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Points on Quantum Projectivizations
Adam Nyman, University of Montana, Missoula, MT
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Memoirs of the American Mathematical Society
2004; 142 pp; softcover
Volume: 167
ISBN-10: 0-8218-3495-9
ISBN-13: 978-0-8218-3495-4
List Price: US$64 Individual Members: US$38.40
Institutional Members: US\$51.20
Order Code: MEMO/167/795

The use of geometric invariants has recently played an important role in the solution of classification problems in non-commutative ring theory. We construct geometric invariants of non-commutative projectivizataions, a significant class of examples in non-commutative algebraic geometry. More precisely, if $$S$$ is an affine, noetherian scheme, $$X$$ is a separated, noetherian $$S$$-scheme, $$\mathcal{E}$$ is a coherent $${\mathcal{O}}_{X}$$-bimodule and $$\mathcal{I} \subset T(\mathcal{E})$$ is a graded ideal then we develop a compatibility theory on adjoint squares in order to construct the functor $$\Gamma_{n}$$ of flat families of truncated $$T(\mathcal{E})/\mathcal{I}$$-point modules of length $$n+1$$. For $$n \geq 1$$ we represent $$\Gamma_{n}$$ as a closed subscheme of $${\mathbb{P}}_{X^{2}}({\mathcal{E}}^{\otimes n})$$. The representing scheme is defined in terms of both $${\mathcal{I}}_{n}$$ and the bimodule Segre embedding, which we construct.

Truncating a truncated family of point modules of length $$i+1$$ by taking its first $$i$$ components defines a morphism $$\Gamma_{i} \rightarrow \Gamma_{i-1}$$ which makes the set $$\{\Gamma_{n}\}$$ an inverse system. In order for the point modules of $$T(\mathcal{E})/\mathcal{I}$$ to be parameterizable by a scheme, this system must be eventually constant. In [20], we give sufficient conditions for this system to be constant and show that these conditions are satisfied when $${\mathsf{Proj}} T(\mathcal{E})/\mathcal{I}$$ is a quantum ruled surface. In this case, we show the point modules over $$T(\mathcal{E})/\mathcal{I}$$ are parameterized by the closed points of $${\mathbb{P}}_{X^{2}}(\mathcal{E})$$.

Graduate students and research mathematicians interested in algebraic geometry.

• Construction of the functor $$\Gamma_n$$
• The representation of $$\Gamma_n$$ for low $$n$$
• The representation of $$\Gamma_n$$ for High $$n$$