Homological properties of extensions of algebras
Kostiantyn Iusenko Instituto de Matemática e Estatística, Univ. de São Paulo, São Paulo, SP, Brazil John William MacQuarrie Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil
Abstract
We consider a class of extensions of associative algebras, which we refer to as “strongly proj-bounded extensions”. We prove that the finiteness of the left global dimension and the support of the Hochschild homology is preserved by strongly proj-bounded extensions, generalizing results of Cibils, Lanzillota, Marcos and Solotar. Moreover, we show that the finiteness of the big left finitistic dimension is preserved by strongly proj-bounded extensions. In order to construct examples, we describe a new class of extensions of algebras of finite relative global dimension, which may be of independent interest. The results apply both for abstract (meaning no topology) and pseudocompact algebras.
中文速览
代数扩张（algebra extension）中同调性质的传递问题一直是代数学的核心难题——已知大代数 A 包含子代数 B，能否从一个的同调"好不好"推断另一个？本文引入"强射影有界扩张"（strongly proj-bounded extension）这一新框架，放宽了 Cibils 等人对"有界扩张"的经典条件，允许扩张"规模更大"且 A 可以拥有比 B 更多的本原幂等元。在此框架下，作者证明：左整体维数（left global dimension）的有限性、Hochschild 同调的支撑有限性，以及大左有限主维数（big left finitistic dimension）的有限性，都在强射影有界扩张下双向保持，从而将 Han 猜想和有限主维数猜想的验证工作在更大范围的代数之间相互转化。这些结果同时适用于抽象代数与伪紧致代数（pseudocompact algebra），后者是有限维代数向无限维情形最自然的推广，与余代数及profinite群的表示论直接相关，因此本文的结论具有相当广泛的适用价值。
原文 arXiv:2108.12923;中英对照 + 大白话阅读 https://aha.fim.ai/paper/2108.12923v3