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Stream: theory: algebraic topology

Topic: cotensor product of Hopf algebroids constructed from BP

Cloudifold (Jan 06 2023 at 14:22):

I am reading Ravenel's green book(Complex Cobordism and Stable Homotopy Groups of Spheres), there is an example in its 306 page:

Let $(A, \Gamma) := (\pi_* BP, BP_* BP) \cong (\mathbb{Z}_{(p)}[v_1, v_2,...], A[t_1,t_2, ...])$, $\Sigma := A[t_{n+1}, t_{n+2}, ...] = \Gamma / (t_1, ... , t_n)$.
The evident map $(A, \Gamma) \to (A, \Sigma)$ is normal.

$D := A \square_{\Sigma} A = \mathbb{Z}_{(p)}[v_1, ..., v_n]$ and $\Phi := A \square_{\Sigma} \Gamma \square_{\Sigma} A = D[t_1, ..., t_n]$.

I am trying to verify these relations about $D$ and $\Phi$.

I know the right $\Sigma$-comodule structure of $A$ and $\Gamma$ come from
$\eta_R : A \to \Gamma \to \Sigma$ and
$\Gamma \xrightarrow{\Delta} \Gamma \otimes_A \Gamma \to \Gamma \otimes_A \Sigma$.
(left comodule structure is similar)
Where $\Delta : \Gamma \to \Gamma \otimes_A \Gamma$ is the coproduct of Hopf algebroid $(A,\Gamma)$.

The theorem 4.1.18 in the book shows how these map is determined, but I don't know how to write $\eta_R$ and $\Delta$ in terms of $v_i$ s and $t_i$ s.