Higher Toda brackets and the Adams spectral sequence in triangulated categories
Higher Toda brackets and the Adams spectral sequence in triangulated categories
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高 Toda 括号和三角类别中的 Adams 谱序列
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
Martin Frankland
中科院分区：
文献类型：
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作者：
J. Christensen;Martin Frankland
亚当斯谱序列可用于任何配备有投射或内射类的三角化类别。更高的户田括号也可以定义在一个三角形的类别，如B观察。Shipley基于J. Cohen的光谱方法。我们提供了一个家庭的定义更高的户田括号，表明他们是等价的Shipley的，并表明他们是自对偶。我们的主要结果是，任何亚当斯谱序列中的亚当斯微分d_r$都可以表示为一个r+1 $-重户田括号和一个r^{\text{th}}$阶上同调运算。我们还展示了在稀疏性假设下结果如何简化，讨论了几个例子，并给出了Heller的一个结果的初等证明，这意味着三重户田括号原则上决定了更高的户田括号.
The Adams spectral sequence is available in any triangulated category equipped with a projective or injective class. Higher Toda brackets can also be defined in a triangulated category, as observed by B. Shipley based on J. Cohen's approach for spectra. We provide a family of definitions of higher Toda brackets, show that they are equivalent to Shipley's, and show that they are self-dual. Our main result is that the Adams differential $d_r$ in any Adams spectral sequence can be expressed as an $(r+1)$-fold Toda bracket and as an $r^{\text{th}}$ order cohomology operation. We also show how the result simplifies under a sparseness assumption, discuss several examples, and give an elementary proof of a result of Heller, which implies that the three-fold Toda brackets in principle determine the higher Toda brackets.