论文标题
关于重复代数的有限维模块的变形理论
On a deformation theory of finite dimensional modules over repetitive algebras
论文作者
论文摘要
令$λ$为代数封闭的字段$ \ mathbf {k} $的基本有限维数代数，让$ \widehatλ$为$λ$的重复代数。在本文中，我们证明，如果$ \ wideHat {v} $是左$ \wideHatλ$-模块，则具有有限维度的$ \ m \ m i \ mathbf {k} $，则$ \ wideHat {v} $具有确定的versal versal vermormation yormation Ring $ r（\widhatλ，\ widehatomentian in commotian nos n isthe n iS n iS notery， $ \ mathbf {k} $ - algebra的残基字段也同构为$ \ mathbf {k} $。我们还证明，只要$ r（\wideHatλ，\ wideHat {v}）$是普遍的，只要$ \ usewise {\ mathrm {end}} _ {\wideHatλ}（\wideHatλ}（\ widehat {v} {v}）= \ m m缩了\ mathbf {k} $，以及在这种情况下，$ r（服用Syzygies。我们将获得的结果应用于有限维数模块，而不是$ 2 $ -KRONECKER代数的重复代数，该代数为对象的变形理论提供了一种替代方法。
Let $Λ$ be a basic finite dimensional algebra over an algebraically closed field $\mathbf{k}$, and let $\widehatΛ$ be the repetitive algebra of $Λ$. In this article, we prove that if $\widehat{V}$ is a left $\widehatΛ$-module with finite dimension over $\mathbf{k}$, then $\widehat{V}$ has a well-defined versal deformation ring $R(\widehatΛ,\widehat{V})$, which is a local complete Noetherian commutative $\mathbf{k}$-algebra whose residue field is also isomorphic to $\mathbf{k}$. We also prove that $R(\widehatΛ,\widehat{V})$ is universal provided that $\underline{\mathrm{End}}_{\widehatΛ}(\widehat{V})=\mathbf{k}$ and that in this situation, $R(\widehatΛ,\widehat{V})$ 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 $\mathbb{P}^1_{\mathbf{k}}$
