2012/06/04 by Thomas Nevins, T. A. Nevins, Nevins, T. A. +3
Mathematics · #14D22 #14D23 #16P40 #16S38 #16W50 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Rings and Algebras (math.RA) #math.AG #math.RA #msc:14D22 #msc:14D23 #msc:16P40 #msc:16S38 #msc:16W50
paper · pdf · doi:10.48550/arxiv.1206.0760
25 pages, comments welcome; v2: substantial revision, several errors corrected. Main results not changed significantly
openalex publication_date 2012/06/04 · arxiv created 2014/04/14 · arxiv updated 2014/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In 2008, Rogalski and Zhang showed that if R is a strongly noetherian connected graded algebra over an algebraically closed field, then R has a canonical birationally commutative factor. This factor is, up to finite dimension, a twisted homogeneous coordinate ring B(X, L, s); here X is the projective parameter scheme for point modules over R, as well as tails of points in qgr-R. (As usual, s is an automorphism of X, and L is an s-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 is a coarse moduli space for 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, s), and that g(R) is, up to finite dimension, a naive blowup on X in the sense of Keeler, Rogalski, and Stafford, and satisfies a universal property. We further show that the point space Y is noetherian.