# Homological dimensions of Banach spaces

## TL;DR

This paper explores the homological dimensions of Banach spaces, providing examples and establishing conditions under which Ext groups vanish or not, including the first examples of spaces with infinite homological dimension.

## Contribution

It introduces the first examples of Banach spaces with infinite homological dimension and compares Ext groups in Banach and quasi-Banach spaces.

## Key findings

- Existence of Banach spaces with non-zero Ext in all degrees
- Hilbert spaces have infinite homological dimension
- Comparison of Ext^2 in Banach and quasi-Banach spaces

## Abstract

The purpose of this paper is to lay the foundations for the study of the problem of when $\Ext^n(X, Y)=0$ in Banach/quasi-Banach spaces. We provide a number of examples of couples $X,Y$ so that $\Ext^n(X,Y)$ is (or is not ) $0$, including the first example of a separable Banach space $\mathscr K$ so that $\Ext^n(\mathscr K, \mathscr K)\neq 0$ for all $n\in \N$. Such space moreover provides the first example of Banach spaces with infinite homological dimension/codimension. We also show that the homological dimension/codimension of Hilbert spaces is infinite. The final section is devoted to compare $\Ext^2(\cdot, \cdot)$ in Banach and Quasi-Banach spaces.