We study Hodge-Tate crystals on the absolute (log-) prismatic site of
OKβ, where OKβ is a mixed characteristic complete
discrete valuation ring with perfect residue field. We first classify
Hodge-Tate crystals by OKβ-modules equipped with certain small
endomorphisms. We then construct Sen theory over a non-Galois Kummer tower, and
use it to classify rational Hodge-Tate crystals by (log-) nearly Hodge-Tate
representations. Various cohomology comparison and vanishing results are proved
along the way.Comment: v1:This preprint contains almost all results of arXiv:2112.10140
(except section 5 there) and all results of arXiv:2201.10136 . Furthermore,
all results are generalized to the log-prismatic setting. Incidentally, this
preprint has a same title as arxiv:2201.10136; we apologize for possible
confusions. v2: improved exposition, very minor revisions. Comments are
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