DG-Semiprimary DG-Algebras, Acyclicity and Hopkins-Levitzki Theorem for DG-Algebras

TL;DR

The work extends the Hopkins-Levitzki paradigm to differential graded algebras by linking the graded structure of $\ker(d)$ to dg-finiteness properties. It introduces the dg-radical $dgrad_2(A)$ and the notion of left $dg$-semiprimary algebras, showing that graded-Noetherian/Artinian conditions on $\ker(d)$ force $dg$-Noetherian/$dg$-Artinian behavior of $(A,d)$ and that acyclic dg-algebras satisfy the Hopkins-Levitzki conclusions. A central achievement is proving that left dg-Artinian, left dg-semiprimary dg-algebras with nilpotent $dgrad_2(A)$ are $dg$-Noetherian and acyclic, alongside a detailed discussion of composition series and the limitations arising from non-semisimplicity in the dg setting. The results bridge graded ring theory with differential graded homological algebra and provide criteria for acyclicity and finite-length phenomena in dg-algebras, while highlighting subtle distinctions from the classical, non-dg case.

Abstract

We study the analogue of the Hopkins-Levitzky Theorem for dg-algebras $(A,d)$. We first consider the Hopkins approach. Here we show that for acyclic dg-algebras with graded-Artinian algebras of cycles $\ker(d)$, we also have that $(A,d)$ is left dg-Noetherian, and we show that acyclic dg-Artinian dg-algebras are dg-Noetherian. Then, studying the Levitzki approach, we consider a definition of a dg-semiprimary algebra. For dg-semiprimary dg-artinian dg-algebras $(A,d)$, we show that all dg-simple dg-modules are acyclic, and so are all dg-modules with finite dg-composition length. We finally show that dg-Artinian dg-semiprimary dg-algebras with nilpotent dg-radical $dgrad_2(A,d)$ are dg-Noetherian and acyclic.

Theorems & Definitions (25)

• Theorem 2.1
• Proposition 2.2
• Corollary 2.3
• Proposition 2.4
• Remark 2.5
• Lemma 2.6
• Theorem 2.7
• Remark 2.8
• Remark 2.9
• Remark 2.10