Canonical Trace Ideal in Commutative Rings

• Canonical trace ideal is defined as the sum of images from homomorphisms of the canonical module, measuring a ring's deviation from being Gorenstein.
• It detects non-Gorenstein loci and classifies rings as nearly or far-flung Gorenstein via comparisons with maximal and conductor ideals.
• This invariant enables explicit computations in numerical semigroup, determinantal, and fiber product rings, linking structural, homological, and combinatorial properties.

The canonical trace ideal plays a critical role in the structure theory of commutative Noetherian rings, especially in the context of Cohen–Macaulay modules and the analysis of how far a given ring is from being Gorenstein. It is a fundamental invariant that encodes fine-grained homological and duality properties, connects to several classes of rings close to Gorenstein, and is closely related to module-theoretic, homological, and combinatorial classifications.

1. Definition of the Canonical Trace Ideal

Let $R$ be a commutative Noetherian ring (local or graded) admitting a canonical module $\omega_R$. The trace ideal of a finitely generated $R$-module $M$ is defined as

$$\operatorname{tr}_R(M) := \sum_{f \in \operatorname{Hom}_R(M, R)} f(M) \subseteq R,$$

the smallest ideal through which every $R$-module homomorphism $M \to R$ factors. For the canonical module, the canonical trace ideal is

$$\operatorname{tr}_R(\omega_R) := \sum_{f \in \operatorname{Hom}_R(\omega_R, R)} f(\omega_R) \subseteq R.$$

If $\omega_R$ can be realized as a fractional ideal of $R$ (automatic if $R$ is generically Gorenstein), then

$$\operatorname{tr}_R(\omega_R) = \omega_R \cdot \omega_R^{-1},$$

where $$\omega_R^{-1} = \{ x \in Q(R) \mid x \omega_R \subseteq R \}$$ (Herzog et al., 2016, Herzog et al., 2021, Miyashita et al., 7 Dec 2025).

The definition generalizes via the following evaluation map: $$\theta: \omega_R^* \otimes_R \omega_R \to R, \quad f \otimes x \mapsto f(x),$$ with image $$\operatorname{tr}_R(\omega_R) = \operatorname{Im}(\theta)$$ (Lyle et al., 7 Sep 2024).

2. Key Properties and Structural Role

The canonical trace ideal is intimately connected to the singularity and Gorenstein properties of the ring.

• Detection of Non-Gorenstein Locus: For local $R$ with maximal ideal $\mathfrak{m}$, $R$ is Gorenstein if and only if $\operatorname{tr}_R(\omega_R) = R$. For each prime $P \in \operatorname{Spec} R$, $R_P$ is Gorenstein if and only if $\operatorname{tr}_R(\omega_R) \nsubseteq P$ (Herzog et al., 2016, Miyashita et al., 7 Dec 2025).
• Nearly Gorenstein Rings: $R$ is called nearly Gorenstein if $$\mathfrak{m} \subseteq \operatorname{tr}_R(\omega_R)$$, i.e., the trace contains the maximal ideal (Herzog et al., 2016, Ficarra, 11 Jun 2024).
• Behavior under Standard Constructions:
  • If $A$, $B$ are standard graded $k$-algebras, then

  $$\operatorname{tr}_{A \times_k B}(\omega_{A \times_k B}) = \operatorname{tr}_A(\omega_A) \oplus \operatorname{tr}_B(\omega_B)$$

  (with modifications if one factor is Gorenstein) (Kumashiro et al., 5 Jun 2025). - For Veronese subrings and Segre products, those constructions preserve nearly Gorenstein properties only under specific conditions (Herzog et al., 2016, Miyashita et al., 7 Dec 2025).

3. Explicit Computations and Invariants

One-Dimensional Local and Numerical Semigroup Rings

For one-dimensional Cohen–Macaulay local domains, special invariants and classification results arise:

• Partial Trace Ideals and the Invariant $h(M)$: For finitely generated $M$ of rank 1, $$h(M) = \min \{ \ell(R/J) \mid \exists M \twoheadrightarrow J \subseteq R \}$$ governs the minimal length of $R/J$ through surjective maps and detects the proximity to Gorenstein (Maitra, 2022).
• Bounds: When specializing to $M = \omega_R$,

  $h(\omega_R) = 0 \iff R \text{ is Gorenstein};$

  Lower and upper bounds in terms of the conductor $\mathfrak{C}$, multiplicity, and integral closure are established, with sharpness in the almost Gorenstein case (Maitra, 2022).

A concrete example is given by $R = k[[t^5, t^6, t^8]]$, where $\operatorname{tr}_R(\omega_R)$ equals the conductor ideal, but $R$ is not almost Gorenstein and $h(\omega_R)$ is strictly less than the lower bound (Maitra, 2022).

Numerical Semigroup Rings

• Canonical Trace and Conductor: In $K[H] = K[t^{n_1}, ..., t^{n_e}]$,

  $$\operatorname{tr}(\omega_{K[H]}) = \omega_{K[H]} \omega_{K[H]}^{-1} \supseteq \text{conductor}$$

  with explicit formulas for the case $e = 3$ in terms of a structure matrix (Herzog et al., 2020, Herzog et al., 2021).

