Nilpotent operators and weighted projective lines
Nilpotent operators and weighted projective lines
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DOI：
10.1515/crelle-2012-0014
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发表时间：
2010-02
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影响因子：
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通讯作者：
D. Kussin;H. Lenzing;H. Meltzer
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文献类型：
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作者：
D. Kussin;H. Lenzing;H. Meltzer
我们展示了奇点理论与C. M. Ringel和M. Schmidmeier最近研究的幂零算子的不变子空间问题之间的惊人联系，这个问题的历史可以追溯到G. Birkhoff。这种联系是通过加权投影线和这些线上向量束的（稳定）类别来建立的。这种设置提供了一种解决子空间问题的新方法。特别地，我们从由三角形奇点x^2+y^3+z^p用Serre构造得到的权型（2,3,p）的加权投影线上的向量束范畴的性质，推导出Ringel和Schmidmeier关于幂零度p的主要结果。因此，对于p=6， Ringel-Schmidmeier分类被管型（2,3,6）的向量束分类所覆盖，从而与Atiyah在光滑椭圆曲线上的向量束分类密切相关。回到一般情况，我们建立了与向量束或幂零算子的不变子空间相关的稳定范畴可以自然地标识为三角化范畴。它们满足Serre对偶性，也有倾斜物体，其自同态环在奇点理论中起作用。事实上，我们因此得到了一个完整的三角形（分数）Calabi-Yau范畴序列，以p为索引，它们自然地形成了一个ade链。
We show a surprising link between singularity theory and the invariant subspace problem of nilpotent operators as recently studied by C. M. Ringel and M. Schmidmeier, a problem with a longstanding history going back to G. Birkhoff. The link is established via weighted projective lines and (stable) categories of vector bundles on those. The setup yields a new approach to attack the subspace problem. In particular, we deduce the main results of Ringel and Schmidmeier for nilpotency degree p from properties of the category of vector bundles on the weighted projective line of weight type (2,3,p), obtained by Serre construction from the triangle singularity x^2+y^3+z^p. For p=6 the Ringel-Schmidmeier classification is thus covered by the classification of vector bundles for tubular type (2,3,6), and then is closely related to Atiyah's classification of vector bundles on a smooth elliptic curve. Returning to the general case, we establish that the stable categories associated to vector bundles or invariant subspaces of nilpotent operators may be naturally identified as triangulated categories. They satisfy Serre duality and also have tilting objects whose endomorphism rings play a role in singularity theory. In fact, we thus obtain a whole sequence of triangulated (fractional) Calabi-Yau categories, indexed by p, which naturally form an ADE-chain.