# Naïve noncommutative blowing up

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

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} ) = ∞.