Question : How to deduce the exactness at $\mathcal{O}_{\mathbb{P}^n}(1)^{n+1}$ of the Euler sequence $0\rightarrow\mathcal{O}_{\mathbb{P}^n}\rightarrow\mathcal{O}_{\mathbb{P}^n}(1)^{n+1}\rightarrow T\mathbb{P}^n\rightarrow0$ on $\mathbb{P}^n$, as a consequence of Euler's theorem $\sum_i x_i\partial_{x_i}F=kF$ on homogeneous functions?
Background : I am reading the book Enumerative Geometry and String theory by S. Katz. In the book, the author defined a subbundle $F$ of $\mathcal{O}_{\mathbb{P}^n}(1)^{n+1}$, consisting sections of the form $(fx_0,\dots,fx_n)$, where $f$ is function on $\mathbb{P}^n$. Then the author defined the map $\mathcal{O}_{\mathbb{P}^n}(1)^{n+1}\to T\mathbb{P}^n:$ $$(l_0,\dots,l_n)\mapsto \sum_i l_i\dfrac{\partial}{\partial x_i}$$ In p.119, the author explicitly said
The subbundle $F$ is indeed mapped to $0$ in $T\mathbb{P}^n$ as a consequence of Euler's formula.
Here, the Euler's formula means the Euler's theorem on homogeneous function, which states that For a homogeneous function $F$ of order $k$, one has the followings : $$\sum_i x_i\partial_{x_i}F=kF$$
Identifying functions $f$ with the section $(fx_0,\dots,fx_n)$, gives the above formulation in the question.
This answer in MO https://mathoverflow.net/a/5216 also asserts similarly.
..., which says that the vector field $\frac{1}{d}\sum x_i\partial_i$ acts trivially on functions. Thus, the quotient must be the actual tangent vector fields, giving us the tangent bundle.
My attempt : I am confused as I cannot follow this result as a consequence of the Euler's formula. The Euler's formula only holds on homogeneous function $F$ of order $k$, while the derivation $\sum_i x_i\dfrac{\partial}{\partial x_i}$ as a section of tangent bundle acts on all (smooth) functions. To conclude the vanishing of $\sum_i x_i\dfrac{\partial}{\partial x_i}$, it is not enough to consider homogeneous functions. Even if one managed to show that it suffices to consider homogeneous functions, those $F$ are merely eigenfunctions with constant $k$ of the Euler vector field $\sum_i x_i\dfrac{\partial}{\partial x_i}$, the right hand side of the Euler's formula is not $0$, I cannot conclude there are linear relations in the set $\{x_0\dfrac{\partial}{\partial x_0},\dots,x_n\dfrac{\partial}{\partial x_n}\}$.
I am aware of similar questions like Understanding the Euler sequence on $\mathbb{P}^n$. However, the main difference is, I am asking for proving the exactness as a consequence of the Euler's formula, instead of proving the exactness by any methods. I am aware of other methods, for example in P.408 of Griffiths-Harris, which used the affine coordinate directly; or the excellent answer in https://mathoverflow.net/a/5218. However I think both methods are not direct consequences of the Euler's formula. Thank you very much!
The way I think about this is that $x_i$ are not coordinates on the projective space, they are coordinates of $\mathbb C^{n+1}$. So the Euler vector fields in the form above are really vector fields descending from $\mathbb C^{n+1}$. So in order to act it on functions, you need to lift the function from $\mathbb P^n$ to $\mathbb C^{n+1}-\{0\}$. The lifted functions are constant along fibers, so if you differentiate the function at a point $p\in\mathbb C^{n+1}-\{0\}$ along the direction $p$ (considered as a vector), then it is zero. This is really just rephrasing Euler's formula for homogeneous degree 0. The holomorphic functions on $\mathbb P^n$ are homogeneous function on $\mathbb C^{n+1}-\{0\}$ of degree 0.