Syzygies, multiplicities and birational algebra

Abstract

In Chapter 1, classical results of Northcott and Rees and of Levin are generalized to show that for a finitely generated module M over a Noetherian local ring R, a proper ideal I of R, and an non-negative integer i, the $i\sp{\rm th}$ Betti number and the $i\sp{\rm th}$ Bass number of $I\sp{n}M$ (or those of $M/I\sp{n}M)$ are given asymptotically by polynomial functions of n. As an application, a new proof is given of a result of Brodmann on the asymptotic stability of the depth of $M/I\sp{n}M$. Based on the results of Huneke and of Lipman, a proof of the Hoskin-Deligne length formula for complete ideals in a two-dimensional regular local ring is given in Chapter 2. Some of the techniques used in the proof generalize to regular local rings of higher dimension and yield a multiplicity formula for finitely supported ideals which has also been independently obtained by Johnston. Chapter 3 initiates a theory of complete modules over two-dimensional regular local rings in analogy with the theory of Zariski for complete ideals. This uses the work of Rees on reductions and integral closures of modules. The main result generalizes Zariski\u27s product theorem for complete ideals to show that the torsion-free tensor product of complete modules is complete. A construction of complete indecomposable modules of arbitrary rank is also given. The proofs are algebraic and depend on some properties of the Buchsbaum-Rim multiplicity of modules