= Cotangent Euler sequence in homogeneous coordinates
{title2=$0\to\Omega^1_{\mathbb P^n}\to\mathcal O(-1)^{\oplus(n+1)}\to\mathcal O\to0$}
For a degree-zero rational function regular on an open subset of <projective space>, each derivative $\partial f/\partial X_i$ is a regular homogeneous rational function of degree $-1$. These derivatives define a <derivation> with values in $\mathcal O(-1)^{\oplus(n+1)}$, hence a map from the <Kähler differential sheaf>. The last map is $(g_i)\mapsto\sum_iX_ig_i$. On $X_i\ne0$, put $y_j=X_j/X_i$ and use frame $X_i^{-1}$. The derivative of $y_j$ has coordinate vector with $1$ at $j$, $-y_j$ at $i$ and zero elsewhere. Those vectors form a free basis of the kernel of $(b_i)\mapsto b_i+\sum_{j\ne i}y_jb_j$, proving exactness directly in every characteristic.
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