Structure of $\mathcal{O}_X$-algebra of $\mathcal{S}*\mathcal{L}$ in Hartshorne II.7.9

Question

In lemma II.7.9 Hartshorne define a graded $\mathcal{O}_X$-algebra $\mathcal{S}'$ from a graded algebra $\mathcal{S}$ and an invertible sheaf $\mathcal{L}$ by set up for each $d\geqslant0$, $$ \mathcal{S}'_d=\mathcal{S}_d\otimes\mathcal{L}^d$$ I can't see why such a definition define an $\mathcal{O}_X$-algebra. I can see that it is a graded $\mathcal{O}_X$-module, I guess that we can give it a structure of non-commutative $\mathcal{O}_X$-algebra but there is clearly a problem of commutativity coming from tensor products. I see that locally, on small enough open set $U$ we have ${\mathcal{S}'}_{|U}\simeq\mathcal{S}_{|U}$ because $\mathcal{L}_{|U}\simeq\mathcal{O}_X$ but it's only local.