Thesis - Tan Dang - Final.pdf (307.7 kB)

Download file# Topics on the Cohen-Macaulay Property of Rees algebras and the Gorenstein linkage class of a complete intersection

We study the Cohen-Macaulay property of Rees algebras of modules of Kähler differentials. When the module of differentials has projective dimension one, it is known that condition $F_1$ is sufficient for the Rees algebra to be Cohen-Macaulay. The converse was proved if the module of differentials is already $F_0$. We weaken the condition $F_0$ globally by assuming some homogeneity condition.

We are also interested in the defining ideal of the Rees algebra of a Jacobian module. If the Jacobian module is an ideal, we prove a formula for computing the defining ideal. Using the formula, we give an explicit description of the defining ideal in the monomial case. From there, we characterize the Cohen-Macaulay property of the Rees algebra.

In the last chapter, we study Gorenstein linkage mostly in the graded case. In particular, we give an explicit example of a class of monomial ideals that are in the homogeneous Gorenstein linkage class of a complete intersection. To do so, we prove a Gorenstein double linkage construction that is analogous to Gorenstein biliaison.

We are also interested in the defining ideal of the Rees algebra of a Jacobian module. If the Jacobian module is an ideal, we prove a formula for computing the defining ideal. Using the formula, we give an explicit description of the defining ideal in the monomial case. From there, we characterize the Cohen-Macaulay property of the Rees algebra.

In the last chapter, we study Gorenstein linkage mostly in the graded case. In particular, we give an explicit example of a class of monomial ideals that are in the homogeneous Gorenstein linkage class of a complete intersection. To do so, we prove a Gorenstein double linkage construction that is analogous to Gorenstein biliaison.