Lemma 50.11.3. We have $H^ q(\mathbf{P}^ n_ A, \Omega ^ p) = 0$ unless $0 \leq p = q \leq n$. For $0 \leq p \leq n$ the $A$-module $H^ p(\mathbf{P}^ n_ A, \Omega ^ p)$ free of rank $1$ with basis element $c_1^{Hodge}(\mathcal{O}(1))^ p$.
Proof. We have the vanishing and and freeness by Lemma 50.11.2. For $p = 0$ it is certainly true that $1 \in H^0(\mathbf{P}^ n_ A, \mathcal{O})$ is a generator.
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 50.11.1. In the proof of Lemma 50.11.2 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 50.11.1 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 $\mathop{\mathrm{Pic}}\nolimits (\mathbf{P}^ n_ A)$, see Section 50.9. 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.
Proof for general $p$ by induction. The base cases $p = 0, 1$ were handled above. Assume $p > 1$. In the proof of Lemma 50.11.2 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 50.9 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. $\square$
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