Erratum: Towards an understanding of ramified extensions of structured ring spectra (Journal of Physical Chemistry DOI: 10.1017/S0305004118000099)

The original version of this paper unfortunately contained a mistake in a calculation that resulted in erroneous statements in Lemma 5.1 and Theorem 5.2. We are grateful to Eva Honing who discovered the mistake. LEMMA 0.1 (replaces Lemma 5.1). There is an isomorphism of graded commutative augmented ku∗-algebras 'Equation Presented' with |s| = 2, where the ku∗-algebra structure on (ku ∧ _koku)∗ is from the left and the augmentation is given by the multiplication m: ku ∧ _koku→ku and by s → 0. Proof. Smashing Wood's cofiber sequence 'Equation Presented' with ku (from the left) over ko gives a split exact sequence (with unit isomorphism 'Equation Presented' ko suppressed) 'Equation Presented'. Let u ∈ π ₂ku be the generator with 'Equation Presented'. Let ul and ur be the images of u in π2(ku ∧ _koku) induced by the left and right inclusion of ku in ku ∧ _koku∗ If s is the unique element in π2(ku ∧ _koku) with 'Equation Presented' and m∗s =0, then (1 ∧ j)∗(ur + 2s)=1∗ j∗u .2=0 and m∗(ur +2s)=u. Since also (1 ∧ j)∗ul =0 and m∗ul =u we must have ur +2s =ul. In ku∗ ⊗ko∗ ku∗, and hence also in π4(ku ∧ _koku), we have that 2u ² _r- 2u ² _l=0. As π∗(ku ∧ _koku) is torsion free, we get u ² _r- u ² _l=0 and therefore 'Equation Presented'. Again, since there is no torsion, this yields s ²- sul =0. THEOREM 0.2 (replaces Theorem 5.2). The Tor spectral sequence 'Equation Presented' collapses at the E ²-page and THH ^ko∗ (ku) is a square zero extension of ku∗: 'Equation Presented' with |yj| = (1 + |u|)(2 j +1)=3(2 j +1). Proof. Lemma 0.1 implies that the 'Equation Presented'. We have a periodic free resolution of ku∗ as a module over ku∗[s]/(s ²- su) 'Equation Presented'. Tensoring this down to ku∗ yields 'Equation Presented'. As ku∗ splits off THH ^ko∗ (ku), the zero column has to survive and cannot be hit by differentials and hence all differentials are trivial. For the E ^∞-term we therefore get 'Equation Presented' for yj in bidegree (2 j +1, 4 j +2) if j >0. So even total degrees occur only in E ^∞ ₀,∗ and odd total degrees occur only in at most one bidegree and we do not need to worry about additive extensions. As yj corresponds to 'Equation Presented', the action of 'Equation Presented' and this is trivial for i ≥ 1 in (ku∗/u){yj} so ui yj is zero in 'Equation Presented'. But E ^∞2 j+1,4 j+2 has all the elements of total degree 6 j +3 in the entire E ^∞-term, so in fact the element in THH ^ko∗ (ku) that yj represents is killed by multiplication by ui for any i ≥ 1. Thus we have no nontrivial products of the ui, i ≥ 1, and the odd dimensional elements. Since the elements of THH ^ko∗ (ku) represented by the yi are all in odd degrees, if there were nonzero products among them they would have to be elements in 'Equation Presented'. But elements in ku∗ are not killed by multiplying by u, whereas the elements represented by the yj are. So there can be no such nontrivial products.