[Paper Review] Algebraic and topological properties of an amalgamated algebra along an ideal

Marco D’Anna, Carmelo Antonio Finocchiaro+1 more|arXiv (Cornell University)|Dec 13, 2013

Rings, Modules, and Algebras6 references3 citations

TL;DR

This paper introduces and studies the amalgamated algebra $A\!\Join^{f}\!J$, a generalization of ring constructions like idealization and $D+M$ rings. It fully describes the prime spectrum of $A\!\Join^{f}\!J$, characterizes when it is local, and provides exact formulas for its embedding dimension, with conditions for Cohen-Macaulay and Gorenstein properties when the ring is local Noetherian.

ABSTRACT

Let $f:A ightarrow B$ be a ring homomorphism and let $J$ be an ideal of $B$. In this paper, we study the amalgamation of $A$ with $B$ along $J$ with respect to $f$, a construction that provides a general frame for studying the amalgamated duplication of a ring along an ideal, introduced by D'Anna and Fontana in 2007, and other classical constructions (such as the $A+ XB[X]$, the $A+ XB[\![X]\!]$ and the $D+M$ constructions). In particular, we completely describe the prime spectrum of the amalgamation and, when it is a local Noetherian ring, we study its embedding dimension and when it turns to be a Cohen-Macaulay ring or a Gorenstein ring.

Motivation & Objective

• To provide a comprehensive algebraic and topological analysis of the amalgamated algebra $A\!\Join^{f}\!J$, a generalization of classical ring constructions such as idealization and $D+M$ rings.
• To fully describe the prime spectrum of $A\!\Join^{f}\!J$ and characterize when it is a local ring.
• To determine the embedding dimension of $A\!\Join^{f}\!J$ when it is a local Noetherian ring, under finiteness conditions on $J$ as an ideal of $f(A)+J$.
• To establish conditions under which $A\!\Join^{f}\!J$ is Cohen-Macaulay or Gorenstein, particularly in the local Noetherian case.
• To correct and refine earlier results by explicitly identifying the missing assumption $B = f(A) + J$ and providing exact embedding dimension formulas under weaker hypotheses.

Proposed method

• Use of pullback constructions to analyze $A\!\Join^{f}\!J$ as a subring of $A \times B$, leveraging known results from commutative algebra.
• Characterization of prime ideals in $A\!\Join^{f}\!J$ via the natural projections $p_A$ and $p_B$, and the kernel structure of these maps.
• Application of the canonical homomorphism $\gamma: A\!\Join^{f}\!J \to (f(A)+J)/J$ to relate the spectrum of $A\!\Join^{f}\!J$ to that of $f(A)+J$.
• Use of Nakayama’s Lemma and module-theoretic arguments to analyze minimal generating sets of the maximal ideal in $A\!\Join^{f}\!J$.
• Explicit computation of the embedding dimension via the minimal generating set of the maximal ideal modulo its square, using the decomposition $\mathcal{G} = \{(m_i, f(m_i)), (0, j_h)\}$.
• Analysis of the Cohen-Macaulay and Gorenstein properties through the study of the canonical module and multiplicity in the local case.

Experimental results

Research questions

• RQ1 What is the complete structure of the prime spectrum of $A\!\Join^{f}\!J$?
• RQ2 Under what conditions is $A\!\Join^{f}\!J$ a local ring?
• RQ3 What is the exact value of the embedding dimension of $A\!\Join^{f}\!J$ when it is a local Noetherian ring, especially when $J$ is finitely generated as an ideal of $f(A)+J$?
• RQ4 When is $A\!\Join^{f}\!J$ a Cohen-Macaulay or Gorenstein ring?
• RQ5 Can the embedding dimension formula be corrected and generalized beyond the assumption $B = f(A) + J$?

Key findings

• The prime spectrum of $A\!\Join^{f}\!J$ is completely described: $\operatorname{Spec}(A\!\Join^{f}\!J)$ is in bijection with the disjoint union of $\operatorname{Spec}(A)$ and $\operatorname{Spec}(f(A)+J)\setminus \operatorname{Spec}(f(A)+J)$, with precise conditions on the fibers.
• The ring $A\!\Join^{f}\!J$ is local if and only if $A$ is local and $f(A)+J$ is local, with the maximal ideal of $A\!\Join^{f}\!J$ being $M^\prime_f = \iota(M) + \{0\} \times J$.
• When $A$ is local with finitely generated maximal ideal $M$ and $J$ is finitely generated as an ideal of $f(A)+J$, the embedding dimension of $A\!\Join^{f}\!J$ is $\operatorname{embdim}(A) + \nu(J)$, where $\nu(J)$ is the minimal number of generators of $J$ in $f(A)+J$.
• The embedding dimension formula is valid even when $B \neq f(A)+J$, provided $J$ is finitely generated as an ideal of $f(A)+J$, correcting a gap in earlier work.
• If $A\!\Join^{f}\!J$ is Cohen-Macaulay, its multiplicity is given by $e(A\!\Join^{f}\!J) = e(A) + e(f(A)+J)$, under suitable conditions.
• The ring $A\!\Join^{f}\!J$ is Cohen-Macaulay (resp. Gorenstein) if and only if $A$ is Cohen-Macaulay (resp. Gorenstein) and $f(A)+J$ is Cohen-Macaulay (resp. Gorenstein), under the assumption that $A$ is local and $J$ is finitely generated over $f(A)+J$.