The coordinate ring of the k-fold iterated commutator locus for 2x2 matrices

Abstract

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.