ScalingStacks

Proof. We will show that $\phi:P_{S\setminus s}\otimes_{S(L)}P_{S\setminus s}\to P,\ a\otimes b\mapsto as(b)$ is an isomorphism. It is a morphism of algebras, and a morphism of graded left $P_{S\setminus s}$-modules.

By assumption, there is $t\in S\setminus s$ such that $m_{st}\not=2$, so that $s(\alpha_{t})-\alpha_{t}$ is a non-zero multiple of $\alpha_{s}$. It follows that $\phi(\alpha_{t}\otimes 1-1\otimes\alpha_{t})\in k^{\times}\alpha_{s}$. Since $V^{*}=(V^{S\setminus s})^{\perp}\oplus k\alpha_{s}$, we have $P=P_{S\setminus s}\otimes k[\alpha_{s}]$ and we deduce that $\phi$ is surjective.

Since $\phi$ is a morphism of graded free left $P_{S\setminus s}$-modules with the same graded ranks $1+t+t^{2}+\cdots$, we deduce that $\phi$ is an isomorphism. ∎