Noncommutative algebraic varieties, their properties and geometric realizations. Lecture 1

We are will discuss a notion of noncommutative and derived algebraic variety. This approach comes from a generalization of derived categories of (quasi)coherent sheaves on usual algebraic varieties and their enhancements. We are going to talk about different properties of noncommutative varieties such as regularity, smoothness and properness. A new operation of gluing of noncommutative varieties will be introduced. We also will define a some notion of geometric realization of noncommutative varieties, and will give explicit examples of such realizations, in the case of special exceptional collections.