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

TL;DR

This paper investigates the homological properties of bi-amalgamated algebras, providing criteria for Cohen-Macaulayness and Gorenstein conditions, with applications to constructing Gorenstein algebroid curves.

Contribution

It offers new characterizations of Cohen-Macaulay and Gorenstein properties for bi-amalgamated algebras, linking them to modules and canonical modules, and applies these to curve singularities.

Findings

Calculated dimension and depth of bi-amalgamated algebras.

Derived necessary and sufficient conditions for Cohen-Macaulayness.

Characterized Gorenstein property via canonical modules.

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 $A\bowtie^{f,g}(J,J')$.