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تسجيل مستخدم جديدOn a deformation theory of finite dimensional modules over repetitive algebras
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Let $Lambda$ be a basic finite dimensional algebra over an algebraically closed field $mathbf{k}$, and let $widehat{Lambda}$ be the repetitive algebra of $Lambda$. In this article, we prove that if $widehat{V}$ is a left $widehat{Lambda}$-module with finite dimension over $mathbf{k}$, then $widehat{V}$ has a well-defined versal deformation ring $R(widehat{Lambda},widehat{V})$, which is a local complete Noetherian commutative $mathbf{k}$-algebra whose residue field is also isomorphic to $mathbf{k}$. We also prove that $R(widehat{Lambda},widehat{V})$ is universal provided that $underline{mathrm{End}}_{widehat{Lambda}}(widehat{V})=mathbf{k}$ and that in this situation, $R(widehat{Lambda},widehat{V})$ is stable after taking syzygies. We apply the obtained results to finite dimensional modules over the repetitive algebra of the $2$-Kronecker algebra, which provides an alternative approach to the deformation theory of objects in the bounded derived category of coherent sheaves over $mathbb{P}^1_{mathbf{k}}$
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Let $mathbf{k}$ be a field of arbitrary characteristic, let $Lambda$ be a finite dimensional $mathbf{k}$-algebra, and let $V$ be a finitely generated $Lambda$-module. F. M. Bleher and the third author previously proved that $V$ has a well-defined ver
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Let $mathbf{k}$ be field of arbitrary characteristic and let $Lambda$ be a finite dimensional $mathbf{k}$-algebra. From results previously obtained by F.M Bleher and the author, it follows that if $V^bullet$ is an object of the bounded derived catego
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