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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)

MSC:

13D10 Deformations and infinitesimal methods in commutative ring theory
16N40 Nil and nilpotent radicals, sets, ideals, associative rings
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References:

[1] I. R. Shafarevich, ”Deformations of commutative algebras of class 2,” Algebra i Analiz,2, No. 6, 178–196 (1990). · Zbl 0727.13006
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