(Euler sequence)
For \(X = \PP^n_A\) there is a short exact sequence of \(\OO_X\)-modules \[ 0 \to \Omega_{X/A} \to \OO_X(-1)\sumpower{n+1} \to \OO_X \to 0 . \]
The sequence is the sheaf-level record of \(\PP^n = (\AA^{n+1} \setminus 0)/\GG_m\): the middle term is the pullback of the cotangent bundle of the affine space, twisted, and the surjection onto \(\OO_X\) is contraction with the Euler vector field, whose kernel is the forms that descend. Dually, \(0 \to \OO_X \to \OO_X(1)\sumpower{n+1} \to T_X \to 0\), and the \(\OO_X\) is the Euler field itself.
Its determinant computes the canonical bundle. Taking top exterior powers in a short exact sequence multiplies them, so \[ \omega_{\PP^n} = \Extpower^n \Omega = \Extpower^{n+1}\qty{\OO(-1)\sumpower{n+1}} \tensor \dualof{\OO} = \OO(-n-1) , \] which is the one computation that the genus formulas, the Fano property of \(\PP^n\), and every application of adjunction in projective space come out of.
The sequence does not split for \(n \geq 1\), and for \(n \geq 2\) the sheaf \(\Omega_{\PP^n}\) is locally free but not a direct sum of line bundles: the Euler sequence is the standard witness that locally free does not mean split.
On \(\PP^1\) with coordinates \(x, y\), the dual Euler sequence is \(0 \to \OO \xrightarrow{(x,\,y)} \OO(1)^{\oplus 2} \to T_{\PP^1} \to 0\). So the cokernel of \((x, y) \colon \OO \to \OO(1)^{\oplus 2}\) is \(T_{\PP^1} \cong \OO(2)\), the determinant of \(\OO(1)^{\oplus 2}\); explicitly \((s, t) \mapsto y s - x t\) identifies it with \(\OO(2)\).