Summer School 2001:
Homological conjectures for finite dimensional algebras

For any two modules M, N over a ring R, Yoneda products endow Ext^*_R(M,M) and Ext^*_R(N,N) with structure of graded rings, and turn Ext^*_R(M,N) into a graded bimodule over them. On the other hand, if S <- R -> T are homomorphisms of commutative rings, then Tor_*^R(S,T) is a graded commutative R-algebra.

The lectures will present applications of these (co)homological structures to the study of commutative local rings and their modules.

In the sequel, R denotes a commutative noetherian ring with unique maximal ideal m and residue field k=R/m, while M, N stand for finite R-modules.

Lecture 1. Differential graded algebra resolutions

The category of commutative rings is naturally contained in the category of commutative DG (= differential graded) algebras. Basic constructions of such DG algebras will be described. It will be sketched how they are used to obtain homological characterizations of regular local rings and of local complete intersections.

Lecture 2. Homotopy Lie algebras

The algebra Ext^*_R(k,k) is naturally the universal enveloping algebra of a graded Lie algebra Pi^*(R), called the homotopy Lie algebra of R. Two constructions of this algebra will be sketched, and its name will be explained by describing its analogy with the rational homotopy Lie algebra of a topological space. Applications include the construction of modules for which the ranks of the free modules in their minimal resolution grow exponentially.

Lecture 3. Support varieties

Complete intersections are characterized by the existense of polynomial bounds on the growth of the ranks of free modules in all free resolutions. It will be shown that the rate of this growth is equal to the dimension of some algebraic variety over k, naturally associated to every R-module. These varieties exhibit behavior similar to that of varieties defined by representations of finite groups. Support varieties will be used to prove vanishing theorems for (co)homology over complete intersections.

Literature