Anan’in, A. Z.; Mavlyutov, A. R. Stability of nilpotency of class 3. (English. Russian original) Zbl 0877.13015 Sib. Math. J. 35, No. 3, 426-438 (1994); translation from Sib. Mat. Zh. 35, No. 3, 480-494 (1994). Let \(C_n\) be the scheme of all associative and commutative algebras of dimension \(n\) over a field \(K\) of characteristic \(\neq 2\). In this paper the authors consider the subscheme, say, \(A_n\) of those algebras that are nilpotent of class 3. They study the question when certain irreducible components of \(A_n\) are also components of \(C_n\). As was shown by 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)] the components of \(A_n\) are given by \(A_{n,r}\) where the generic algebra \(N\) in \(A_{n,r}\) satisfies \(\dim N^2=r\). Moreover, \(A_{n,r}\) is a component of \(C_n\) if \(3\leq r\leq(d+1)(d+2)/6\). The authors prove the following two results: (1) If \(d\geq 9\) and \(d(d+1)/9\leq r\leq[d/3] (d-3)\), then there exists a point \(N\in A_{n,r}\) such that all regular tangent vectors to \(C_n\) at \(N\) are tangent to \(A_{n,r}\). By a regular tangent vector they mean, roughly speaking, that it occurs as tangent vector along a smooth curve in \(C_n\). (2) If \(d=4t>4\) and \(r=5t^2-2t\) then \(A_{n,r}\) is a component of \(C_n\). Reviewer: H.Flenner (Bochum) Cited in 1 ReviewCited in 2 Documents MSC: 13D10 Deformations and infinitesimal methods in commutative ring theory 16N40 Nil and nilpotent radicals, sets, ideals, associative rings Keywords:nilpotent algebra; nilpotent of class 3 Citations:Zbl 0743.13009; Zbl 0727.13006 PDFBibTeX XMLCite \textit{A. Z. Anan'in} and \textit{A. R. Mavlyutov}, Sib. Math. J. 35, No. 3, 426--438 (1994; Zbl 0877.13015); translation from Sib. Mat. Zh. 35, No. 3, 480--494 (1994) Full Text: DOI References: [1] I. R. Shafarevich, ”Deformations of commutative algebras of class 2,” Algebra i Analiz,2, No. 6, 178–196 (1990). · Zbl 0727.13006 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.