Artinian Gorenstein Algebra Isomorphism

• Artinian Gorenstein algebras are finite-dimensional, commutative local algebras characterized by a one-dimensional socle, underpinning their duality properties.
• The hypersurface criterion leverages nil-polynomials to establish affine equivalence, providing an efficient invariant for determining algebra isomorphism.
• Macaulay inverse systems, symmetric decompositions, and group-orbit methods work in tandem to offer robust computational tools for classifying these algebras.

An Artinian Gorenstein algebra is a commutative, associative local algebra over a field (characteristic zero or sufficiently large) with a unique maximal ideal and having finite dimension greater than one. The defining property is that the socle (annihilator of the maximal ideal) is one-dimensional, equivalent to the non-degeneracy of the trace pairing. Recent advances have yielded explicit, computable criteria for algebra isomorphism, grounded in the geometry of associated algebraic hypersurfaces and the structure of Macaulay inverse systems. These geometric and duality-based characterizations connect deep properties of such algebras to group actions, canonical forms, and stratifications, providing practical invariants and algorithms for classification.

1. Definitions and Fundamental Properties

Let $A$ be a local, commutative, Artinian $k$-algebra ($\dim_k A > 1$), with maximal ideal $\mathfrak m$. The socle is $\Soc(A) = \{ u \in A : u \mathfrak m = 0 \}$, and $A$ is Gorenstein if $\dim_k \Soc(A) = 1$. For socle degree $v$ (the maximal $p$ with $\mathfrak m^p \ne 0$), $\Soc(A) = \mathfrak m^v$. The non-degeneracy condition is equivalently that any $k$-linear projection $\tau: A \to \Ann(\mathfrak m)$ yields a non-degenerate bilinear form via $(a, c) \mapsto \tau(ac)$ (Isaev, 2012, Isaev, 2015).

2. Hypersurface Criterion for Isomorphism

To each such $A$ and admissible linear projection $\pi: \mathfrak m \to \Ann(\mathfrak m)$ (with $\ker \pi \supset 1$), construct $K = \ker \pi \cap \mathfrak m$ so that $\mathfrak m = K \oplus \Ann(\mathfrak m)$. The nil-polynomial

$$P_\pi(u) = -\pi(\exp u) = -\sum_{m=2}^v \frac{\pi(u^m)}{m!}, \quad \exp(u) = \sum_{m=0}^v \frac{u^m}{m!}$$

defines an algebraic hypersurface $$S_\pi = \{ u + P_\pi(u) : u \in K \} \subset \mathfrak m$$. The main result is: $A \cong \widetilde{A}$ if and only if $S_\pi$ and $S_{\widetilde{\pi}}$ are affinely equivalent via some affine bijection $$\mathfrak A: \mathfrak m \rightarrow \widetilde{\mathfrak m}$$; moreover, if the map is linear, it is already an algebra isomorphism (Isaev, 2012, Isaev, 2015). Necessity follows since any isomorphism induces affine equivalence by translation; sufficiency proceeds via block-diagonal reduction and compatibility of the polynomial components on $K$ and $\Ann(\mathfrak m)$.

3. Macaulay Inverse Systems and Duality

An Artinian Gorenstein algebra $A$ with embedding dimension $k$ may be represented as $A \cong k[x_1,\dots,x_k]/I$, with an associated Macaulay inverse system—a degree-$v$ polynomial $g(y_1,\ldots,y_k)$ such that $I = \Ann(g) = \{ f \in k[x_1,\ldots,x_k] : f(\partial/\partial y) g = 0 \}$. The nil-polynomial $P_\pi$ restricted to suitable subspaces recovers precisely the inverse system: for a complement $L$ to $\mathfrak m^2$ inside a hyperplane in $\mathfrak m$, the restriction $Q(y) = \pi(\exp(y_1 e_1 + \cdots + y_k e_k))$ gives $\Ann(Q) = I$ (Isaev, 2012, Isaev, 2015). The affine-hypersurface criterion thus offers a computational alternative to inverse system comparison, as the latter requires checking for equivalence under $(\mathbb{C}^* \ltimes GL(k))$.

