Noncommutative Zariski Geometries and Their Classical Limit

Abstract

We undertake a case study of two series of nonclassical Zariski geometries. We show that these geometries can be realised as representations of certain noncommutative C * -algebras and introduce a natural limit construction which for each of the two series produces a classical U (1)-gauge field over a 2-dimensional Riemann surface.

Generalised Imaginaries and Galois Cohomology

“…The same situation is observed (Section 4.1) in the case of non-standard Zariski structures defined in [23,10], if one assumes finiteness of the group action used in their definition; there again the definability of the structure in an algebraically closed field is equivalent to eliminability of a certain generalised imaginary.…”

“…Hrushovski and Zilber further prove (Theorem C) that there exists a variety X and a group extension such that D(X, G) is not interpretable in an algebraically closed field. More examples of structures of the form D(X, G) were considered in [23].…”

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The objective of this article is to characterise elimination of finite generalised imaginaries as defined in [9] in terms of group cohomology. As an application, I consider series of Zariski geometries constructed [10,23,22] by Hrushovski and Zilber and indicate how their non-definability in algebraically closed fields is connected to eliminability of certain generalised imaginaries.

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Perfect Infinities and Finite Approximation

“…The classical meaning of a coordinate algebra comes from the algebra of co-ordinate functions on the object, that is functions ψ : M → F as in 2.6, of a certain class. The most natural algebra of functions for Zariski geometries seems to be the algebra F In [16], in order to see the rest of the structure we extended F[M] by introducing auxiliary semi-definable functions, which satisfy certain equations but are not uniquely defined by these equations. The F-algebra H(M) of semi-definable functions contains the necessary information about M but is not canonically defined.…”

“…In [16] one more important observation was made. The examples of nonclassical Zariski geometries come in uniform families with variations within a family given by the size of the fibre of the covering map p. It is natural to ask what can be seen when the size of the fibre tends to infinity.…”