Commutative Algebra

Commutative algebra studies commutative rings and modules over these rings, which are the basic objects of algebraic geometry and algebraic number theory. Homological algebra is the study of homology theories from algebraic topology, giving us powerful algebraic tools for applications in many fields of mathematics.

These notes follow the graduate course MA5204 Graduate Algebra IIA, offered in AY2018/19 Semester 2 by Prof Chin Chee Whye. The main text for this course is H. Matsumara, Commutative Ring Theory (1989).

• A Refresher in Commutative Rings
• Commutative Algebra (I): Radicals
• Commutative Algebra (II): Formal Power Series and Polynomials
• Homological Algebra (I): The Five Lemma and the Snake Lemma
• Homological Algebra (II): Chain Complexes
• Commutative Algebra (III): Finite Modules
• Commutative Algebra (IV): Finitely Presented Modules
• Commutative Algebra (V): Composition Series
• Commutative Algebra (VI): Chain Conditions
• Commutative Algebra (VII): Noetherian Rings
• Commutative Algebra (VIII): Spectrum of a Ring
• Commutative Algebra (IX): The Induced Map on the Spectrum

(In progress, last updated 21 Mar 2019)