On the cohomology of Drinfel'd's p-adic symmetric domain

There are, by now, three approaches to the de-Rham cohomology of Drinfel'd's p-adic symmetric domain: the original work of Schneider and Stuhler, and more recent work of Iovita and Spiess, and of de Shalit. In the first part of this paper we compare all three approaches and clarify a few points which remained obscure. In the second half we give a short proof of a conjecture of Schneider and Stuhler, previously proven by Iovita and Spiess, on a Hodge-like decomposition of the cohomology of p-adically uniformized varieties.