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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2509.10121 |
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| _version_ | 1866917068703531008 |
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| author | Smoktunowicz, Agata |
| author_facet | Smoktunowicz, Agata |
| contents | 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]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_10121 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On flat deformations and their applications Smoktunowicz, Agata Rings and Algebras 16S80 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]. |
| title | On flat deformations and their applications |
| topic | Rings and Algebras 16S80 |
| url | https://arxiv.org/abs/2509.10121 |