math.RA — Rings and Algebras

Actual images and local fibers in the second commutator layer

Contributed by Shengyong Pan

Let $K$ be an algebraically closed field of characteristic zero and let $J$ be the commutator ideal of the free associative algebra. We prove that, for every proper multilinear polynomial $f\in J^2\setminus J^3$ and every $n\ge3$, the set of values of $f$ on $M_n(K)$ is $\mathfrak{sl}_n(K)$ if $\tr f\equiv0$, and $M_n(K)$ otherwise. In the first case the evaluation map is smooth at an input with nonzero rank-one square-zero value. In the second case it is flat at such an input, with a local complete-intersection fiber of the expected dimension. The theorem allows arbitrary proper multilinear corrections of the same degree in $J^3$. We use restrictions to two active variables with commuting passive inputs and detect the first nonzero trace term on the exact fiber of the traceless projection.