CW-complexes and minimal Hilbert vector of graded Artinian Gorenstein algebras
Armando Capasso
TL;DR
The paper addresses how to identify SGAG algebras with minimal Hilbert vectors for fixed socle degree $d$ and codimension $n$, introducing the standard locus and the Full Perazzo locus to organize these algebras. It develops a novel CW-complex correspondence, via Construction 1, that associates to a homogeneous polynomial $f$ a finite CW-complex $\zeta_f$, allowing the annihilator $ ext{Ann}(f)$ to be constrained by the CW-structure and skeletons of a universal complex $P(m)$. Leveraging these topological tools, the author proves that the Hilbert function values $h_k$ are minimized on the Full Perazzo components, by showing that $ ext{Ann}(f)_k$ attains its maximal possible dimension there and that the FP locus has pure dimension $ au(n,d-1)-1$ and sits inside minimal-dimension components of the standard locus. Consequently, the Full Perazzo Conjecture is established: FP algebras realize the minimal Hilbert vector among SGAG algebras of given codimension and socle degree. The work bridges commutative algebra and topological combinatorics, yielding new insights into Lefschetz properties and the structure of Hilbert vectors for Artinian Gorenstein algebras.
Abstract
I introduce a geometric interpretation of the set of standard graded Artinian Gorenstein algebras of codimension n and degree d: the standard locus, which is a subset of the projective space of degree d polynomials in n variables, and I characterize it. Under opportune hypothesis, I prove that the locus of full Perazzo polynomials is the union of the minimal dimensional irreducible components of the standard locus and it is a pure dimensional subset. On the other hand, I associate to any homogeneous polynomial a topological space, which is a CW-complex. Using all these sets, I prove that the Hilbert function restricted to the standard locus has minimal values on any irreducible component of the domain. I apply all this to the Full Perazzo Conjecture and I prove it.