Naïve noncommutative blowing up

D. S. Keeler, D. Rogalski, J. T. Stafford

    Research output: Contribution to journalArticlepeer-review

    Abstract

    Let B(X, ℒ, σ) be the twisted homogeneous coordinate ring of an irreducible variety X over an algebraically closed field k with dim X ≤ 2. Assume that c ∈ X and σ ∈ Aut(X) are in sufficiently general position. We show that if one follows the commutative prescription for blowing up X at c, but in this noncommutative setting, one obtains a noncommutative ring R = R(X, c,ℒ, σ) with surprising properties. (1) R is always Noetherian but never strongly Noetherian. (2) If R is generated in degree one, then the images of the R-point modules in qgr-A are naturally in one-to-one correspondence with the closed points of X. However, in both qgr-R and gr-R, the R-point modules are not parametrized by a projective scheme. (3) While qgi-R has finite cohomological dimension, dimk H1 (script O signR) = ∞.
    Original languageEnglish
    Pages (from-to)491-546
    Number of pages55
    JournalDuke Mathematical Journal
    Volume126
    Issue number3
    DOIs
    Publication statusPublished - 15 Feb 2005

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