On some approaches towards non-commutative algebraic geometry

Abstract

The works of R. Descartes, I. M. Gelfand and A. Grothendieck have convinced us that commutative rings should be thought of as rings of functions on some appropriate (commutative) spaces. If we try to push this notion forward we reach the realm of Non-commutative Geometry. The confluence of ideas comes here mainly from three seemingly disparate sources, namely, quantum physics, operator algebras (Connes-style) and algebraic geometry. Following the title of the article, an effort has been made to provide an overview of the third point of view. Since na\" ive efforts to generalize commutative algebraic geometry fail, one goes to the root of the problem and tries to work things out "categorically". This makes the approach a little bit abstract but not abstruse. However, an honest confession must be made at the outset - this write-up is very far from being definitive; hopefully it will provide glimpses of some interesting developments at least.