The Stacks project

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.

Comments (4)

Comment #1791 by Keenan Kidwell on

Shouldn't "as the image of " actually be "as the image of ?"

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 times" make sense if ? It could be clearer to define , then can be defined for .

Comment #6636 by on

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

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View Definition 17.25.6 as pdf