Naïve blowups and canonical birationally commutative factors

T. A. Nevins, S. J. Sierra

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)1125-1161
Number of pages37
JournalMathematische Zeitschrift
Issue number3-4
StatePublished - Aug 26 2015


  • Canonical birationally commutative factor
  • Naive blowup algebra
  • Noncommutative Hilbert scheme
  • Point space

ASJC Scopus subject areas

  • Mathematics(all)


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