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## 2007.205: Naïve noncommutative blowing up

2007.205: D. S. Keeler, D. Rogalski and J. T. Stafford (2005) Naïve noncommutative blowing up. Duke Mathematical Journal, 126 (3). pp. 491-546. ISSN 0012-7094

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## Abstract

Let B(X,\mathscr{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 \in X and σ \in 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,$\mathscr{L}$,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-R 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 qgr-R has finite cohomological dimension dim_k H^1 ( \mathscr{O} ) = ∞.

Item Type: Article MSC 2000 > 14 Algebraic geometryMSC 2000 > 16 Associative rings and algebras 2007.205 Mrs Louise Healey 22 November 2007