On flat deformations and their applications

We say that a formal deformation from an algebra $N$ to algebra $A$ is strongly flat if for every real number $e$ there is a real number $0<s<e$ such that this deformation specialised at $t=s$ gives an algebra isomorphic to $A$. We show that every strongly flat deformation from a finite-dimensional $C$-algebra $N$ to a semisimple $C$-algebra $A$ specialised at $t=s$ for all sufficiently small real numbers $s>0$ gives an algebra isomorphic to $A$. It is shown that all semisimple algebras which can be obtained as a specialisation of such a deformation are isomorphic. We also show that every strongly flat deformation $\mathcal N=N\{t\}$ from a finite-dimensional $\mathbb C$-algebra $N$ to a semisimple $\mathbb C$-algebra $A$ specialised at $t=s$ for all sufficiently small real numbers $s>0$ gives an algebra isomorphic to $A$. A remark by Joachim Jelisiejew is also included which allows us to obtain this result as an application of Gabriel's theorem [6]. We also give a characterisation of semisimple algebras $A$ to which a given algebra $N$ cannot be deformed to. This gives a partial answer to a question of Michael Wemyss on Acons [26]. We also give a partial answer to question 6.5 from [1].