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My question comes from the paper of Tamafumi's "On Equivariant Vector Bundles On An Almost Homogeneous Variety" (it can be downloaded freely in

Question 1 Let $X = X(\Delta)$ be a toric variety with $\Delta$ being a complete fan. Let $\sigma$ be a maximal cone. And $\mathcal(E)$ a locally free sheave over $X$. Tamafumi said that then $E|_{U_{\sigma}}$ is associated to a projective $A_{\sigma}$-module. Why?

I know free module must be projective. But locally free sheaf depends on the open cover. Who can tell me why $E|_{U_{\sigma}}$ is associated to a projective $A_{\sigma}$-module?

Thank you very much!

Question 2 In the Theorem 3.5., this theorem required $M$ to be a projctive $A_{\sigma}$-module of "rank $r$". I can't understand the meaning of "rank $r$". Does it mean that $M$ is a direct summand of a free $A_{\sigma}$-module of rank $r$?

Thank you for your help!

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