Algebra extensions and derived-discrete algebras

TL;DR

The paper analyzes how derived-discreteness and piecewise heredity behave under algebra extensions. It develops a framework using separable functors to compare derived categories of extensions, proving that split extensions preserve derived-discreteness from $B$ to $A$, while separable extensions with $B_A$ projective preserve from $A$ to $B$, with analogous results for piecewise hereditary algebras. It leverages strong global dimension and projective-resolution arguments to relate properties across extensions, and applies these results to base-field extensions and skew group algebras, recovering compatibility with the known classification and highlighting natural counterexamples. Overall, it provides a cohesive method to transfer the dichotomic classification of derived-discrete algebras across common algebra-constructs.

Abstract

Let $φ\colon A\rightarrow B$ be an algebra extension. We prove that if $φ$ is split, the derived-discreteness of $A$ implies the derived-discreteness of $B$; if $φ$ is separable and the right $A$-module $B$ is projective, the converse holds. We prove an analogous statement for piecewise hereditary algebras.

Theorems & Definitions (30)

• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Lemma 2.6