A duality theorem in Hopf algebras and its application to Morava \(K\)-theory of \(B\mathbb{Z}/p^r\)
Let \(A\) be a bicommutative Hopf algebra over a ring \(R\), \(S\) a commutative \(R\) algebra, \(A^*= \Hom_{R\text{-module}} (A,R)=\) the dual of \(A\), and \(G(A)(S)= \Hom_{R\text{-algebra}} (A,S)\). The authors show that for certain \(R\), \(S\) there is an isomorphism \(G(A)(S)\equiv G(A^*)(S)\). This is applied to the particular case of the Hopf algebra \(A=\overline{K(n)}^0_{O_L} (B\mathbb{Z}/p^r)\) where \(O_L\) is the ring of integers in a finite extension of the \(p\)-adic numbers and \(\overline{K(n)}^*_{O_L}(\;)=\overline{K(n)}^*(\;)\otimes O_L\) denotes p-adic Morava K-theory of period 2 with \(O_L\) coefficients. Using this it is shown that the group of homotopy classes of ring spectra maps \((B\mathbb{Z}/p^r)_+ \rightarrow \overline{K(n)}_{O_L} \) is isomorphic to \((\mathbb{Z}/p^r)^n\).
scientific article; zbMATH DE number 1062464