4. Symmetric Decomposition of the Associated Graded Algebra

For $A^* = \mathrm{Gr}(A)$, the chain of ideals $$A^* = C_A(0) \supset C_A(1) \supset \cdots \supset C_A(j-1) = 0$$ gives subquotients $Q_A(a) = C_A(a)/C_A(a+1)$, each a reflexive $A^*$-module with a perfect symmetric pairing. The resulting symmetric decomposition of the Hilbert function,

$H(A) = \sum_{a = 0}^{j - 1} H_A(a)$

(where $H_A(a)$ is the Hilbert function of $Q_A(a)$) serves as an isomorphism invariant and detects fine structure, such as interior zeroes indicating non-cyclic modules (Iarrobino et al., 2018). The Macaulay dual generator $F$ admits normal forms such that each degree block $F_{j-a}$ involves only the least number of variables, aligning variable blocks with the recursive structure of $Q_A(a)$.

5. Group-Orbit Methods, Normal Forms, and Classification Algorithms

The isomorphism problem translates into orbit equivalence under $G = \Aut(S) \ltimes S^*$ acting on the dual $P$ of $S = k[[\alpha_1, \ldots, \alpha_n]]$ via $$(\varphi, s) \cdot f(\theta) = f(s \varphi(\theta))$$. For generators $f$, $g$ in $P$, $A \cong S/\Ann(f)$ and $B \cong S/\Ann(g)$ are isomorphic precisely when $f$ and $g$ are $G$-equivalent (Jelisiejew, 2015). Explicit formulas for the action involve contraction: $$\varphi^\vee(f) = \sum_{a \in \mathbb{N}^n} x^{[a]} (D^a \lrcorner f)$$ where $\varphi(\alpha_i) = \alpha_i + D_i$, and $x^{[a]}$ denotes divided-power monomials. Analyzing tangent spaces of orbits gives a handle on deformation, rigidity, and obstruction theory.

Tests for isomorphism involve reducing generators to normal forms, computing symmetric decompositions, and matching dual generator blocks up to individual $GL(n_a)$ actions per stratification level (Iarrobino et al., 2018). For instance, algebras with Hilbert function $(1,3,3,3,1)$ exhibit finitely many isomorphism types, explicitly enumerated, while functions like $(1,2,2,2,1,1,1)$ yield parametric families indexed by field elements (Jelisiejew, 2015).

6. Applications and Illustrative Examples

The hypersurface criterion allows efficient distinction in families where traditional inverse systems require intricate coordinate changes. For example, the one-parameter family $A_t = k[x,y]/(2x^3 + t y^3, t x^2y^2 + 2y^5)$ admits an explicit classification: $A_r \cong A_s$ if and only if $s = \pm r$, computable directly via nil-polynomials and hypersurface equivalence (Isaev, 2012). In nonnegatively graded cases, affine equivalence of hypersurfaces reduces to linear equivalence, further streamlining the process.

Connected sums and non-canonical cases are detected by analyzing the decomposition invariants and dual generator blocks—if, for instance, $H(j-2)$ displays a single nonzero entry, the algebra decomposes (after appropriate change of variables) as a connected sum (Iarrobino et al., 2018).

7. Computational Perspective and Classification Schemes

By leveraging the hypersurface, symmetric decomposition, and group-orbit approaches, classification is tractable for small Hilbert function profiles, either finitely or parametrically. The synthesis of methods—coordinate projection, stratification, duality, and group action—has led to effective computational criteria, supplanting purely combinatorial or ad hoc methods in many cases (Isaev, 2012, Isaev, 2015, Iarrobino et al., 2018, Jelisiejew, 2015).

Approach	Main Invariant	Computational Steps
Affine hypersurface	Nil-polynomial $P_\pi$, $S_\pi$	Compute $P_\pi$, test affine eq.
Macaulay inverse system	Polynomial $g$ in dual variables	Transform $g$, check $GL(k)$ eq.
Symmetric decomposition	Hilbert function tuples $D(A)$	Decompose, analyze blocks
Group-orbit method	$G$-orbit representatives	Apply automorphisms and units

Each framework provides explicit techniques and invariants, with the hypersurface approach often yielding streamlined computations, the inverse system encoding classical Gorenstein duality, and the symmetric decomposition detecting connected sums and non-cyclic phenomena.