Nagata Idealization in Commutative Algebra

• Nagata idealization is a ring construction that adjoins a module as a square-zero ideal, enabling explicit control over algebraic and homological invariants.
• Both classical and twisted versions preserve key properties like Cohen–Macaulay, Gorenstein, and complete intersection, critical for constructing rings with exotic behaviors.
• Extensions of Nagata idealization use combinatorial and topological techniques to analyze graded structures, Lefschetz properties, and stability in Artinian Gorenstein algebras.

Nagata idealization, also called the trivial extension, is a construction in commutative algebra that adjoins a module as a square-zero ideal inside a ring, yielding a new ring with controlled algebraic and homological properties. This process has inspired significant generalizations, including topological and homological variants and “counterpart” constructions that realize the idealization within domains, and has found applications ranging from the explicit construction of rings with exotic properties to the paper of Artinian Gorenstein and level algebras via bigraded structures.

1. Classical Nagata Idealization

Let $S$ be a commutative ring and $K$ an $S$-module. The classical Nagata idealization, denoted $S \ltimes K$, is the additive group $S \oplus K$ with multiplication

$$(s_1, k_1)\cdot(s_2, k_2) = (s_1 s_2,\, s_1 k_2 + s_2 k_1).$$

Here, $0 \oplus K$ is a square-zero ideal: $(0, k_1) \cdot (0, k_2) = (0, 0)$. The ring $S$ embeds as $S \ltimes 0 \subset S \ltimes K$, and the projection onto $K$ induces an $S$-module structure. Notable properties include:

• The ideal $0 \oplus K$ is nilpotent of index two.
• $S \ltimes K$ is Noetherian if and only if $S$ is Noetherian and $K$ is finitely generated.
• If $S$ is Cohen–Macaulay and $K$ is a maximal Cohen–Macaulay module, then $S \ltimes K$ is Cohen–Macaulay; similar preservation holds for Gorenstein and complete intersection properties, subject to suitable constraints (Olberding, 2012).

Nagata idealization serves as a mechanism to construct commutative rings with prescribed homological properties, furnish counterexamples, and facilitate the paper of square-zero extensions in various contexts.

2. Counterpart and “Twisted” Nagata Idealization

To overcome the inherent nilpotency in $S \ltimes K$ and construct domains, Olberding introduced a “counterpart” construction based on derivations. Here, for a ring $S$, an $S$-module $K$, and a multiplicative set $C \subset S$ of nonzerodivisors, a subring $R \subseteq S$ is said to be twisted by $K$ along $C$ by a $C$-linear derivation $D : S_C \to K_C$ if:

• $R = S \cap D^{-1}(K)$,
• $D(S_C)$ generates $K_C$ as an $S_C$-module,
• for every $c \in C$, $S = \ker D + cS$.

When $S$ is a domain, $K$ is torsion-free, and $C = S \setminus \{0\}$, this is termed strongly twisted (Olberding, 2012). In this case:

• $R$ is a domain whenever $S$ is;
• $R \subset S$ is a subintegral, quadratic, integral extension with the same total ring of fractions;
• $S$ is contained in the normalization of $R$ and is finite over $R$ only if $R = S$.

This construction enables the realization of many domain-theoretic and homological behaviors locally indistinguishable (analytically) from the classical Nagata idealization. In the local case, the $\mathfrak{m}$-adic completion satisfies $\widehat{R} \cong \widehat{S} \ltimes \widehat{K}$, and thus, all homological invariants may be traced back to classical idealization data.

3. Homological Invariants and Finiteness Criteria

The structural properties of rings formed via Nagata idealization and its counterpart constructions are explicitly controlled by the module $K$ and the derivation $D$. The key criteria include:

• Noetherianity: For $S$ a domain and $R$ strongly twisted by torsion-free $K$, $R$ is Noetherian if and only if $S$ is Noetherian and $K/aK$ is finitely generated for every $0 \neq a \in \ker D$.
• Cohen–Macaulay, Gorenstein, Complete Intersection, Hypersurface: For $S$ quasilocal and Cohen–Macaulay, $R$ is:
  • Cohen–Macaulay $\iff$ $K$ is a maximal Cohen–Macaulay $S$-module,
  • Gorenstein $\iff$ $S$ has a canonical module $\omega_S$ and $K \cong \omega_S$,
  • a complete intersection $\iff$ $S$ is and $K \cong S$,
  • a hypersurface $\iff$ $S$ is regular and $K \cong S$.

This isomorphism at the completion level ensures that local invariants (e.g., multiplicity, embedding dimension, depth) are inherited directly from the classical idealization (Olberding, 2012).

