= Cotangent-sheaf cohomology of projective space
{title2=$H^q(\mathbb P_k^n,\Omega^1)$}
For $n\geq1$, the cotangent form of the <Euler sequence> is $0\to\Omega^1_{\mathbb P^n/k}\to\mathcal O(-1)^{\oplus(n+1)}\to\mathcal O\to0$. The <cohomology of twisting sheaves on projective space> and the <long exact sequence in sheaf cohomology> give $H^1(\mathbb P_k^n,\Omega^1)\cong k$, with every other cohomology group zero. Therefore the <Euler characteristic of a coherent sheaf> is $-1$. The nonzero class is the connecting image of the constant section $1$.
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