Twisted multi-homogeneous coordinate rings
Given two line bundles \(L\) and \(M\) on a projective scheme \(X\), one defines the bi-homogeneous coordinate ring \(B(X;L,M)=\bigoplus_{i,j\geq 0}H^{0}(X,L^{\otimes i}\otimes M^{\otimes j})\). The author gives an analogous construction in the setting of non-commutative algebraic geometry. To this end the line bundles are replaced by invertible bimodules and, to accommodate an analog of commutativity of the tensor product, a bimodule isomorphism \(M\otimes L\rightarrow L\otimes M\) is added as part of the structure. In fact this construction is carried out for a multi-homogeneous coordinate ring with an arbitrary number of invertible bimodules. This allows, under a suitable ampleness condition on the bimodules, to prove an analog of Serre's theorem giving an equivalence between the category of quasi-coherent sheaves and the category of graded modules modulo torsion submodules. In general, multi-homogeneous coordinate rings are not noetherian. A criterion for the ascending chain condition to hold is given. This gives, as applications, the noetherian property for the Rees algebra (which is the analog of blow up) and for the tensor product of homogeneous coordinate rings (which correspond to the fiber product).
graded modules modulo torsion submodules
scientific article; zbMATH DE number 1423878