Midwest Algebra, Geometry and their Interactions Conference MAGIC05 University of Notre Dame, Notre Dame October 7-11, 2005 Contracted ideals by Maria Evelina ROSSI, Università di Genova Abstract: The class of contracted ideals in a local regular ring of dimension 2 plays an important role in the original work of Zarisky, more recent contributions are due to several authors including Cutkosky, Huneke, Lipman, Rees, Sally, Tessier. Any complete ideal is contracted and its reduction number is 1, which in turns implies that the associated graded ring, the Rees algebra and the Fiber Cone are Cohen-Macaulay and their structure is well understood. These results do not hold in general for contracted ideals or for complete ideals in higher dimension. The goal of this talk is to present results concerning these problems obtained in collaboration with A. Conca, E. De Negri and A.V. Jayanthan. We study the depth, the Hilbert function and the defining equations of various graded algebras associated with an m-primary contracted ideal of a local regular ring (R,m) of dimension 2. In the homogeneous case, we give several equivalent conditions for an ideal to be contracted, in particular to be componentwise linear or m-full. This interpretation gives new ideas in order to approach the study of classes of complete ideals in higher dimension.