• Residue Invariant: The colength

  $$\mathrm{res}(H) = \dim_K K[H] / \operatorname{tr}(\omega_{K[H]})$$

  quantifies the failure to be Gorenstein; nearly Gorenstein rings correspond to $\mathrm{res}(H) \le 1$ (Herzog et al., 2020).

• Far-Flung Gorenstein Rings: The case $\operatorname{tr}_R(\omega_R)$ equals the conductor ideal $R : R$ is termed "far-flung Gorenstein." Such rings are classified in terms of combinatorics of pseudo-Frobenius numbers and satisfy extremal properties regarding multiplicity and type (Herzog et al., 2021).

Codimension Two and Determinantal Rings

• For $R = S/I$, with $I$ perfect of height two and $R$ generically Gorenstein, the canonical trace is generated by the $(\mu(I) - 2)$-minors of a Hilbert–Burch matrix $A$:

$$\operatorname{tr}_R(\omega_R) = I_{\mu(I)-2}(A) \cdot R$$

(Ficarra et al., 2022, Ficarra, 11 Jun 2024).

• This holds for determinantal rings and their specializations (Ficarra et al., 2022). Classification for nearly Gorenstein monomial ideals of height 2 is achieved via this approach (Ficarra, 11 Jun 2024).

4. Homological and Functorial Characterizations

The canonical trace ideal admits deep homological descriptions:

• Via Annihilators: Several equalities hold:

$$\operatorname{tr}_R(\omega_R) = \operatorname{ann}_R \operatorname{Ext}^i_R(\omega_R, \text{mod }R) = \operatorname{ann}_R \operatorname{Ext}^1_R(\omega_R, \Omega^d \omega_R), \ldots$$

for all $i > 0$, providing a tight link to the ring's Ext modules (Dao et al., 2020).

• Homological Vanishing and Near-Gorensteinness: In type 2 rings, $R$ is nearly Gorenstein if and only if $R$ is generically Gorenstein and $$\mathfrak{m} \operatorname{Ext}_R^i(\omega_R, R) = 0$$ for $1 \leq i \leq \dim R$ (Dao et al., 2020).
• Dao–Kobayashi–Takahashi Criterion: For numerical semigroup rings of minimal multiplicity, $$\mathfrak{m} \operatorname{Ext}_R^i(\omega_R, R) = 0$$ for all $i > 0$ if and only if $R$ is nearly Gorenstein, but the equivalence may fail in higher embedding dimension (Lyle et al., 7 Sep 2024).

5. Applications: Teter Rings and Classifications

The canonical trace ideal enables fine classifications of rings beyond the almost Gorenstein and nearly Gorenstein cases:

• Teter Property: A Cohen–Macaulay ring is Teter if there exists an injection $\varphi: \omega_R \hookrightarrow R$ with embedding dimension of the cokernel at most the dimension. Necessary and sufficient criteria for Teter property are provided in terms of the degree of the trace and the minimal number of generators of $\omega_R$ (Miyashita et al., 7 Dec 2025).
• Codimension–Type Bounds: For nearly Gorenstein rings under further hypotheses (e.g., level rings), the Cohen–Macaulay type is bounded above by the codimension (Miyashita et al., 7 Dec 2025).
• Fiber Products and Stanley–Reisner Rings: The canonical trace for fiber products is given as the direct sum of the traces in each factor, enabling combinatorial and topological characterizations of when Stanley–Reisner rings are nearly Gorenstein or possess the Teter property (Kumashiro et al., 5 Jun 2025).

6. Summary Table: Key Canonical Trace Phenomena

Phenomenon	Necessary and Sufficient Condition	Reference
Gorenstein ring	$\operatorname{tr}_R(\omega_R) = R$	(Herzog et al., 2016)
Nearly Gorenstein ring	$$\mathfrak{m} \subseteq \operatorname{tr}_R(\omega_R)$$	(Herzog et al., 2016, Ficarra, 11 Jun 2024)
Far-flung Gorenstein (dim 1)	$\operatorname{tr}_R(\omega_R) = R : R$ (conductor)	(Herzog et al., 2021)
Codimension-two, generically Gorenstein	$(n-2)$-minors of Hilbert–Burch matrix generate $\operatorname{tr}_R(\omega_R)$	(Ficarra et al., 2022, Ficarra, 11 Jun 2024)
Teter property (graded, codim = type)	Level, type $=$ codimension, lowest-degree trace contains non-zerodivisor	(Miyashita et al., 7 Dec 2025)

7. Impact and Open Directions

The study of the canonical trace ideal has generated precise invariants for characterizing various classes of rings approaching Gorensteinness, facilitated explicit computations for numerical semigroup and determinantal rings, and provided tools to relate ring-theoretic, homological, and combinatorial invariants. Open questions remain regarding the full range of possible residues and types for given families, expansion to higher-dimensional settings, and deeper homological characterizations—especially in the context of Teter and nearly Gorenstein properties—for broader classes of non-Gorenstein and generalized Cohen–Macaulay rings (Miyashita et al., 7 Dec 2025, Dao et al., 2020, Maitra, 2022).