Endomorphisms of the binomial coalgebra. (Q1877763)

Let \(R\) be a commutative unital ring and \(B_1\) a free \(R\)-module with an infinitely countable base \(e_n\), \(n\geqslant 0\). Then \(B_1\) is a coalgebra with comultiplication \[ \Delta(e_n)=\sum_{i=0}^n{n\choose i}e_i\otimes e_{n-i} \] and counit \(\varepsilon(e_n)=\delta_{n0}\). For any prime \(p\) define a function \(s_p\) on the set of non-negative integers such that \(s_p(0)=0\) and \(s_p(lp+m)=s_p(l)+m\) where \(0\leqslant m<p\). If \(a\) is a non-negative integer then denote by \(P_a\) the submodule of \(B_1\) spanned by all \(e_n\) with \(s_p(n)=a\). If \(\varphi\) is an \(R\)-module endomorphism of \(R_1\) and \(x\in P_a\) then put \[ \varphi(a)=\sum_{b\geqslant 0}\varphi_a^b(x),\quad \varphi_a^b(x)\in P_b. \] The collection of maps \(\varphi_a^b(x)\) defines a map \(\Theta\) from \(R\)-coalgebra endomorphisms of \(B_1\) to \(\prod_{l\geqslant 1}\hom(P_l,P_1)\), namely, \(\Theta(\phi)=(\phi^1_l)_{l\geqslant 1}\). If \(R\) is a reduced ring of prime characteristic \(p\) then \(\Theta\) is bijective. Let \(HR=R^{\mathbb{N}\cup 0}\) with Hurwitz multiplication \((gh)_n=\sum_{k=0}^ng_kh_n\). There is a bijective map from the set of continuous endomorphisms from \(HR\) to \(\prod_{l\geqslant 1}\hom(P_l,P_1)\) sending \(\chi\) to \(\Theta(\varphi)\) where \(\varphi\) is the endomorphism of the coalgebra \(B_1\) which is adjoint to \(\chi\).