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• W2104156951 abstract "We show that if a Hopf algebra has finite dimensional primitives and a primitive lies in arbitrarily long finite sequences of divided powers then it lies in an infinite sequence of divided powers. Introduction. K. Newman has shown us a counter-example to [3, Theorem 2, p. 521]. The theorem states that if H is a cocommutative Hopf algebra over a perfect field k of characteristic p > 0 and k is the unique simple subcoalgebra of H then a primitive element 1x of Hlies in a sequence of divided powers 1 = Ox, 1x, . . . pn+l lx if and only if 1x has coheight n, for n = 0, 1, .. ., oo. (The sequence should be considered infinite if n = oo.) The proof given in [3, p. 524] correctly shows that the existence of the sequence of divided powers implies that 1x has the desired coheight for all n. The proof there also correctly shows that for finite n if lx has coheight n then the desired sequence of divided powers exists. The error seems to be the assertion K1 H1/(U ker Fi) [3, p. 254, lines 23-24]. Newman's example shows that 1x may have infinite coheight-so lies in arbitrarily long finite sequences of divided powers-and yet 1x lies in no infinite sequence of divided powers. We show here that with the further assumption that the primitives of H are finite dimensional then 1x having infinite coheight implies that 1x lies in an infinite sequence of divided powers. This result is important because it plays a key role in the proof of Jacobson's conjecture in [1]. We explain this in more detail at the end of the present paper. 1. Suppose C is a cocommutative coalgebra with a unique simple subcoalgebra which is one dimensional. We identify this simple subcoalgebra with k and so consider kc C. Then the primitive elements of C, P(C){c E C I Ac =1 0 c + c 0 1}. HEYNEMAN'S THEOREM. If dim P(C) < oo then C satisfies the minimum condition and descending chain condition for subcoalgebras. Proof. For a vector space U the coalgebra Sh(U) is defined in [4, p. 244, p. 254]. By [4, p. 254, 12.1.1 ] there is a finite-dimensional space U and an injective coalgebra Received by the editors April 7, 1970. AMS 1969 subject classifications. Primary 1452; Secondary 1450." @default.
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• W2104156951 date "1971-01-01" @default.
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• W2104156951 title "Weakening a theorem on divided powers" @default.
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• W2104156951 doi "https://doi.org/10.1090/s0002-9947-1971-0279162-1" @default.
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