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Naïve noncommutative blowing up. (English) Zbl 1082.14003

In the article [D. Rogalski, Adv. Math. 184, 289–341 (2004; Zbl 1068.16038)], the author constructed in a purely algebraic way a class of noncommutative surfaces with surprising properties. The aim of the paper under review is to give a geometric construction of these and more general algebras with similar properties.
The basic idea of the construction is a noncommutative analog of the blow up in a closed point in sufficiently general position. Similarly to the construction of bimodule algebras, the authors start with an automorphism \(\sigma\) of an irreducible variety \(X\) of dimension at least two. For a closed point \(c\in X\) in sufficiently general position (with respect to \(\sigma\)) and a \(\sigma\)-ample invertible sheaf \(\mathcal L\) on \(X\), they define a sheaf \(\mathcal R\) of graded algebras. The main objects of study are the algebra \(R\) of global sections of \(\mathcal R\) and the category qgr-\(R\) of finitely generated \(R\)-modules modulo torsion.
It is shown that \(R\) is always Noetherian, but never strongly Noetherian, i.e. there is a commutative Noetherian algebra \(C\) such that \(R\otimes C\) is not Noetherian. The category qgr-\(R\) is shown to be equivalent to qgr-\(\mathcal R\) and independent of the sheaf \(\mathcal L\). It is proved that simple objects in qgr-\(R\) are in bijective correspondence with closed points in \(X\), but \(R\)-point modules are not parametrized by a scheme of locally finite type. Finally, some cohomological properties of \(R\) are analyzed. In particular, qgr-\(R\) has finite cohomological dimension, and for smooth \(X\) finite homological dimension, but the group Ext\(^1(R,R)\) is infinite dimensional.
Reviewer: Andreas Cap (Wien)

MSC:

14A22 Noncommutative algebraic geometry
16P40 Noetherian rings and modules (associative rings and algebras)
16W50 Graded rings and modules (associative rings and algebras)
16S38 Rings arising from noncommutative algebraic geometry
18E15 Grothendieck categories (MSC2010)

Citations:

Zbl 1068.16038
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References:

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