Definition 17.25.6. Let $(X, \mathcal{O}_ X)$ be a ringed space. Given an invertible sheaf $\mathcal{L}$ on $X$ and $n \in \mathbf{Z}$ we define the $n$th tensor power $\mathcal{L}^{\otimes n}$ of $\mathcal{L}$ as the image of $\mathcal{O}_ X$ under applying the equivalence $\mathcal{F} \mapsto \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{L}$ exactly $n$ times.

Comment #1791 by Keenan Kidwell on

Shouldn't "as the image of $\mathcal{O}_X$" actually be "as the image of $\mathcal{L}$?"

Comment #1823 by on

Nope. But this definition could be stated better. Any suggestions?

Comment #6635 by Peter Goetz on

How does "applying the equivalence ... exactly $n$ times" make sense if $n < 0$? It could be clearer to define $\scr{L}^{\otimes -1} = \scr{Hom}_{\scr{O}_X}(\scr{L}, \scr{O}_X)$, then $\scr{L}^{\otimes n}$ can be defined for $n < 0$.

Comment #6636 by on

Of course you are right, but what I meant was that if $n = -1$ for example, then apply the inverse of the functor.

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