Homological dimensions of K"othe algebras

TL;DR

This paper calculates various homological dimensions of K"othe algebras based on their defining K"othe set, providing a comprehensive understanding of their algebraic properties.

Contribution

It introduces explicit formulas for homological dimensions of K"othe algebras in terms of the K"othe set, advancing the classification of these algebras.

Findings

Global dimension expressed via K"othe set

Weak global dimension characterized

Bidimension and weak bidimension formulas derived

Abstract

Given a metrizable K"othe algebra $\lambda(P)$, we compute the global dimension, the weak global dimension, the bidimension, and the weak bidimension of $\lambda(P)$ in terms of the K"othe set $P$.