Theorem
The Euler sequence is the following short exact sequence of sheaves on \(\mathbb P^n\): \[0 \to \mathcal O_{\mathbb P^n} \xrightarrow{f} \mathcal O_{\mathbb P^n}(1)^{\oplus n+1}\xrightarrow{g} \mathcal T\mathbb P^n\to 0.\] Writing \([x_0:...:x_n]\) for homogeneous coordinates on \(\mathbb P^n\) the maps are defined as follows.
Map 1 If \(c\) is a locally constant function on \(\mathbb P^n\) then \[f:c \mapsto (c\cdot x_0, c\cdot x_1, ..., c\cdot x_n)\] i.e. \(f\) is multiplication of the set of linear monomials \((x_0,...,x_n)\) by \(c\).
Map 2 For a set \(l_i(x)\) of degree 1 homogeneous functions on \(\mathbb P^n\), \[g:(l_0(x),...,l_n(x) \mapsto l_0(x)\frac{\partial}{\partial x_0} + ... + l_n(x)\frac{\partial}{\partial x_n}.\]