Points on Quantum Projectivizations
Seiten
2003
American Mathematical Society (Verlag)
978-0-8218-3495-4 (ISBN)
American Mathematical Society (Verlag)
978-0-8218-3495-4 (ISBN)
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The use of geometric invariants has played an important role in the solution of classification problems in non-commutative ring theory. This title constructs geometric invariants of non-commutative projectivizataions, a significant class of examples in non-commutative algebraic geometry.
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 [/textbf{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 ${/mathbb{P}}_{X^{2}}(/mathcal{E})$.
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 [/textbf{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 ${/mathbb{P}}_{X^{2}}(/mathcal{E})$.
Introduction Compatibilities on squares Construction of the functor $/Gamma_n$ Compatibility with descent The representation of $/Gamma_n$ for low $n$ The bimodule Segre embedding The representation of $/Gamma_n$ for High $n$ Bibliography Index.
| Erscheint lt. Verlag | 1.3.2004 |
|---|---|
| Reihe/Serie | Memoirs of the American Mathematical Society |
| Verlagsort | Providence |
| Sprache | englisch |
| Themenwelt | Naturwissenschaften ► Physik / Astronomie |
| ISBN-10 | 0-8218-3495-9 / 0821834959 |
| ISBN-13 | 978-0-8218-3495-4 / 9780821834954 |
| Zustand | Neuware |
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