arxiv: 2604.00266 · v1 · submitted 2026-03-31 · 🧮 math.AC

Recognition: unknown

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

Abuzer G\"und\"uz, Efe G\"urel

Pith reviewed 2026-05-08 02:20 UTC · model gemini-3-flash-preview

classification 🧮 math.AC MSC 13H1013A1514H20

keywords bi-amalgamated algebraCohen-Macaulay ringGorenstein ringalgebroid curvesfiber productcanonical module

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The pith

Bi-amalgamated algebras provide a systematic way to construct and characterize Gorenstein algebroid curves

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes how specific structural properties, such as being Cohen-Macaulay or Gorenstein, are preserved or created when merging two ring extensions into a single bi-amalgamated algebra. By treating these complex constructions as fiber products of simpler rings, the authors provide formulas to calculate their dimension and depth. This framework allows researchers to design algebraic curves with specific types of singularities by choosing the right combination of base rings and ideals.

Core claim

The authors define the bi-amalgamated algebra as a subring of a product of two rings and prove that its homological properties are determined by the relationship between the constituent ideals. They establish that the algebra is Gorenstein if and only if a specific isomorphism exists between the canonical modules of its two components. This result is then applied to curve theory, showing that bi-amalgamations can generate new classes of Gorenstein algebroid curves from simpler geometric data.

What carries the argument

The bi-amalgamated algebra $A \bowtie^{f,g} (J,J')$, a ring construction that merges a base ring $A$ with two other rings through specific maps and ideals. It functions as a generalized fiber product that allows for the transfer of homological properties from the components to the combined structure.

If this is right

Complex curve singularities can be decomposed into simpler algebraic components for easier analysis.
The Gorenstein property of a combined algebra can be verified by checking the alignment of canonical modules in its constituent parts.
New models of Cohen-Macaulay rings can be generated by selecting appropriate ideals $J$ and $J'$ in a local ring.
The calculation of ring depth and dimension for these structures is reduced to a set of verifiable conditions on the mapping homomorphisms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This construction could potentially be used to model the 'gluing' of two different geometric varieties along a shared sub-structure while controlling the resulting singularity type.
The methods used here for curves might extend to higher-dimensional varieties, providing a tool for constructing Gorenstein varieties in arbitrary dimensions.

Load-bearing premise

The analysis assumes that the maps and ideals involved are compatible enough to allow the algebra to be treated as a standard fiber product of rings.

What would settle it

The discovery of a bi-amalgamated algebra that satisfies the canonical module isomorphism defined in Theorem 4.3 but fails to be Gorenstein.

Figures

Figures reproduced from arXiv: 2604.00266 by Abuzer G\"und\"uz, Efe G\"urel.

**Figure 1.** Figure 1: The value semigroup of the bi–amalgamated algebroid curve A ▷◁f,g (J, J′ ) view at source ↗

**Figure 2.** Figure 2: The value semigroup of the amalgamated algebroid curve A ▷◁f J. 5. A remark on the universally catenary property Bi–amalgamations naturally generalize some classic pull–back constructions of polynomial and power series rings. In the previous section, we have also seen an application of the bi–amalgamated algebra construction to curve singularities. In this section, we describe the bi–amalgamated algebra as… view at source ↗

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

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

A solid extension of amalgamated ring theory that provides clear characterizations for Cohen-Macaulay and Gorenstein properties using fiber products.

read the letter

This paper provides a complete homological characterization of bi-amalgamated algebras, a ring construction that generalizes several known types like the 'amalgamated duplication' and the standard fiber product. The punchline is that the authors successfully pin down the Cohen-Macaulay and Gorenstein properties for these rings, which are notoriously tricky to characterize in general but become tractable here thanks to a well-chosen structural assumption.

What the paper does well is leverage the fiber product structure. By assuming 'Condition C'—where the pre-images of the target ideals $J$ and $J'$ coincide in the base ring $A$—the authors are able to treat the bi-amalgamation as a pullback. This allows them to use standard homological tools to derive the depth and dimension (Theorems 3.2 and 3.6). Theorem 4.3 is particularly useful; it gives a necessary and sufficient condition for the Gorenstein property by looking at the canonical modules of the constituent components. The authors also deserve credit for Section 5, where they apply these results to algebroid curves. This moves the paper from abstract algebra into more concrete singularity theory, showing exactly how these constructions can be used to build Gorenstein curve singularities.

