{'fraction_non_alphanumeric': 0.04495504495504495, 'fraction_numerical': 0.012987012987012988, 'mean_word_length': 4.725714285714286, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Liftings of formal groups and the Artinian completion of BP

Let BP denote the localization at υn, of the Brown-Peterson spectrum (associated to the prime p). There is a related ring spectrum E(n) with homotopy ring (as a quotient ring of in fact the cohomology theory E(n)*( ) is determined via a Conner-Floyd type isomorphism from on finite complexes, and moreover E(n) and BP are in the same Bousfield class (see [2, 14]). Although it is known (essentially from [17]) that BP cannot be a product of suspensions of E(n) in a multiplicative sense, D. Ravenel conjectured that such a splitting might occur after suitable completion of these spectra (see the introduction to [14]). This question was the original motivation of the present paper; however in proving Ravenel's conjecture we were naturally led to the consideration of some fundamental results in the theory of liftings of formal group laws and ‘change of ring’ results for Ext groups occurring in connection with the work of [10, 11, 12].