eBook ISBN:  9781470403935 
Product Code:  MEMO/167/795.E 
List Price:  $68.00 
MAA Member Price:  $61.20 
AMS Member Price:  $40.80 
eBook ISBN:  9781470403935 
Product Code:  MEMO/167/795.E 
List Price:  $68.00 
MAA Member Price:  $61.20 
AMS Member Price:  $40.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 167; 2004; 142 ppMSC: Primary 14; Secondary 18
The use of geometric invariants has recently played an important role in the solution of classification problems in noncommutative ring theory. We construct geometric invariants of noncommutative projectivizataions, a significant class of examples in noncommutative 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 \(\Gamma_{i} \rightarrow \Gamma_{i1}\) 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 \({\mathbb{P}}_{X^{2}}(\mathcal{E})\).
ReadershipGraduate students and research mathematicians interested in algebraic geometry.

Table of Contents

Chapters

1. Introduction

2. Compatibilities on squares

3. Construction of the functor $\Gamma _n$

4. Compatibility with descent

5. The representation of $\Gamma _n$ for low $n$

6. The bimodule Segre embedding

7. The representation of $\Gamma _n$ for high $n$


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The use of geometric invariants has recently played an important role in the solution of classification problems in noncommutative ring theory. We construct geometric invariants of noncommutative projectivizataions, a significant class of examples in noncommutative 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 \(\Gamma_{i} \rightarrow \Gamma_{i1}\) 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.

Chapters

1. Introduction

2. Compatibilities on squares

3. Construction of the functor $\Gamma _n$

4. Compatibility with descent

5. The representation of $\Gamma _n$ for low $n$

6. The bimodule Segre embedding

7. The representation of $\Gamma _n$ for high $n$