12 ANDREW BAKER since ψ(ti) i Hi Hk−i Hk Hk. Therefore i ti ρ(wi) = j,r,s aj,r,str ts wj, and comparing the coefficients of the left hand ti, we obtain ρ(wi) = j,s aj,i,sts wj Hk W. This shows that each wi Wk, so the coproduct restricted to Wk satisfies ρWk H Wk. Example 3.4. Let p be an odd prime and let P∗ = Fp[ζk : k 1] be the (graded) polynomial sub-Hopf algebra of the mod p dual Steenrod algebra A∗ with coaction ψζn = n r=0 ζr ζpr n−r , where ζ0 = 1. Then (P∗, Fp) is unipotent since the subspaces P(n)∗ = Fp[ζk : 1 k n] satisfy the conditions of Lemma 3.3. This shows that P∗ is a local ring. If p = 2, this also applies to the mod 2 dual Steenrod algebra and implies that A is a local ring. For details on the next example, see the books by Ravenel and Wilson [13, 16]. Unfortunately the sub-Hopf algebra K(n)∗(E(n)) is commonly denoted K(n)∗K(n) in the earlier literature, but at the behest of the referee we refrain from perpetuating that usage. Example 3.5. Let p be an odd prime and let K(n) be the n-th p-primary Morava K-theory. Then K(n)∗ = Fp[vn,vn −1 ], with vn K(n)2(pn−1). There is a graded Hopf algebra over K(n)∗, Γ(n)∗ = K(n)∗(E(n)) = K(n)∗[tk : k 1]/(vntpn vp n t : 1), where tk Γ(n)2(pk−1) and E(n) is a Johnson-Wilson spectrum. In fact Γ(n)∗ is a proper sub-Hopf algebra of K(n)∗(K(n)). Using standard formulae, it follows that the K(n)∗-subspaces Γ(n, m)∗ = K(n)∗(t1,...,tm) Γ∗ satisfy the conditions of Lemma 3.3, therefore Γ(n, m)∗ is unipotent. When p = 2, the Hopf algebra Γ(n)∗ is also unipotent even though K(n) is not homotopy commutative. Here is a major source of examples which includes the algebraic ingredients used in [1] to prove the existence of a Landweber filtration for discrete comodules over the Hopf algebroid of Lubin-Tate theory. For two topologised objects X and Y we denote the set of all continuous maps X −→ Y by Mapc(X, Y ).
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