Cardinal sequences

In this article we characterize all those sequences of cardinals of length ω1 which are cardinal sequences of some (locally) compact scattered space (or, equivalently, a superatomic Boolean algebra). This extends the similar results from [R. La Grange, Concerning the cardinal sequence of a Boolean algebra, Algebra Universalis, 7 (1977) 307–313] for such sequences of countable length. For ordinals between ω1 and ω2 we can only give a sufficient condition for a sequence of that length to be a cardinal sequence of a compact scattered space. This condition is, arguably, the most one can expect in ZFC. In any case, ours is a significant extension of the sufficient conditions given in [J.C. Martinez, A consistency result on thin-tall superatomic Boolean algebras, Proc. Amer. Math. Soc. 115 (1992) 473–477] and [J. Bagaria, Locally generic Boolean algebras and cardinal sequences, Algebra Universalis 47 (2002) 283–302].