Notation: Tensor of a sheaf and residue field This is a simple question of notation. Let $k(p)$ be the residue field of a point $p$ on $\mathbb{P}^{N}$. How is defined (and where is) the sheaf $\mathcal{O}_{\mathbb{P}^{N}}(1) \otimes k(p)$? Also, what could mean the tensor $H^{0}(\mathbb{P}^{N},\mathcal{O}_{\mathbb{P}^{N}}(1)) \otimes \mathcal{O}_{\mathbb{P}^{N}}$? Thank you.

a) The sheaf $\mathcal{O}_{\mathbb{P}^{N}}(1) \otimes k(p)$ is the sky-scraper sheaf on $\mathbb{P}^{N}$ whose only non-zero fiber is over $p$, the value of that fiber being the one-dimensional $k(p)$-vector space $L(p)^\ast $, where $L(p)\subset k^{n+1}$ is the line represented by the point $p$ and the asterisk means "dual vector space".
The notation $\mathcal{O}_{\mathbb{P}^{N}}(1) \otimes k(p)$ may also denote just the vector space $L(p)^\ast $ or may denote the sheaf with that fiber over the one-point scheme $\operatorname {Spec } (k(p))$ .

b) The notation $H^{0}(\mathbb{P}^{N},\mathcal{O}_{\mathbb{P}^{N}}(1)) \otimes \mathcal{O}_{\mathbb{P}^{N}}$ denotes the trivial vector bundle (= locally free sheaf) of rank $n+1$ with fiber $H^{0}(\mathbb{P}^{N}, \mathcal{O}_{\mathbb{P}^{N}}(1))$ .