The stable cooperations of Morava $K$-Theory and the fiber product of automorphism groups of formal group laws

There are many previous studies on the Hopf algebra $K(n{)}_{\ast }(K(n))$, the stable cooperations of $n$th Morava $K$-theory at an odd prime. Whereas the main part of $K(n{)}_{\ast }(K(n))$ corepresents the group-valued functor consisting of strict automorphisms of the Honda formal group law of height $n$, relations between the whole structure of $K(n{)}_{\ast }(K(n))$ including the exterior part and formal group laws have not been investigated well. Firstly, we constitute a functor $C(-)$ which is given by the fiber product of two natural homomorphism between subgroups of automorphisms of formal group laws, and the Hopf algebra $C_{\ast }$ corepresenting $C(-)$. Next, we construct a Hopf algebra homomorphism ${\kappa }^{\ast }:C_{\ast }\to K(n{)}_{\ast }(K(n))$ naturally. To relate $C_{\ast }$ to $K(n{)}_{\ast }(K(n))$, we use stable comodule algebras which are introduced by Boardman. From the algebra structure of $K(n{)}_{\ast }(K(n))$ which is given by W"urgler and Yagita, we see that ${\kappa }^{\ast }$ is an isomorphism. Since we formulate $C_{\ast }$ by using formal group laws, the isomorphism ${\kappa }^{\ast }$ clarifies relationship between the Hopf algebra structure of $K(n{)}_{\ast }(K(n))$ including the exterior algebra part and formal group laws.