 Twisting sheaf on projective space

The twisting sheaf $\mathcal{O}(d)$ on $\mathrm{Proj}k[x_0,\dots ,x_n]$ is associated with the shifted graded module $k[x_0,\dots ,x_n](d)$. For $d\ge 0$, its global sections are the homogeneous polynomials of degree $d$.

On projective space the twisting sheaf $\mathcal{O}(d)$ is the $d$th tensor power of $\mathcal{O}(1)$ for $d\ge 0$, and the corresponding dual power for $d<0$. Twisting another sheaf means tensoring it with this line bundle.