We have the vanishing and and freeness by Lemma \ref{lemma-twisted-hodge-cohomology-projective-space}. For $p = 0$ it is certainly true that $1 \in H^0(\mathbf{P}^n_A, \mathcal{O})$ is a generator. \medskip\noindent Proof for $p = 1$. Consider the short exact sequence $$ 0 \to \Omega \to \mathcal{O}(-1)^{\oplus n + 1} \to \mathcal{O} \to 0 $$ of Lemma \ref{lemma-euler-sequence}. In the proof of Lemma \ref{lemma-twisted-hodge-cohomology-projective-space} we have seen that the generator of $H^1(\mathbf{P}^n_A, \Omega)$ is the boundary $\xi$ of $1 \in H^0(\mathbf{P}^n_A, \mathcal{O})$. As in the proof of Lemma \ref{lemma-euler-sequence} we will identify $\mathcal{O}(-1)^{\oplus n + 1}$ with $\bigoplus_{j = 0, \ldots, n} \mathcal{O}(-1)\text{d}T_j$. Consider the open covering $$ \mathcal{U} :  \mathbf{P}^n_A = \bigcup\nolimits_{i = 0, \ldots, n} D_{+}(T_i) $$ We can lift the restriction of the global section $1$ of $\mathcal{O}$ to $U_i = D_+(T_i)$ by the section $T_i^{-1} \text{d}T_i$ of $\bigoplus \mathcal{O}(-1)\text{d}T_j$ over $U_i$. Thus the cocyle representing $\xi$ is given by $$ T_{i_1}^{-1} \text{d}T_{i_1} - T_{i_0}^{-1} \text{d}T_{i_0} = \text{d}\log(T_{i_1}/T_{i_0}) \in \Omega(U_{i_0i_1}) $$ On the other hand, for each $i$ the section $T_i$ is a trivializing section of $\mathcal{O}(1)$ over $U_i$. Hence we see that $f_{i_0i_1} = T_{i_1}/T_{i_0} \in \mathcal{O}^*(U_{i_0i_1})$ is the cocycle representing $\mathcal{O}(1)$ in $\Pic(\mathbf{P}^n_A)$, see Section \ref{section-first-chern-class}. Hence $c_1^{Hodge}(\mathcal{O}(1))$ is given by the cocycle $\text{d}\log(T_{i_1}/T_{i_0})$ which agrees with what we got for $\xi$ above. \medskip\noindent Proof for general $p$ by induction. The base cases $p = 0, 1$ were handled above. Assume $p > 1$. In the proof of Lemma \ref{lemma-twisted-hodge-cohomology-projective-space} we have seen that the generator of $H^p(\mathbf{P}^n_A, \Omega^p)$ is the boundary of $c_1^{Hodge}(\mathcal{O}(1))^{p - 1}$ in the long exact cohomology sequence associated to $$ 0 \to \Omega^p \to \wedge^p\left(\mathcal{O}(-1)^{\oplus n + 1}\right) \to \Omega^{p - 1} \to 0 $$ By the calculation in Section \ref{section-first-chern-class} the cohomology class $c_1^{Hodge}(\mathcal{O}(1))^{p - 1}$ is, up to a sign, represented by the cocycle with terms $$ \beta_{i_0 \ldots i_{p - 1}} = \text{d}\log(T_{i_1}/T_{i_0}) \wedge \text{d}\log(T_{i_2}/T_{i_1}) \wedge \ldots \wedge \text{d}\log(T_{i_{p - 1}}/T_{i_{p - 2}}) $$ in $\Omega^{p - 1}(U_{i_0 \ldots i_{p - 1}})$. These $\beta_{i_0 \ldots i_{p - 1}}$ can be lifted to the sections $\tilde \beta_{i_0 \ldots i_{p -1}} = T_{i_0}^{-1}\text{d}T_{i_0} \wedge \beta_{i_0 \ldots i_{p - 1}}$ of $\wedge^p(\bigoplus \mathcal{O}(-1) \text{d}T_j)$ over $U_{i_0 \ldots i_{p - 1}}$. We conclude that the generator of $H^p(\mathbf{P}^n_A, \Omega^p)$ is given by the cocycle whose components are \begin{align*} \sum\nolimits_{a = 0}^p (-1)^a \tilde \beta_{i_0 \ldots \hat{i_a} \ldots i_p} & = T_{i_1}^{-1}\text{d}T_{i_1} \wedge \beta_{i_1 \ldots i_p} + \sum\nolimits_{a = 1}^p (-1)^a T_{i_0}^{-1}\text{d}T_{i_0} \wedge \beta_{i_0 \ldots \hat{i_a} \ldots i_p} \\ & = (T_{i_1}^{-1}\text{d}T_{i_1} - T_{i_0}^{-1}\text{d}T_{i_0}) \wedge \beta_{i_1 \ldots i_p} + T_{i_0}^{-1}\text{d}T_{i_0} \wedge \text{d}(\beta)_{i_0 \ldots i_p} \\ & = \text{d}\log(T_{i_1}/T_{i_0}) \wedge \beta_{i_1 \ldots i_p} \end{align*} viewed as a section of $\Omega^p$ over $U_{i_0 \ldots i_p}$. This is up to sign the same as the cocycle representing $c_1^{Hodge}(\mathcal{O}(1))^p$ and the proof is complete.