2×2矩阵的k次迭代交换子轨迹的坐标环
The coordinate ring of the k-fold iterated commutator locus for 2x2 matrices
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中文总结 AI 辅助
该文研究2×2矩阵的k次迭代交换子轨迹的坐标环，在特征≠2的域上证明其对应理想的四个性质，给出其准素分解及证明方法。
中文摘要 AI 辅助
对于2×2矩阵A₁,…,Aₖ，记[A₁,…,Aₖ]为左规范迭代交换子[[…[[A₁,A₂],A₃],…,Aₖ]]，Iₖ为3k个变量的简约坐标多项式环Rₖ中，切割出其消失轨迹的理想。我们在特征≠2的任意域上，对所有k≥2证明了四个相互独立的结果（而非一个捆绑结论）：Iₖ的余维数为2；Iₖ恰有3个极小生成元；Rₖ/Iₖ是Cohen-Macaulay环；Iₖ是根理想。最后一个结果结合基于非包含论证的显式分支计数，可导出准素分解定理：Iₖ = P₂ ∩ … ∩ Pₖ是恰含k-1个素理想的无冗余准素分解，遵循显式递归块关联模式。证明将Iₖ识别为显式2×3矩阵的2×2子式的理想（即行列式理想，而非仅看似行列式的理想），分别运用经典行列式理想理论（Bruns-Vetter）和每个分支上的显式秩-2雅可比见证（Serre准则）证明代数性与根性。
英文摘要
For $2 \times 2$ matrices $A_1,\dots,A_k$, write $[A_1,\dots,A_k]$ for the left-normed iterated commutator $[\dots[[A_1,A_2],A_3],\dots,A_k]$, and $I_k$ for the ideal, in the $3k$-variable reduced-coordinate polynomial ring $R_k$, cutting out its vanishing locus. We prove, for every $k \geq 2$ over any field of characteristic $\neq 2$, and as four independently-established results rather than one bundled claim: $I_k$ has codimension 2; $I_k$ has exactly 3 minimal generators; $R_k / I_k$ is Cohen-Macaulay; and $I_k$ is radical. The last of these, together with an explicit component count resting on a non-containment argument, assembles into the Primary Decomposition Theorem: $I_k = P_2 \cap \cdots \cap P_k$ is an irredundant primary decomposition into exactly $k-1$ primes, following an explicit recursive block-involvement pattern. The proof identifies $I_k$ as the ideal of $2 \times 2$ minors of an explicit $2 \times 3$ matrix (a determinantal ideal, not merely one that looks determinantal), and invokes classical determinantal-ideal theory (Bruns-Vetter) and an explicit rank-2 Jacobian witness on every component (Serre's criterion) for the algebraic and radicality halves respectively.
发表机构
- Matematiska Institutionen, Linköpings Universitet（林雪平大学数学系）
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