Linkage and Hilbert functions

Paolo Mantero, Purdue University

Abstract

This work mainly deals with two long-standing open questions. The first one, from linkage theory, is fundamental in the sense that it asks whether one can identify distinguished ideals in any even linkage class (such elements are called 'minimal'). The second question was raised by Vasconcelos. Roughly speaking, the question asks whether it is true, under some assumptions to avoid trivial counterexamples, that the Macaulayness of an R-ideal I and its square I2 imply that R/I is Gorenstein. As for the first question, we introduce (and motivate) definitions of minimality for Cohen-Macaulay ideals of arbitrary height in any Gorenstein local ring. Minimal ideals in the even linkage class of I play a role similar to the one of complete intersection ideals in the case of licci ideals. We show that minimal ideals have the best properties among all ideals in the even linkage class. Under special assumptions, we prove that minimal elements exist and are 'unique from a homological point of view'. We also exhibit classes of ideals that are minimal. In the main result concerning the second question, we prove that the question of Vasconcelos has a positive answer for special classes of ideals. However, we exhibit examples proving that, in general, the question has a negative answer. The examples also prove the sharpness of our main result. We now explain in more detail the structure of this work. Chapter 1 is an expository introduction. In Chapter 2 we recall preliminaries of Commutative Algebra and we introduce linkage via complete intersections. While this material is usually presented in the local setting, we chose not to restrict to the local case. In Chapter 3 we review the basics of generic and universal links, tools that have been introduced by Huneke and Ulrich to study algebraic linkage. Also in this chapter we work in rings that need not be local. In Chapter 4 we introduce the notions of faithful equivalence and strong faithful equivalence, we exploit their own properties and the properties that these equivalences preserve. In Chapter 5 we use these notions to define two concepts of minimality for Cohen-Macaulay ideals in a Gorenstein local ring. We also recall a small bit of deformation theory and use a notion of rigidity over Noetherian local rings that ensures the existence and uniqueness (up to faithful equivalence) of minimal elements in several situations. We exhibit here several classes of examples and situations where we are able to apply our results. Furthermore, we bound the number of steps needed to link special m-primary ideals to a minimal element of the even linkage class. In Chapter 6 we introduce a much weaker concept of faithful equivalence ('weak faithful equivalence'), and we show that this notion is still able to capture interesting homological properties. In Chapter 7 we employ it to give yet another definition of minimality for ideals in an even linkage class. We are able to prove the existence of these weakly minimal elements in a greater variety of situations. Furthermore, we are able to prove that several special classes of ideals are weak minimal in the even linkage class. In Chapter 8 we study Vasconcelos' question. We remark here that this chapter is based on joint work with Y. Xie. We show that one can reduce the original question to the case of non-degenerate prime ideals. We then study special classes of ideals for which we can prove that Vasconcelos' question has a positive answer. In contrast, we also exhibit a computer-generated example, found with the help of J. C. Migliore, for which the question has a negative answer. This example proves also the sharpness of our main result. Finally, we conjecture that similar 'negative' examples exist in more generality.

Degree

Ph.D.

Advisors

Ulrich, Purdue University.

Subject Area

Applied Mathematics|Mathematics

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