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arXiv 1603.04293math.RTmath.RA
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$τ$ -刚性模的一个约化定理

A reduction theorem for $τ$-rigid modules

Florian Eisele, Geoffrey Janssens, Theo Raedschelders

发表信息：
Math. Z. (2018), 1-37
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摘要：

本文证明了一个定理，它给出了有限维代数$A$上的支撑$τ$ -倾斜模与$A/I$上的支撑$τ$ -倾斜模之间的一个双射，其中$I$是由$A$的中心与$A$的根相交生成的理想.这种双射是明确的，并且行为良好。我们给出了各种推论，特别侧重于块的群环的有限群。特别是，我们证明了有$τ$ -倾斜有限野生块与一个以上的简单模块。然后，我们继续分类的所有支持$τ$倾斜模块的所有代数的二面体，半二面体和四元数型，定义的埃德曼，其中包括所有驯服块的群环。注意，由于这些代数是对称的，这与分类所有基本的两项倾斜复形是一样的，并且结果是驯服块最多有$32$不同的基本两项倾斜复形。我们通过使用前面提到的归约定理来做到这一点，该定理将问题归约为仅取决于基域$k$的10个不同的代数，所有这些代数恰好都是弦代数。为了处理这十个代数，我们给出了一个组合分类的所有$τ$ -刚性模（不一定是对称的）弦代数。

英文摘要：

We prove a theorem which gives a bijection between the support $τ$-tilting modules over a given finite-dimensional algebra $A$ and the support $τ$-tilting modules over $A/I$, where $I$ is the ideal generated by the intersection of the center of $A$ and the radical of $A$. This bijection is both explicit and well-behaved. We give various corollaries of this, with a particular focus on blocks of group rings of finite groups. In particular we show that there are $τ$-tilting finite wild blocks with more than one simple module. We then go on to classify all support $τ$-tilting modules for all algebras of dihedral, semidihedral and quaternion type, as defined by Erdmann, which include all tame blocks of group rings. Note that since these algebras are symmetric, this is the same as classifying all basic two-term tilting complexes, and it turns out that a tame block has at most $32$ different basic two-term tilting complexes. We do this by using the aforementioned reduction theorem, which reduces the problem to ten different algebras only depending on the ground field $k$, all of which happen to be string algebras. To deal with these ten algebras we give a combinatorial classification of all $τ$-rigid modules over (not necessarily symmetric) string algebras.

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