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In his book Algebraic Geometry, Hartshorne introduces the twisting sheaf on a scheme $X = \operatorname{Proj} $ of a graded algebra $S$. He does this by taking the associated sheaf on $X$ of the $S$-module $S(n)$. This is on pg. 117 of my edition.

I don't know what the module $S(n)$ is. Is it the submodule of $S$ generated by the component of homogeneous elements of degree $n$?

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    $\begingroup$ It's defined somewhere in the book. In general when you see the "bracket $n$" it means that you're shifting the degree by $n$. This is, $S(n)_d= S_{d+n}$. $\endgroup$
    – user347489
    Commented Apr 18, 2018 at 3:10

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As stated in the comments, the definition is $S(n)_d = S_{n+d}$. For your reference, the definition is on page 50 in Chapter I section 7. (This is not the same as the module generated by $S_n$ mentioned in the question, even in the ungraded sense. In special cases such as regular module over a polynomial ring, it is.)

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    $\begingroup$ OK ive found it. thanks $\endgroup$
    – basket
    Commented Apr 18, 2018 at 5:50

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