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 \({\mathbb{P}}_{X^{2}}(\mathcal{E})\).

Readership

Graduate students and research mathematicians interested in algebraic geometry.

Table of Contents

• 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