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Generic noncommutative surfaces. (English) Zbl 1068.16038

The idea of interpreting certain noncommutative \(\mathbb{N}\)-graded algebras over a field \(k\) as coordinate rings of noncommutative projective schemes has emerged about a decade ago and has opened up a new subject area, now known as noncommutative projective geometry. Numerous ring-theoretic problems have been solved using this approach: for example, graded domains of dimension 2, which correspond to noncommutative curves, have been classified, and so have noncommutative analogues of the projective plane \(\mathbb{P}^2\). The survey article by J. T. Stafford and M. Van den Bergh [Bull. Am. Math. Soc., New Ser. 38, No. 2, 171-216 (2001; Zbl 1042.16016)] documents the considerable progress of the classification theory of graded algebras of dimension 3. In the paper under review, the author constructs examples of algebras of dimension \(\geq 3\) which provide answers to a number of open questions in the literature.
Given a generic Zhang twist \(S=\bigoplus_{i\geq 0}S_i\) of a polynomial ring in \(t+1\) variables, \(t\geq 2\), over an algebraically closed field \(k\), let \(R\) denote any subalgebra of \(S\) which is generated by a generic subspace of \(S_1\) with codimension one. Then \(R\) has the following properties. (1) \(R\) is Noetherian, connected, finitely \(\mathbb{N}\)-graded, finitely generated in degree 1, and a certain technical homological condition, denoted \(\chi_1\), is satisfied (thus assuring the validity of the noncommutative version of Serre’s Theorem due to M. Artin and J. J. Zhang [Adv. Math. 109, No. 2, 228-287 (1994; Zbl 0833.14002)]), but the conditions \(\chi_i\) fail for all \(i\geq 2\), which answers a question posed by J. T. Stafford and J. J. Zhang [Math. Proc. Camb. Philos. Soc. 116, No. 3, 415-433 (1994; Zbl 0821.16026)]. (2) \(R\) is not strongly Noetherian, that is, \(R\otimes_kB\) is not necessarily a Noetherian ring for every commutative Noetherian \(k\)-algebra \(B\), thus putting an end to some hopes that all finitely generated \(\mathbb{N}\)-graded Noetherian \(k\)-algebras might be strongly Noetherian and that the point modules would therefore be naturally parametrized by a commutative projective scheme over \(k\), thus forming a nice geometric object. (3) \(R\) is the first example of a connected, finitely \(\mathbb{N}\)-graded maximal order for which \(\chi_i\) fails for \(i\geq 2\).
This very interesting article will have considerable impact on what one can expect typical noncommutative surfaces to look like.

MSC:

16S38 Rings arising from noncommutative algebraic geometry
16W50 Graded rings and modules (associative rings and algebras)
14A22 Noncommutative algebraic geometry
16P40 Noetherian rings and modules (associative rings and algebras)
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References:

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