Extension properties of Stone-Čech coronas and proper absolute extensors

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We characterize, in terms of X, extensional dimension of the Stone-Čech corona βX X of locally compact and Lindelöf space X. The non-Lindelöf case case is also settled in terms of extending proper maps with values in Iτ L, where L is a finite complex. Further, for a finite complex L, an uncountable cardinal τ and a Zτ-set X in the Tychonov cube Iτ we find necessary and sufficient condition, in terms of Iτ X, for X to be in the class AE([L]). We also introduce a concept of a proper absolute extensor and characterize the product [0,1)× Iτ as the only locally compact and Lindelöf proper absolute extensor of weight τ> ω which has the same pseudocharacter at each point.