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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