Naïve blowups and canonical birationally commutative factors

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In 2008, Rogalski and Zhang (Math Z 259(2):433–455, 2008) showed that if R is a strongly noetherian connected graded algebra over an algebraically closed field \({\Bbbk }\), then R has a canonical birationally commutative factor. This factor is, up to finite dimension, a twisted homogeneous coordinate ring \(B(X, \mathcal {L}, \sigma )\); here X is the projective parameter scheme for point modules over R, as well as tails of points in \({{\mathrm{Qgr-\!}}}R\). (As usual, \(\sigma \) is an automorphism of X, and \(\mathcal {L}\) is a \(\sigma \)-ample invertible sheaf on X). We extend this result to a large class of noetherian (but not strongly noetherian) algebras. Specifically, let R be a noetherian connected graded \({\Bbbk }\)-algebra, where \({\Bbbk }\) is an uncountable algebraically closed field. Let \(Y_\infty \) denote the parameter space (or stack or proscheme) parameterizing R-point modules, and suppose there is a projective variety X that corepresents tails of points. There is a canonical map p:\(Y_\infty \rightarrow X\). If the indeterminacy locus of \(p^{-1}\) is 0-dimensional and X satisfies a mild technical assumption, we show that there is a homomorphism g:\(R \rightarrow B(X,\mathcal {L},\sigma )\), and that g(R) is, up to finite dimension, a naive blowup on X in the sense of Keeler et al. (Duke Math J 126(3):491–546, 2005), Rogalski and Stafford (J Algebra 318(2):794–833, 2007) and satisfies a universal property. We further show that the point space \(Y_\infty \) is noetherian.