| Abstract: Thanks to classical work of Zariski and Grothendieck, the problem of studying singularities in algebraic geometry can be translated into an algebraic problem about commutative rings. In this framework, regular rings correspond to non-singular algebraic varieties or schemes, and various algebraic methods have been developed to determine how far a ring is from being regular. In particular, starting from the pioneering work of Auslander, Buchsbaum and Serre, homological algebra is an essential tool to study singularities algebraically. In this talk, which will be of expository nature and targeted to graduate students, I will discuss Cohen-Macaulay singularities from an algebraic perspective, by illustrating the rich homological and ideal-theoretic properties which -- quoting Mel Hochster -- make life ``really worth living in a Cohen-Macaulay ring''. |