# Euler exact sequence of sheaves on $\mathbb P^n(\mathbf C)$

Reading this article by A. Caldararu I'm having troubles with Prop. 3.2 at page 11. Maybe it's a lack of prerequisites, and surely a trivial question for those who are inside the topic, but I'm not able to find anything about what follows:

In particular, how do I have to interpret the sentence

a basis of $H^0(P^n, \cal O^{n+1})$ is given by a dual basis of $y_0, ... ,y_n$ [...]

Is there some kind of duality between $H^0(P^n, \cal O^{n+1})$ and $H^0(P^n, \cal O(1))$?

Edit: Maybe it's better to make my question more precise. It's not difficult to notice that there exists an identification $H^0(P^n, \mathcal O^{n+1})\cong H^0(P^n, \mathcal O)^{n+1}\cong \mathbf C^{n+1}$; and according to Caldararu's argument $H^0(P^n, \cal O(1))$ (which is the homogeneous degree-1 part of the coordinate ring of $P^n(\mathbf C)$) should coincide with the dual of the former space: I'm not very comfortable in doing a linear-algebra parallel now (degree-1 homogeneous polynomials = covectors), but maybe it's only a psychological reason which prevents me to do so.

Hence, a slightly less difficult question for you is: am I correct in thinking this way?

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You should probably tell us what $y_0$, $\dots,$y_n\$ are. – Mariano Suárez-Alvarez Dec 28 '11 at 19:31
Sure, I forgot the policy for comfortable reading. Just edited with the Proposition I need to understand. – tetrapharmakon Dec 28 '11 at 19:45