Abstract
The theory of algebraic varieties gives an algebraic interpretation of differential geometry, thus of our physical world. To treat, among other physical properties, the theory of entanglement, we need to generalize the space parametrizing the objects of physics. We do this by introducing noncommutative varieties.
Highlights
Differential geometry is simplified by applying algebraic geometry
All holomorphic functions are interpreted as polynomials, and completed to power series
We show how our results can be stated without mentioning deformation theory, and we show one way of interpreting noncommutative varieties of matrix polynomial algebras
Summary
Differential geometry is simplified by applying algebraic geometry. All holomorphic functions are interpreted as polynomials, and completed to power series. We have used infinitesimal methods, or rather deformation theory of modules, to construct varieties as moduli of the closed points. A generalization to deformation theory over commutative rings to noncommutative, have made the definition of noncommutative varieties possible. In this short text, we show how our results can be stated without mentioning deformation theory, and we show one way of interpreting noncommutative varieties of matrix polynomial algebras. We will show how these varieties can (and must) be constructed using deformation theory
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