Donovan猜想，带交换亏群的块和离散赋值环
Donovan's conjecture, blocks with abelian defect groups and discrete valuation rings
摘要：
对于具有交换亏群的块，我们给出了Donovan猜想的拟单群的一个约化，该亏群是关于一个合适的离散赋值环$\mathcal{O}$定义的.其结果是，多诺万的猜想持有$\mathcal{O}$ -块与阿贝尔缺陷群的素数两个，并使用最近的工作法雷尔和凯萨，任意素数多诺万的猜想$\mathcal{O}$ -块与阿贝尔缺陷群减少到边界的Cartan不变量块的quasisimple群体的缺陷。一个结果的独立利益是，在一般情况下（即为任意缺陷群）多诺万的猜想为$\mathcal{O}$ -块是一个后果的预测界限上的$\mathcal{O}$ -Frobenius数和嘉当不变量，作为证明了凯萨块定义在一个代数闭域。
英文摘要：
We give a reduction to quasisimple groups for Donovan's conjecture for blocks with abelian defect groups defined with respect to a suitable discrete valuation ring $\mathcal{O}$. Consequences are that Donovan's conjecture holds for $\mathcal{O}$-blocks with abelian defect groups for the prime two, and that, using recent work of Farrell and Kessar, for arbitrary primes Donovan's conjecture for $\mathcal{O}$-blocks with abelian defect groups reduces to bounding the Cartan invariants of blocks of quasisimple groups in terms of the defect. A result of independent interest is that in general (i.e. for arbitrary defect groups) Donovan's conjecture for $\mathcal{O}$-blocks is a consequence of conjectures predicting bounds on the $\mathcal{O}$-Frobenius number and on the Cartan invariants, as was proved by Kessar for blocks defined over an algebraically closed field.
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