The main soft spot is the reliance on Condition C. As noted in the stress test, if the maps and ideals don't align perfectly, the fiber product isomorphism fails, and the proofs in this paper don't carry through. The authors are honest about this requirement, and it is the standard way to handle these constructions, but it does mean the results apply only to a specific 'balanced' class of bi-amalgamations. Additionally, the final remarks on catenary properties are quite thin and don't add much beyond what one would expect from existing fiber product theory.

This paper is for specialists in commutative algebra and those working on the classification of singularities. The math is sound, the citations are appropriate, and the results fill a clear gap in the literature on ring constructions. It definitely deserves a serious referee.

I recommend sending this for peer review.

Referee Report

2 major / 3 minor

Summary. This manuscript investigates the homological properties of bi-amalgamated algebras, denoted as $A \bowtie^{f,g} (J, J')$, which generalize the amalgamated algebra construction. The authors establish that under 'Condition C' ($f^{-1}(J) = g^{-1}(J') = I$), the bi-amalgamated algebra is isomorphic to a fiber product of rings. Leveraging this structure, the paper provides explicit formulas for the Krull dimension and depth of these algebras. Furthermore, it characterizes the Cohen-Macaulay (CM) and Gorenstein properties in terms of the properties of the constituent rings and their canonical modules. The final section applies these results to the theory of algebroid curves, providing a construction for Gorenstein singularities.

Significance. The paper provides a rigorous and useful extension of the amalgamated algebra framework, which has been a productive area of commutative algebra over the last two decades. By focusing on the bi-amalgamated case, the authors offer more flexibility in constructing rings with prescribed homological properties. The explicit characterization of the Gorenstein property (Theorem 4.3) is particularly significant as it offers a concrete path to generating Gorenstein rings via fiber products, a task that is often non-trivial. The application to algebroid curves demonstrates that the construction is not merely a formal exercise but has genuine utility in singularity theory. The derivations are sound, building on established fiber product theory (e.g., D'Anna, 2006) with clear logical progression.

major comments (2)

[§3.2, Theorem 3.2 & §3.6, Theorem 3.6] The characterization of the Cohen-Macaulay property for the bi-amalgamated algebra $R \times_T S$ (where $T = A/I$) relies on the assumption that the depth of the fiber product is determined by the components. In general, $\text{depth}(R \times_T S) = \min\{\text{depth}(R), \text{depth}(S), \text{depth}(T)+1\}$. Theorem 3.6 identifies conditions for CM-ness, but it assumes $\dim(A/I) = \dim(A \bowtie) - 1$ or $A/I=0$. The manuscript should more explicitly justify why the case $\dim(T) < \dim(R)-1$ does not need to be considered or how the depth of $T$ is controlled in the general definition of Condition C.
[§4.3, Theorem 4.3] The Gorenstein characterization using the canonical module $\omega_{A \bowtie}$ is the central result of the paper. However, for a fiber product $R \times_T S$ to be Gorenstein, the structure of the maps $\pi_1: R \to T$ and $\pi_2: S \to T$ is critical. Specifically, if $T \neq 0$, the Gorenstein property often forces $T$ to be a Gorenstein ring of a specific dimension and the maps to satisfy certain lifting properties. The authors should clarify if Theorem 4.3 implicitly requires $T$ to be a field (as in the curve applications) or if it holds for any $A/I$ satisfying Condition C.

minor comments (3)

[Section 2, Proposition 2.2] In the proof that $A \bowtie$ is a local ring, it would be helpful to explicitly state the form of the unique maximal ideal in terms of the maximal ideals of $A$, $B$, and $C$ to assist the reader in verifying the locality condition.
[§5.1, Example 5.1] The application to algebroid curves is well-motivated. It would be beneficial to add a brief remark on whether this construction can produce non-complete intersection Gorenstein curves, as this would further highlight the significance of the bi-amalgamation method over simpler constructions.
[General] There are minor notation inconsistencies regarding the ideal $I = f^{-1}(J) = g^{-1}(J')$. In some instances, it is treated as an ideal of $A$, and in others, its image in $f(A)$ is used. Explicitly stating the maps in the fiber product diagram in Section 2 would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment of our work and for their insightful comments regarding the Cohen-Macaulay and Gorenstein properties. We have addressed both major comments by providing further justification for our dimensional assumptions and clarifying the structural requirements on the base ring T in the fiber product construction.

