Commutative Algebra

Commutative algebra is the study of commutative rings and modules over them. It serves as a foundation of algebraic geometry and algebraic number theory. Typical examples of commutative rings are the ring of integers Z and the ring of polynomials $k[x_1, …, x_n]$ in $n$ variables over a field $k$ (we will encounter lots of more interesting examples). Geometrically, the latter can be thought of as the ring of polynomial functions on the $n$-dimensional space. Such a geometric perspective is very useful, enabling us to use our visual intuition when working with the polynomial ring. It turns out that one can view any commutative ring geometrically: to a commutative ring $R$ we can associate a topological space ${\rm Spec}(R)$, called its spectrum, whose geometry reflects the algebraic properties of $R$. Thus (unlike elsewhere in algebra, e.g. for noncommutative rings), in commutative algebra one can mix and match the precision and formality of algebra with geometric intuition.

The goal of the course is to introduce the basic concepts of commutative algebra, such as modules, local rings, valuation rings, Noetherian rings, completion, the Krull dimension etc., emphasizing the geometric meaning behind them when possible. Amore detailed list of topics can be found below.