Let \(k\) be a field, let \(n\ge0\), and put \(X=\PP_k^n\) and \(\Omega_X^p=\bigwedge^p\Omega_{X/k}^1\). Show that \[ H^q(X, \Omega_X^p) = \begin{cases} 0 & p \neq q \\ k & p = q \end{cases} \] for \(0 \leq p, q \leq n\).
Solution
Put \(E=\OO_X(-1)^{\oplus(n+1)}\) and \(\Omega^0=\OO_X\). All cohomology is coherent-sheaf cohomology for the Zariski topology.
1. For \(1\le p\le n\), there is a short exact sequence\[ 0\longrightarrow\Omega_X^p\longrightarrow \OO_X(-p)^{\oplus\binom{n+1}{p}} \longrightarrow\Omega_X^{p-1}\longrightarrow0. \]
The Euler sequence is \[ 0\longrightarrow\Omega_{X/k}^1\longrightarrow E \xrightarrow{e}\OO_X\longrightarrow0 \] [@Har10a, Theorem II.8.13]. It splits locally because its quotient is locally free. Contraction by \(e\) defines a map \(\bigwedge^pE\to\bigwedge^{p-1}E\). Locally write \(E=\Omega^1\oplus\OO\sigma\) with \(e(\sigma)=1\). Contraction kills \(\bigwedge^p\Omega^1\) and sends \(\sigma\wedge\alpha\) to \(\alpha\) for \(\alpha\in\bigwedge^{p-1}\Omega^1\). Its image is therefore \(\Omega^{p-1}\) and its kernel is \(\Omega^p\). The contraction is defined globally, so these local sequences glue. Finally, the monomial wedge basis gives \(\bigwedge^pE\cong\OO(-p)^{\oplus\binom{n+1}{p}}\).
2. For \(1\le p\le n\), one has \(H^0(X,\Omega_X^p)=0\), and the boundary maps give isomorphisms\[ H^q(X,\Omega_X^{p-1})\xrightarrow{\cong}H^{q+1}(X,\Omega_X^p) \qquad(q\ge0). \]
For \(1\le p\le n\), every cohomology group of \(\OO_X(-p)\) vanishes by T-IJW1K [@Har10a, Theorem III.5.1]. Its degree-zero group vanishes because the twist is negative, its intermediate groups always vanish, and its top-degree group vanishes because \(-p>-n-1\). Groups above degree \(n\) vanish as well. Thus the middle term of step 1 has zero cohomology in all degrees. Its long exact sequence gives the stated zero group and boundary isomorphisms.
3. The requested cohomology groups are\[ \boxed{H^q(\PP_k^n,\Omega_X^p)\cong \begin{cases} k,&p=q,\\ 0,&p\ne q, \end{cases} \qquad 0\le p,q\le n.} \]
For \(p=0\), one has \(H^0(X,\OO_X)=k\) and \(H^q(X,\OO_X)=0\) for \(q>0\), again by the projective-space cohomology calculation. For \(p>0\), step 2 gives zero in degree zero and shifts every positive-degree group down by one in both indices. If \(q<p\), iteration reaches a degree-zero group of a positive exterior power, which is zero. If \(q\ge p\), it reaches \(H^{q-p}(X,\OO_X)\), which is \(k\) exactly when \(q=p\) and is otherwise zero. This proves the formula when \(n\ge1\). For \(n=0\), only \((p,q)=(0,0)\) is in the stated range and \(X=\Spec k\), so the formula holds directly.
4. The diagonal groups have concrete generators \(\gamma_0=1\) and \(\gamma_p=\partial_p(\gamma_{p-1})\), where \(\partial_p\) is the boundary map from step
2.
Each boundary is an isomorphism, so these elements are nonzero generators. For an explicit representative, use the standard affine cover \(U_i=D_+(x_i)\), ordered by \(i\). Let \(\sigma_i=x_i^{-1}e_i\in\Gamma(U_i,E)\), where \(e_i\) is the formal generator of the \(i\)th summand. Then \(e(\sigma_i)=1\). Under the Euler injection, \(\sigma_j-\sigma_i\) corresponds to \(d\log(x_j/x_i)\).
In the exterior sequence of step 1, the cochain \[ \sigma_{i_0}\wedge\cdots\wedge\sigma_{i_{p-1}} \] lifts the representative of \(\gamma_{p-1}\) on the corresponding intersection. Its alternating Čech boundary is \[ \sum_{a=0}^p(-1)^a \sigma_{i_0}\wedge\cdots\wedge\widehat{\sigma_{i_a}}\wedge\cdots\wedge\sigma_{i_p} =(\sigma_{i_1}-\sigma_{i_0})\wedge\cdots\wedge(\sigma_{i_p}-\sigma_{i_0}). \] The equality follows by expansion: all terms with two copies of \(\sigma_{i_0}\) vanish. Thus \(\gamma_p\) is represented by \[ d\log(x_{i_1}/x_{i_0})\wedge\cdots\wedge d\log(x_{i_p}/x_{i_0}). \] The affine cover and its intersections are affine, so these Čech classes compute the stated sheaf cohomology [@Har10a, Theorem III.4.5]. In particular, the calculation uses no characteristic-zero or analytic comparison theorem.
5. Q.E.D.
Steps <1>1–<1>3 prove all the required values, and step 4 specifies generators for the one-dimensional groups.