Cohen-Macaulay and Gorenstein Properties of Bi-Amalgamated Algebras with Applications to Algebroid Curves

Abstract

Let $A \bowtie^{f,g} (J,J')$ be the bi--amalgamation of a commutative ring $A$ with $(B,C)$ along the ideals $(J,J')$ with respect to the ring homomorphisms $(f,g)$. In this article, we study the basic homological properties of the bi--amalgamated algebra construction. We first calculate the dimension and depth of the bi--amalgamated algebra under fairly general circumstances and derive necessary and sufficient conditions for Cohen--Macaulayness in terms of maximal and big Cohen--Macaulay modules of $A$. Furthermore, we characterize the Gorenstein property of the bi--amalgamated algebra through the canonical modules of $f(A)+J$ and $g(A)+J'$. We apply our results to the theory of curve singularities by constructing Gorenstein algebroid curves through bi--amalgamated and amalgamated algebras. We also give a brief remark concerning the universally catenary property of $A\bowtie^{f,g}(J,J')$.

Figures (2)

• Figure 1: The value semigroup of the bi--amalgamated algebroid curve $A \bowtie^{f,g} (J, J')$.
• Figure 2: The value semigroup of the amalgamated algebroid curve $A \bowtie^f J$.

Theorems & Definitions (51)

• Proposition 1.1
• Lemma 2.1
• proof
• Corollary 2.1
• proof
• Lemma 2.2
• proof
• Corollary 2.2
• Theorem 2.1
• proof