read point-by-point responses

Referee: [§3.2, Theorem 3.2 & §3.6, Theorem 3.6] The characterization of the Cohen-Macaulay property for the bi-amalgamated algebra $R \times_T S$ (where $T = A/I$) relies on the assumption that the depth of the fiber product is determined by the components. In general, $\text{depth}(R \times_T S) = \min\{\text{depth}(R), \text{depth}(S), \text{depth}(T)+1\}$. Theorem 3.6 identifies conditions for CM-ness, but it assumes $\dim(A/I) = \dim(A \bowtie) - 1$ or $A/I=0$. The manuscript should more explicitly justify why the case $\dim(T) < \dim(R)-1$ does not need to be considered or how the depth of $T$ is controlled in the general definition of Condition C.

Authors: The referee is correct that the depth of the fiber product is bounded by $\text{depth}(T)+1$. Our focus on the case $\dim(T) = d-1$ (where $d = \dim(A \bowtie)$) is motivated by the fact that if $\dim(T) < d-1$, then $\text{depth}(T) + 1 \le \dim(T) + 1 < d$. Consequently, in such cases, the bi-amalgamated algebra $A \bowtie$ can never be Cohen-Macaulay, as its depth would be strictly less than its dimension. Thus, Theorem 3.6 covers all cases where the Cohen-Macaulay property is actually attainable. We have added a remark after Theorem 3.6 to explicitly state this justification and clarify the behavior of the depth when $\dim(T) < d-1$. revision: yes
Referee: [§4.3, Theorem 4.3] The Gorenstein characterization using the canonical module $\omega_{A \bowtie}$ is the central result of the paper. However, for a fiber product $R \times_T S$ to be Gorenstein, the structure of the maps $\pi_1: R \to T$ and $\pi_2: S \to T$ is critical. Specifically, if $T \neq 0$, the Gorenstein property often forces $T$ to be a Gorenstein ring of a specific dimension and the maps to satisfy certain lifting properties. The authors should clarify if Theorem 4.3 implicitly requires $T$ to be a field (as in the curve applications) or if it holds for any $A/I$ satisfying Condition C.

Authors: Theorem 4.3 does not strictly require $T$ to be a field; it holds for any $A/I$ of dimension $d-1$ where the canonical module of the fiber product can be described as a fiber product of canonical modules (using the duality theory for fiber products established by D'Anna). However, the referee is correct that the resulting condition $A \bowtie \cong \omega_{A \bowtie}$ imposes very strong constraints on $T$ and the maps $\pi_i$ when $T$ is not a field. In the algebroid curve application in Section 5, $T$ is indeed the residue field, which simplifies these conditions significantly. We have added a remark following Theorem 4.3 to clarify these structural implications and to state that while the theorem is general, the Gorenstein property in fiber products is most naturally realized when $T$ is a Gorenstein ring (often a field). revision: yes

Circularity Check

0 steps flagged

No circularity identified

full rationale

The paper provides a rigorous mathematical derivation of the homological properties of bi-amalgamated algebras. The authors identify a structural condition ('Condition C') under which the bi-amalgamated algebra is isomorphic to a fiber product of rings (Lemma 2.2). Based on this isomorphism, they apply established results from commutative algebra and the theory of fiber products (e.g., D'Anna 2006, Kabbaj et al. 2014) to characterize the Cohen-Macaulay and Gorenstein properties. The logic is linear and deductive: the construction is defined, its relationship to known structures is proven, and then the properties of those known structures are specialized to the case of bi-amalgamations. There are no instances where a result is simply a renaming of an input or where a conclusion is forced by a self-citation to an unverified theorem. The characterization of the Gorenstein property in Theorem 4.3 involves non-trivial conditions on the canonical modules of the constituent rings, which are derived, not assumed. The application to algebroid curves in Section 5 serves as a valid demonstration of the theory's utility rather than a circular justification.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper operates entirely within the standard framework of commutative algebra and ring theory, using no ad-hoc parameters or unverified new physical/mathematical entities.

axioms (2)

standard math The dimension of a local ring is equal to the length of its longest chain of prime ideals (Krull dimension).
Used in Theorem 3.2 to calculate the dimension of the bi-amalgamation.
standard math Depth of a module is defined by the length of the longest regular sequence.
Crucial for characterizing the Cohen-Macaulay property in Section 3.

pith-pipeline@v0.9.0 · 6277 in / 1658 out tokens · 24094 ms · 2026-05-08T02:20:00.656166+00:00 · methodology

discussion (0)

Reference graph

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