# Restriction of locally free sheaf associated projective modules

My question comes from the paper of Tamafumi's "On Equivariant Vector Bundles On An Almost Homogeneous Variety" (it can be downloaded freely in http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.nmj/1118795362).

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 $r$?