Nilpotency of class 3 stable in a certain class of algebras
Let \(C_n\) be an affine scheme of associative commutative algebras of dimension \(n\) and let \(A_n\) be the subscheme of \(C_n\) consisting of algebras nilpotent of degree 3. Given a scheme \(S\), the symbol \(S^{\text{red}}\) denotes the related reduced scheme. The author continues a description of the components of \(C_n^{\text{red}}\) that lie in \(A_n^{\text{red}}\), initiated by \textit{I.~R.~Shafarevich} [Leningr. Math. J. 2, No. 6, 1335-1351 (1991); translation from Algebra Anal. 2, No. 6, 178-194 (1990; Zbl 0727.13006)] and by \textit{A.~Z.~Anan'in} and \textit{A.~R.~Mavlyutov} [Sib. Math. J. 35, No. 3, 426-438 (1994); translation from Sib. Mat. Zh. 35, No. 3, 480-494 (1994; Zbl 0877.13015)]. In the cited articles it was proven that every component \(A_{n,r}\) of \(A_n^{\text{red}}\) is determined by a number \(r=\dim N^2\) for a generic algebra \(N\) of \(A_{n,r}\); this number satisfies the inequalities \(1\leq r\leq d(d+1)/2\), where \(d=n-r\). Moreover, \(A_{n,r}\) is not a component of \(C_n^{\text{red}}\) for \(r>(d^2-1)/3\) and \(r=1,2\), but \(A_{n,r}\) is a component for certain values of \(r\) in the interval \([3,(d^2-1)/3]\). Let \(C_{n,r}\) denote the subscheme of \(C_n\) which consists of algebras \(A\) with \(\dim A^2\leq r\). The author proves that \(A_{n,r}\) is a component of \(C_{n,r}^{\text{red}}\) and that, for every \(A\in C_{n,r-1}\), the tangent space \(T_AC_{n,r}\) coincides with the tangent space \(T_AC_n\).
scientific article; zbMATH DE number 1432399