Topics in Algebraic Geometry

Abstract

The key objective of this yearlong project was to learn serveral concepts in commuta- tive algebra and algebraic geometry. We started with learning concepts in commuta- tive algebra, which included primary decomposition, Noetherian and Artinian Rings, regular local rings and dimension theory of Noetherian local rings. After that we studied concepts in algebraic geometry, which included varieties and morphisms of varieties, sheaves and schemes. We concluded by stating the Riemann-Roch theorem and how that result is used to solve Riemann-Roch problem for the curves.