4. Homological Examples, Stable Domains, and Applications

Several illustrative classes arise from Nagata idealization methods:

• Domains with Isolated Singularities: For $S = k[X, Y]_{(X, Y)}$ and $K = V^n$ with $V = k[X]_{(X)}$, one constructs $R \subset S$ of embedding dimension $2 + n$, multiplicity 1, analytic ramification, and isolated singularity (Olberding, 2012).
• Stable Rings: For integrally closed $S$ of Prüfer, Dedekind, or Krull type, the twisted subring $R$ is finitely stable, one-dimensional stable, or almost Krull with local-stability at height one primes, respectively. Specific constructions allow the prescription of the number of generators of maximal ideals, controlling stability further.
• Generalized and Local Behaviors: Analytically, for $R$, localizations $R/rR$ resemble idealizations, and the process embeds $K$ “in the normalization” rather than as a square-zero ideal.

These approaches extend the pullback-of-derivation frameworks of Ferrand–Raynaud and Goodearl–Lenagan to all dimensions and offer new constructions of analytically ramified Noetherian domains.

5. Nagata Idealization in Graded Artinian Gorenstein Algebras and Topology

Nagata idealization admits a significant role in the theory of standard graded Artinian Gorenstein algebras. For $A$ a graded Artinian Gorenstein algebra and $M$ a graded $A$-module, the trivial extension $A \ltimes M$ is again Gorenstein and supports rich Lefschetz phenomena.

CW-complex Model: Capasso, De Poi, Ilardi, et al. construct CW-complexes $P(m)$ whose $(d-1)$-cells naturally index degree-$d$ monomials in $m$ variables. They establish a dictionary between finite CW-subcomplexes $A_f$ of $P(m)$ (generated by monomials $g_i$ in a generalized Nagata polynomial $f = \sum_{i=0}^n x_i^{d_1} g_i(u)$) and the bigraded structure of the associated Artinian Gorenstein algebra $A = T / \mathrm{Ann}(f)$ (Capasso et al., 2020). The Hilbert function and generators of $\mathrm{Ann}(f)$ are controlled by the combinatorics and topology (skeletons and intersections) of $A_f$:

• $$\dim_K A_{(i,j)} = \begin{cases} f_j, & i=0, \ n f_j, & 1 \leq i \leq d_1-1, \ f_{d_2-j}, & i = d_1, \end{cases}$$ where $f_j$ is the number of $(j-1)$-cells in $A_f$.

This topological encoding generalizes earlier simplicial-complex models and unifies the paper of square-free and arbitrary monomials in the context of Lefschetz properties, Hessian rank, and Hilbert functions.

6. Higher-Order and Bigraded Nagata Idealizations: Lefschetz Properties

Cerminara, Gondim, Ilardi, and Maddaloni developed “higher-order” Nagata idealizations to paper graded Artinian algebras associated to bihomogeneous polynomials of bidegree $(d_1, d_2)$ (Cerminara et al., 2018). The resulting Gorenstein algebra $A = Q'/\mathrm{Ann}(f)$, where $f = \sum_i F_i(x) G_i(u)$, has socle bidegree $(d_1, d_2)$ and exhibits:

• Weak Lefschetz Property (WLP): Holds if $d_1 \geq d_2$, regardless of the number of variables; this is certified via analysis of multiplication by a generic linear form and mixed Hessian computations.
• Strong Lefschetz Property (SLP): SLP may fail when $d_1 < d_2$ due to higher Hessian vanishing.
• Combinatorial Structure: In square-free monomial cases, the algebra is governed by the face numbers of associated simplicial complexes, and the annihilator ideal admits an explicit combinatorial description.

This links geometric (Nagata hypersurface scroll structure), combinatorial, and algebraic invariants, and enables the construction and classification of Artinian Gorenstein algebras with desirable Lefschetz properties.

7. Combinatorial and Polynomial Extensions: Nagata Extensions

Nagata’s original “polynomial” extension construction leads to the Nagata ring $R(X)$, formed by localizing $R[X]$ at the multiplicative set of content-invertible polynomials, and to associated extensions $R(X) \subset S(X)$ (Picavet et al., 2015). Key results include:

• Preservation of Invariants: The length of chains of intermediate rings ($\ell[R, S]$) and the Dobbs–Mullins invariant are preserved under Nagata extension: $\ell[R, S] = \ell[R(X), S(X)]$, $A(S/R) = A(S(X)/R(X))$ for FCP extensions.
• Finiteness Properties (FIP): The FIP is preserved under Nagata extension if and only if the base extension is FIP and arithmetic (i.e., localizations are chained).
• Structural Analysis: The classification relies on field-theoretic decompositions, t-closure, and infra-integral splits, and admits further paper in lattice-theoretic and arithmetic contexts.

This combinatorial perspective supplements module-based idealizations, extending the Nagata paradigm in polynomial and algebraic settings and informing the structure of extensions in positive characteristic or non-Noetherian contexts.