Naïve blowups and canonical birationally commutative factors

• Published in: Mathematische Zeitschrift Vol. 280; no. 3-4; pp. 1125 - 1161
• Main Authors: Nevins, T. A., Sierra, S. J.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2015
• Subjects: Mathematics; Mathematics and Statistics; 16W50; Naive blowup algebra; Canonical birationally commutative factor; 14D23; 14D22; 16P40; Noncommutative Hilbert scheme; Point space; 16S38
• ISSN: 0025-5874; 1432-1823
• DOI: 10.1007/s00209-015-1470-3

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.