Geometric Rigidity in Moduli Stacks of Algebras

TL;DR

This paper develops a geometric framework for understanding algebra law moduli spaces, introducing a deformation complex and quadratic map that elucidate rigidity phenomena in algebraic structures.

Contribution

It constructs an intrinsic deformation complex and a quadratic map to analyze rigidity and deformation obstructions in moduli stacks of algebra laws.

Findings

The deformation complex encodes first-order classes and obstructions.

The quadratic map controls second-order lifts and rigidity.

Zariski openness of orbits is characterized by the quadratic map.

Abstract

We study quadratic moduli schemes $X$ of algebra laws on a fixed vector space $W$ under the transport-of-structure action of $GL(W)$ on $Hom(W^{\otimes 2},W)$. We construct an intrinsic three-term deformation complex on $X$ whose fibers encode transverse first-order classes and primary obstructions, and whose cohomology agrees on the operadic loci with the standard low-degree deformation cohomology (\`a la Gerstenhaber and Nijenhuis--Richardson). We then define a canonical quadratic map $\kappa^{inc}_{2,\mu}\colon H^2_{inc}(\mu)\to H^3_{inc}(\mu)$ that controls second-order lifts modulo isotriviality. If $\mu$ is smooth point in a reduced component and $(\kappa^{inc}_{2,\mu})^{-1}(0)=\{0\}$, then the $G$-orbit of $\mu$ is Zariski open in that component. This provides a coordinate-free explanation of Richardson-type geometric rigidity even when the second deformation cohomology does not vanish.