equ qcoh sheaf and families RJI57TR8

equ qcoh sheaf and families

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Proposition 1.10. [17, Theorem 4.9] The category of Σ-families is equivalent to the category of torus equivariant quasi-coherent sheaves on a smooth toric variety X. (pdf) (Wang, 2023, p. 75)

For any sheaf \(\mathcal{F}\) on \(X\), and cone \(\sigma\), there is a module \(F^{\sigma}\), and admits a \(T\)-action. Therefore it is a \(M\)-graded \(k[S_{\sigma}]\)-module \(F^{\sigma}=\oplus_{m\in M} F^{\sigma}_{m}\).