# Rees algebras and fiber cones of modules

In the first part of this thesis, we study Rees algebras of modules. We investigate their Cohen-Macaulay property and their defining ideal, using

*generic Bourbaki ideals*. These were introduced by Simis, Ulrich and Vasconcelos in [65], in order to characterize the Cohen-Macaulayness of Rees algebras of modules. Thanks to this technique, the problem is reduced to the case of Rees algebras of ideals. Our main results are the following.In Chapters 3 and 4 we consider a finite module

*E*over a Gorenstein local ring*R*. In Theorem 3.2.4 and Theorem 4.3.2, we give sufficient conditions for*E*to be of linear type, while Theorem 4.2.4 provides a sufficient condition for the Rees algebra*R(E)*of*E*to be Cohen-Macaulay. These results rely on properties of the residual intersections of a generic Bourbaki ideal*I*of*E*, and generalize previous work of Lin (see [46, 3.1 and 3.4]). In the case when*E*is an ideal, Theorem 4.2.4 had been previously proved independently by Johnson and Ulrich (see [39, 3.1]) and Goto, Nakamura and Nishida (see [20, 1.1 and 6.3]).In Chapter 5, we consider a finite module

*E*of projective dimension one over*k*[X_{1}, . . . , X_{n}]. Our main result, Theorem 5.2.6, describes the defining ideal of*R(E)*, under the assumption that the presentation matrix φ of*E*is*almost linear*, i.e. the entries of all but one column of φ are linear. This theorem extends to modules a known result of Boswell and Mukundan on the Rees algebra of almost linearly presented perfect ideals of height 2 (see [5, 5.3 and 5.7]).The second part of this thesis studies the Cohen-Macaulay property of the special fiber ring

*F(E)*of a module*E*. In Theorem 6.2.14, we prove that the generic Bourbaki ideals of Simis, Ulrich and Vasconcelos allow to reduce the problem to the case of fiber cones of ideals, similarly as for Rees algebras. We then provide sufficient conditions for*F(E)*to be Cohen-Macaulay. Our Theorems 6.2.15, 6.1.3 and 6.2.18 are module versions of results proved for the fiber cone of an ideal by Corso, Ghezzi, Polini and Ulrich (see [10, 3.1] and [10, 3.4]) and by Monta˜no (see [47, 4.8]), respectively.