Noncommutative Algebra and Noncommutative Geometry (1404.0126v3)

Abstract: Divided into three parts, the first marks out enormous geometric issues with the notion of quasi-freenss of an algebra and seeks to replace this notion of formal smoothness with an approximation by means of a minimal unital commutative algebra's smoothness. The second part of this text is then, devoted to the approximating of properties of nc. schemes through the properties of two uniquely determined (classical) schemes estimating the nc. scheme in question in a maximal way from the inside and through the minimal scheme approximating the nc. scheme in question from the outside. The very brief final par of this exposition, aims to understand and distil the properties at work in constructing any "scheme-like" object over an "appropriate" category, purely out of philosophical interest.