On flat deformations and their applications
Agata Smoktunowicz
TL;DR
The paper investigates when a finite-dimensional $ ext{C}$-algebra can be deformed flatly to a semisimple algebra and what the possible targets are. It introduces strongly flat deformations and a $K$-algebra construction $E$ that captures the deformation data, using polynomial identities to compare structures and show that small-specialization algebras must match the semisimple target. A key result is that a strongly flat deformation from $N$ to a semisimple $A$ of the same dimension forces $N_s\, oralls ext{ sufficiently small }s>0$ to be isomorphic to $A$, and it provides criteria to obstruct deformations to certain semisimple algebras, with concrete applications to contraction algebras and Acons (addressing questions by Wemyss). The work also integrates Gabriel's theorem to show that, in certain flat-deformation scenarios, the target semisimple algebras are uniquely determined up to isomorphism, and it develops notions like the "type" of an algebra to formalize these obstructions and their consequences.
Abstract
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].