= Twisting sheaf on Proj
{title2=$\mathcal O_{\operatorname{Proj}S}(n)=\widetilde{S(n)}$}
= Twisting sheaves on Proj
{synonym}
With the <graded shift> convention $S(n)_d=S_{n+d}$, this is the <sheaf associated with a graded module> $S(n)$. Under the <degree-one generation condition for Proj>, it is an <invertible sheaf>: on $D_+(f)$ with $\deg f=1$, multiplication by $f^n$ freely generates its local module for every integer $n$. Without that hypothesis, it need not be invertible. The <twisting sheaf on projective space> is the standard special case. https://stacks.math.columbia.edu/tag/01MM[Stacks Project, Section 27.10] records the hypotheses and local trivializations.
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