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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 Link).

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!

Glorfindel
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Peter Hu
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  • I'm having trouble understanding the notation. Is $U_\sigma$ an affine open subset isomorphic to $\operatorname{Spec} A_\sigma$? – Daniel Jun 27 '22 at 21:06
  • @Daniel just so you're aware, this question is 10 years old and the asker has not visited the site for at least 6 years. The reason it was edited and bumped today was for replacement of a dead link. – KReiser Jun 27 '22 at 21:30
  • For future users coming across this question, assuming $U_\alpha\cong\operatorname{Spec} A_\alpha$, the first duplicate explains how a locally free sheaf on $U_\alpha$ corresponds to a projective module over $A_\alpha$ and the second duplicate explains various definitions of the rank of a coherent sheaf. – KReiser Jun 27 '22 at 22:28

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