Homological properties of extensions of algebras
Kostiantyn Iusenko, John W. MacQuarrie
TL;DR
The paper extends the framework of bounded extensions to strongly proj-bounded extensions $B\subseteq A$, showing that key homological finiteness properties—namely the left global dimension, Hochschild homology, and the big left finitistic dimension—are preserved under such extensions. It develops a comprehensive relative homological algebra toolkit and proves a Jacobi-Zariski-type mechanism in this relative setting, enabling precise control of homological invariants across extensions. A constructive quiver-based method is presented to produce extensions with finite relative global dimension, yielding explicit upper bounds $\mathrm{gldim}(A,B)\le n-1$ and analogous bounds for the enveloping algebra, along with several corollaries and examples. The results apply to both abstract and pseudocompact algebras, broadening the applicability to contexts such as GPS and potential reductions of Han’s and Finitistic Dimension conjectures, and providing dual perspectives for coalgebras via pseudocompact dualities.
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.