Extension properties of Stone-\v{C}ech coronas and proper absolute extensors

We characterize, in terms of $X$, extensional dimension of the Stone-\v{C}ech corona $\beta X \setminus X$ of locally compact and Lindel\"{o}f space $X$. The non-Lindel\"{o}f case case is also settled in terms of extending proper maps with values in $I^{\tau}\setminus L$, where $L$ is a finite complex. Further, for a finite complex $L$, an uncountable cardinal $\tau$ and a $Z_{\tau}$-set $X$ in the Tychonov cube $I^{\tau}$ we find necessary and sufficient condition, in terms of $I^{\tau}\setminus X$, for $X$ to be in the class $\operatorname{AE}([L])$. We also introduce a concept of a proper absolute extensor and characterize the product $[0,1)\times I^{\tau}$ as the only locally compact and Lindel\"{o}f proper absolute extensor of weight $\tau > \omega$ which has the same pseudocharacter at each point.