In these lectures, we shall discuss some module-theoretics aspects of local algebra and their relations to neighboring disciplines. Traditionally, a great deal of research has been focused on Cohen - Macaulay local rings. It has been long recognized that, over such rings, Cohen - Macaulay modules play a special role. As two extremes, this class includes maximal Cohen - Macaulay (mCM) modules and modules of finite length. Especially amenable to homological methods is the case when the local ring is Gorenstein, i.e., of finite injective dimension over itself. This finiteness condition allows to realize mCM's as high enough syzygy modules, and thus introduces the first major theme to be discussed in the lectures: the relationships between a module and the "tail" of its projective resolution.
One such relationship is reflected by the concepts of (maximal) Cohen - Macaulay approximation and of hulls of finite injective dimensions, introduced by Auslander and Buchweitz in the mid-80's. In the first lecture, we shall discuss various ways of constructing mCM approximations and hulls. This includes the pitchfork construction of Auslander - Buchweitz and the gluing construction, appearing in joint work with Herzog. The latter is, in fact, an algebraic reformulation of a simple-minded homotopy-theoretic construct. Minimal MCM approximations are unique and thus give rise to various invariants of modules. One of them, the delta-invariant of Auslander seems to be of particular importance. The vanishing of that invariant has been the focus of research in the last decade, and we shall discuss some recent developments.
We then intend to show how mCM approximations can be used to study isolated singularities. This direction targets criteria for the quasihomogeneity of isolated singularities (over a field of characteristic zero). The distinct feature of this class of algebras is the presence of a certain module of finite length. This module is the transpose of the module of Kaehler differentials. In the hypersurface case, this module becomes a finite- dimensional algebra and is called the moduli algebra (or transgradient algebra) of the singularity. The famous criterion of K. Saito asserts that an isolated singularity is quasihomogeneous if and only if the defining equation belongs to its own jacobian ideal (this is the ideal generated by the partial derivatives). This condition is identical to one of the criteria for the vanishing of the delta-invariant of the moduli algebra (as a module over the singularity).
Another module-theoretic approach to the problem of quasi-homogeneity, at least in dimension two, is to compare the module of Zariski differentials with the mCM approximation of the maximal ideal. The latter is called the Auslander module of the singularity. In the case when the singularity is a rational double point, and therefore is of type A, D, or E, the indecomposable non-projectives are described by the simple roots of the corresponding Dynkin diagram. The unique projective can be thought of as the maximal root, and the Auslander module corresponds to the simple root(s) adjacent to the maximal root in the extended Dynkin diagram. A conjecture of the speaker asserts that a two-dimensional singularity is quasihomogeneous if and only if the module of Zariski differentials and the Auslander module are isomorphic. The "only if" part has been established, but the other direction is wide open. There are reasons to believe that this conjecture is the "next" step after the Zariski - Lipman conjecture.
We shall also discuss how the above approach generalizes to higher dimensions. Two modules of finite length come to the forefront: the residue field and the transpose of Kaehler differentials. Another conjecture of the speaker asserts that the singularity is quasihomogeneous if and only if the tails of the projective resolutions of the two modules are the same up to a shift. The gluing construction mentioned above allows to prove the "only if" part for complete intersections. A heuristic explanation of this phenomenon is nothing more complicated than the Euler identity for quasihomogeneous polynomials.
Returning to the delta-invariant, it is of interest to define it not just for Gorenstein rings but for arbitrary local rings, or even for more general algebraic structures. In the second lecture, we shall show how this can be done. The definition is based on a remarkable construction of Pierre Vogel that greatly generalizes and simplifies the definition of Tate cohomology. The new invariant, called the ksi-invariant, allows to generalize many results originally proved for Gorenstein rings. Sometimes the proofs are identical to the old ones, sometimes modifications are needed, and sometimes completely different techniques are required. We show how this new invariant can be used in local algebra to study the syzygy modules of the residue field. More precisely, we establish a new regularity criterion for local rings: a local ring is non-regular if and only if the ksi-invariant of each of the syzygy module of the residue field is zero. Besides providing non-trivial new information, this result links some seemingly unrelated results established earlier. The proof, including the definition of the ksi-invariant, will be done without any reference to high-powered machinery. It is accessible to any student who has had or is having a first course in homological algebra.
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Last modified: 08-06-1998