Jacobian ideals, resolutions, and the relation types of parameters
Abstract
Given a d-dimensional complete noetherian local ring R of equicharacteristic and a finitely generated R-module M, C. Huneke has conjectured a relation between the Jacobian ideal of R and the Fitting ideals in an arbitrary free resolution of M. In Chapter 1, we show that the conjecture holds if R is a Cohen-Macaulay ring of characteristic 0. By using the work of Seheja and Storch on universal finite differential modules, we obtain a similar result for non-Cohen-Macaulay rings. Moreover, we generate some results of Dieterich, Popescu and Roczen about annihilator ideal of the functor Ext, namely, $J\sp{k}$Ext$\sbsp{R}{d+1}$(,) = 0 for some $k > 0,$ where J is the Jacobian ideal of R. In case R is a Cohen-Macaulay ring, k can be chosen to be 1. In Chapter 2, we study two invariants "relation type" and "postulation number" associated with ideals generated by systems of parameters over noetherian local rings. The main result is that for some 2-dimensional rings, both the relation type and the postulation number of parameter ideals are uniformly bounded. The proof also shows that if dim R = 2 and depth R = 1 then the relation type is exactly 2 more than the postulation number for every parameter ideal.
Degree
Ph.D.
Advisors
Huneke, Purdue University.
Subject Area
Mathematics
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