Homological dimensions of Banach spaces

The purpose of this paper is to lay the foundations for the study of the problem of when $\operatorname{Ext}^n(X, Y)=0$ in Banach spaces. We provide a number of examples of couples $X$, $Y$ such that $\operatorname{Ext}^n(X,Y)$ is (or is not) $0$. We show that $\operatorname{Ext}^n(\mathcal K, \mathcal K)\neq 0$ for all $n\in \mathbb{N}$ when $\mathcal K$ is the Kadec space. In particular, both the projective and injective dimensions of $\mathcal K$ are infinite.
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