dorsal/arxiv
View SchemaGeometric Rigidity in Moduli Stacks of Algebras
| Authors | Atabey Kaygun |
|---|---|
| Categories | |
| ArXiv ID | 2601.05715vv1 |
| URL | https://arxiv.org/abs/2601.05715 |
| License | http://arxiv.org/licenses/nonexclusive-distrib/1.0/ |
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.
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"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.",
"arxiv_id": "2601.05715",
"authors": [
"Atabey Kaygun"
],
"categories": [
"math.AG"
],
"license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
"title": "Geometric Rigidity in Moduli Stacks of Algebras",
"url": "https://arxiv.org/abs/2601.05715",
"version": "v1"
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