Commutative algebra is the study of commutative rings and their modules, and forms the basis for the machinery used in algebraic geometry and number theory, as well as much of algebraic topology.
Topics we will study include commutative rings; ideals, modules, and exact sequences; localization; the Nullstellensatz, Nakayama’s lemma, and integrality; flat families; discrete valuation rings; and other topics as time permits. Where appropriate, we will interpret the commutative algebra results in the light of algebraic geometry.
Students should have completed the standard algebra courses on groups, rings, and fields (comparable to the courses Algebra 1, Algebra 2 and Algebra 3 given at Leiden University). No knowledge of algebraic geometry is assumed.