Ideals in triangulated categories: phantoms, ghosts and skeleta
Ideals in triangulated categories: phantoms, ghosts and skeleta
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DOI：
10.1006/aima.1998.1735
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发表时间：
1998-06
影响因子：
1.7
通讯作者：
Daniel Christensen
中科院分区：
文献类型：
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作者：
Daniel Christensen
我们开始表明，在一个三角范畴，指定一个投射类相当于指定一个理想I的态射与某些性质，如果我有这些性质，那么它的每一个权力。我们证明了一个射影类如何导致一个亚当斯谱序列，并给出了一些结果，这种谱序列的收敛和崩溃。我们用它来研究各种理想。在稳定同伦类别中，我们研究了幻影映射、骨架幻影映射、超幻影映射和幽灵。（鬼是一个映射，它导致同伦群的零映射。我们表明，鬼导致一个稳定的模拟的Lusternik-Schnirelmann类别的空间，我们计算这个稳定的模拟低维真实的射影空间。我们还给出了鬼和Hopf和Kervaire不变量问题之间的关系。对于A ∞环谱上的A ∞模，鬼谱序列是一个泛系数谱序列.从幻影投射类出发，我们导出了有限谱的滤图的广义Milnor序列，由此得出从X到Y的幻影映射群总是可以描述为alim <$1群.最后两节集中于代数例子。在阿贝尔范畴的导出范畴中，我们研究了诱导同调群的零映射的映射的理想，并为Kelly关于这类映射的合成的消失的一个结果找到了一个自然的背景.我们还解释了纯正合序列如何与幻映射在派生类的一个环，并给出一个例子表明，幻映射可以组成非平凡。
We begin by showing that in a triangulated category, specifying a projective class is equivalent to specifying an ideal I of morphisms with certain properties and that if I has these properties, then so does each of its powers. We show how a projective class leads to an Adams spectral sequence and give some results on the convergence and collapsing of this spectral sequence. We use this to study various ideals. In the stable homotopy category we examine phantom maps, skeletal phantom maps, superphantom maps, and ghosts. (A ghost is a map which induces the zero map of homotopy groups.) We show that ghosts lead to a stable analogue of the Lusternik–Schnirelmann category of a space, and we calculate this stable analogue for low-dimensional real projective spaces. We also give a relation between ghosts and the Hopf and Kervaire invariant problems. In the case ofA∞modules over anA∞ring spectrum, the ghost spectral sequence is a universal coefficient spectral sequence. From the phantom projective class we derive a generalized Milnor sequence for filtered diagrams of finite spectra, and from this it follows that the group of phantom maps fromXtoYcan always be described as alim←1group. The last two sections focus on algebraic examples. In the derived category of an abelian category we study the ideal of maps inducing the zero map of homology groups and find a natural setting for a result of Kelly on the vanishing of composites of such maps. We also explain how pure exact sequences relate to phantom maps in the derived category of a ring and give an example showing that phantoms can compose non-trivially.