Parseval-Rayleigh identities for graded Artinian Gorenstein algebras

Abstract: We formulate and prove Parseval-Rayleigh identities for graded Artinian Gorenstein algebras over fields of positive characteristic. Specializing the general result, we provide an alternative proof of the Parseval-Rayleigh identities of generic Artinian reductions of Stanley-Reisner rings of oriented simplicial spheres.

• The paper introduces a universal framework using minimal free resolutions and the Frobenius action to establish Parseval-Rayleigh identities in graded Artinian Gorenstein algebras.
• It generalizes known results from complete intersections and Stanley–Reisner rings by explicitly constructing a universal element ε that governs these identities.
• The findings enhance the understanding of Lefschetz properties and offer novel computational strategies for exploring homological and combinatorial features in positive characteristic.

Parseval–Rayleigh Identities in Graded Artinian Gorenstein Algebras

Introduction and Context

The paper "Parseval–Rayleigh identities for graded Artinian Gorenstein algebras" (2604.27631) systematically develops an abstract framework for Parseval–Rayleigh type identities in the setting of graded Artinian Gorenstein algebras over a field of positive characteristic. These identities, originally motivated by connections to Lefschetz properties, have been crucial in the study of combinatorial and homological aspects of such algebras. Prior work established these identities for select cases (notably generic complete intersections and Stanley–Reisner rings of oriented simplicial spheres); this paper generalizes the construction to arbitrary graded Artinian Gorenstein $k$-algebras, also extending validity to positive characteristic.

The manuscript's technical backbone is the use of homological algebra—in particular, minimal free resolutions and the Frobenius action—to formalize and prove universal Parseval–Rayleigh identities. This generality offers a new conceptual lens for exploring structural and computational aspects of Artinian Gorenstein algebras, including new families outside established classes.

Main Theorem and Homological Construction

Let $k$ be a field of characteristic $p > 0$, $S = k[x_0, ..., x_{m-1}]$ a polynomial ring, and let $I = (g_0, ..., g_{n-1}) \subset S$ be a homogeneous Artinian Gorenstein ideal, providing $R = S/I$ with socle degree $s$. Given a $k$-linear isomorphism $vol : R_s \to k$ and corresponding distinguished generator $\nu \in R_s$ with $k$0, the Frobenius twist produces the ideal $k$1 and algebra $k$2, of socle degree $k$3. The main theorem asserts the existence and uniqueness of an element $k$4 such that $k$5, where $k$6 is the volume form on $k$7 normalized analogously.

For any degree $k$8 polynomial $k$9, the Parseval–Rayleigh identity reads:

$p > 0$0

where $p > 0$1 is the set of degree $p > 0$2 monic monomials, and $p > 0$3 denotes the Kronecker pairing on monomials.

The proof leverages canonical choices in the minimal free resolution of $p > 0$4 and its Frobenius twist, extraction of socle generators, and explicit construction of $p > 0$5 via induced chain maps between resolutions. The independence of $p > 0$6's class from homotopic ambiguities is established, as is the crucial property $p > 0$7 by tracing the compatibility of chain maps and local duality.

Specialization and Explicit Cases

Complete Intersections

For $p > 0$8 a complete intersection generated by $p > 0$9, the algebraic and homological structure simplifies: both $S = k[x_0, ..., x_{m-1}]$0 and $S = k[x_0, ..., x_{m-1}]$1 are Koszul complexes, and explicit formulas for socle generators and all morphisms are accessible. In this case, $S = k[x_0, ..., x_{m-1}]$2, and the general identity recovers the established Parseval–Rayleigh identities as in [AdiCI].

Stanley–Reisner Rings of Oriented Simplicial Spheres

For oriented simplicial spheres, the generic Artinian reduction of the Stanley–Reisner ring with chosen volume matches the Kustin–Miller normalization. The explicit $S = k[x_0, ..., x_{m-1}]$3 takes the form $S = k[x_0, ..., x_{m-1}]$4, with $S = k[x_0, ..., x_{m-1}]$5 linear forms in the variables defined over the rational function field $S = k[x_0, ..., x_{m-1}]$6. Homological arguments demonstrate that this recovers the known Parseval–Rayleigh identities, but the generality and method crucially differ from earlier proofs by reducing the assertion to properties of the universal element $S = k[x_0, ..., x_{m-1}]$7.

Non-Complete-Intersection Example

The paper provides a concrete computation of $S = k[x_0, ..., x_{m-1}]$8 for a minimal non-complete-intersection Gorenstein algebra (the Buchsbaum–Eisenbud height-3 example), detailing the explicit minimal free resolutions, differentials, chain maps, and associated volumes. This example lies outside the field of prior explicit treatments, highlighting the method's broad applicability.

Implications and Prospects

The generalization of Parseval–Rayleigh identities to all graded Artinian Gorenstein algebras has several implications:

• Lefschetz Properties: The framework affirms that sufficient algebraic identities underpin the strong Lefschetz property in diverse contexts, potentially enabling sharp structural results in positive characteristic.
• Computational Homological Algebra: The recipe for constructing $S = k[x_0, ..., x_{m-1}]$9 via resolutions and chain maps is algorithmic, offering symbolic computation routes for explicit invariants in previously inaccessible algebras.
• Combinatorial Commutative Algebra: The approach suggests that a unified homological mechanism governs identities observed in toric, simplicial, and more general combinatorial settings.

This perspective may stimulate further cross-fertilization between combinatorics, commutative algebra, and representation theory, where Artinian Gorenstein algebras serve as testing grounds for duality and symmetry phenomena. The method could be extended to address analogous questions for other classes of (possibly non-Artinian or non-Gorenstein) algebras and to positive characteristic syzygetic phenomena.

Conclusion

This paper (2604.27631) advances the understanding of Parseval–Rayleigh identities in Artinian Gorenstein algebras, unifying disparate cases under a homologically robust, positive-characteristic-compatible formalism. The existence and explicit computation of the universal element $I = (g_0, ..., g_{n-1}) \subset S$0 controlling these identities clarify their algebraic origin, broaden their applicability, and cement their role in Lefschetz-type statements. The implications for both theory and computation are significant, suggesting multiple avenues for further inquiry in commutative algebra and algebraic geometry.