequence, we obtain that Auslander–Reiten conjecture is left and right symmetric for a certain Noetherian algebra. [ABSTRACT FROM AUTHOR]

We consider stratifying ideals of finite dimensional algebras in relation with Morita contexts. A Morita context is an algebra built on a data consisting of two algebras, two bimodules and two morphisms. For a strongly stratifying Morita context - or equivalently for a strongly stratifying ideal - we show that Han's conjecture holds if and only if it holds for the diagonal subalgebra. The main tool is the Jacobi-Zariski long exact sequence. One of the main consequences is that Han's conjecture holds for an algebra admitting a strongly (co-)stratifying chain whose steps verify Han's conjecture. If Han's conjecture is true for local algebras and an algebra Λ admits a primitive strongly (co-)stratifying chain, then Han's conjecture holds for Λ. [ABSTRACT FROM AUTHOR]

We prove that if B\subseteq A is an extension of finite dimensional algebras such that the projective dimension of A/B as a B-bimodule is finite, then the finitistic dimension of B is finite whenever the finitistic dimension of A is finite. We exhibit examples demonstrating that the algebra B appearing in such an extension can be more complicated than A. [ABSTRACT FROM AUTHOR]

Co-Gorenstein algebras were introduced by Beligiannis in [A. Beligiannis, The homological theory of contravariantly finite subcategories: Auslander–Buchweitz contexts, Gorenstein categories and co-stabilization, Comm. Algebra28(10) (2000) 4547–4596]. In [S. Kvamme and R. Marczinzik, Co-Gorenstein algebras, Appl. Categorical Struct.27(3) (2019) 277–287], the authors propose the following conjecture (co-GC): if Ω n (m o d A) is extension closed for all n ≤ 1 , then A is right co-Gorenstein, and they prove that the generalized Nakayama conjecture implies the co-GC, also that the co-GC implies the Nakayama conjecture. In this paper, we characterize the subcategory Ω ∞ (m o d A) for algebras of Ω n -finite representation type. As a consequence, we characterize when a truncated path algebra is a co-Gorenstein algebra in terms of its associated quiver. We also study the behavior of Artin algebras of Ω ∞ -infinite representation type. Finally, an example of a non-Gorenstein algebra of Ω ∞ -infinite representation type and an example of a finite dimensional algebra with infinite ϕ -dimension are presented. [ABSTRACT FROM AUTHOR]

Let G be a classical group defined over a local field F of characteristic zero. For any irreducible admissible representation \pi of G(F), which is of Casselman-Wallach type if F is archimedean, we extend the study of spectral decomposition of local descents by Jiang and Zhang [Algebra Number Theory 12 (2018), 1489–1535] for special orthogonal groups over non-archimedean local fields to more general classical groups over any local field F. In particular, if \pi has a generic local L-parameter, we introduce the spectral first occurrence index {\mathfrak {f}}_{\mathfrak {s}}(\pi) and the arithmetic first occurrence index {\mathfrak {f}}_{{\mathfrak {a}}}(\pi) of \pi and prove in this paper that {\mathfrak {f}}_{\mathfrak {s}}(\pi)={\mathfrak {f}}_{{\mathfrak {a}}}(\pi). Based on the theory of consecutive descents of enhanced L-parameters developed by Jiang, Liu, and Zhang [Arithmetic wavefront sets and generic L-packets, arXiv:2207.04700], we are able to show in this paper that the first descent spectrum consists of all discrete series representations, which determines explicitly the branching decomposition problem by means of the relevant arithmetic data and extends the main result (Jiang and Zhang [Algebra Number Theory 12 (2018), 1489–1535], Theorem 1.7) to broader generality. [ABSTRACT FROM AUTHOR]

We prove that a virtually periodic object in an abelian category gives rise to a non-vanishing result on certain Hom groups in the singularity category. Consequently, for any artin algebra with infinite global dimension, its singularity category has no silting subcategory, and the associated differential graded Leavitt algebra has a non-vanishing cohomology in each degree. We verify the Singular Presilting Conjecture for singularly-minimal algebras and ultimately-closed algebras. We obtain a trichotomy on the Hom-finiteness of the cohomologies of differential graded Leavitt algebras. [ABSTRACT FROM AUTHOR]

Let \Gamma \subseteq \mathbb {N} be a numerical semigroup. In this paper, we prove an upper bound for the Betti numbers of the semigroup ring of \Gamma which depends only on the width of \Gamma, that is, the difference between the largest and the smallest generator of \Gamma. In this way, we make progress towards a conjecture of Herzog and Stamate [J. Algebra 418 (2014), pp. 8–28]. Moreover, for 4-generated numerical semigroups, the first significant open case, we prove the Herzog-Stamate bound for all but finitely many values of the width. [ABSTRACT FROM AUTHOR]

Given a space with a circle action, we study certain cocyclic chain complexes and prove a theorem relating cyclic homology to S¹ -equivariant homology, in the spirit of celebrated work of Jones. As an application, we describe a chain level refinement of the gravity algebra structure on the (negative) S¹ -equivariant homology of the free loop space of a closed oriented smooth manifold, based on work of Irie on chain level string topology and work of Ward on an S¹ -equivariant version of operadic Deligne’s conjecture. [ABSTRACT FROM AUTHOR]

Let A be a finite dimensional hereditary algebra over an algebraically closed field k. In this paper, we study the tilting quiver of A from the viewpoint of τ -tilting theory. First, we prove that there exists an isomorphism between the support τ -tilting quiver Q(s τ -tilt A) of A and the tilting quiver Q(tilt A ¯ ) of the duplicated algebra A ¯ . Then, we give a new method to calculate the number of arrows in the tilting quiver Q(tilt A) when A is representation-finite. Finally, we study the conjecture given by Happel and Unger, which claims that each connected component of Q(tilt A) contains only finitely many non-saturated vertices. We provide an example to show that this conjecture does not hold for some algebras whose quivers are wild with at least four vertices. [ABSTRACT FROM AUTHOR]