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 K, then R has a canonical birationally commutative factor. This factor is, up to finite dimension, a twisted homogeneous coordinate ring B(X,L,σ); here X is the projective parameter scheme for point modules over R, as well as tails of points in Qgr-R. (As usual, σ is an automorphism of X, and L is a σ-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 K-algebra, where K is an uncountable algebraically closed field. Let Y∞ 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∞ → 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 → B(X,L,σ), 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∞ is noetherian.
- Canonical birationally commutative factor
- Naive blowup algebra
- Noncommutative Hilbert scheme
- Point space
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