_{∞}-ring structures for Tate spectra
_{∞}-ring structures for Tate spectra
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泰特光谱的_{∞}-环结构
DOI：
10.1090/s0002-9939-96-03194-2
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发表时间：
1996
影响因子：
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通讯作者：
J. McClure
中科院分区：
文献类型：
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作者：
J. McClure
令 G 为紧李群，kG 为 G 谱（如 [3，第 I.2 节] 中定义）。 Greenlees 和 May ([2]) 定义了一个相关的 G 谱 t(kG)，称为 Tate 谱。他们观察到，如果 kG 是环 G 谱，则 t(kG) 上存在诱导环 G 谱结构，并且如果 kG 是同伦交换的，则 t(kG) 也将是同伦交换的（参见 [2，命题 3.5]）。因此，很自然地要问 kG 上的等变 E∞ 环结构是否会导致 t(kG) 上的等变 E∞ 环结构（我们稍后会回忆一下定义）。我们对这个问题提供正面和负面的答案。从积极的一面来看，我们表明 t(kG) 继承的结构比等变 E∞ 环结构稍弱，但对于大多数实际目的来说应该足够了。为了解释这一点，让我们回顾一下[3，示例 VII.1.4]，每个 G 宇宙 U 都与一个等变操作数 L(U) 相关联。让我们固定一个完整的 G 宇宙 U，并让 V 表示平凡的 G 宇宙 U。等变 E∞-环结构被定义为等价于 L(U) 的等变操作数的动作（参见 [3，定义 VII.2.1 和 VII.1.2 以及备注 VII.1.3]）。让我们将 E′ ∞ 环结构定义为等效于 L(V ) 的等变操作数的动作；由于 G 对 L(V ) 的作用微不足道，我们可以通过说 E′ ∞ 环结构是通过 G 映射操作的非等变 E∞ 的作用来重新表述这一点。由于存在操作数 L(V ) → L(U) 的映射，因此等变 E∞ 结构专用于 E′ ∞ 结构。另一方面，[3]的Remark VII.2.5表明，如果kG是E ′ ∞环谱，则定点谱(kG) H具有（非等变的）E∞环结构，该结构随着H的变化而保持一致；这可能是与应用程序最相关的一点。我们的积极成果是：
Let G be a compact Lie group and kG a G spectrum (as defined in [3, Section I.2]). Greenlees and May ([2]) have defined an associated G-spectrum t(kG) called the Tate spectrum. They observe that if kG is a ring G-spectrum then there is an induced ring G-spectrum structure on t(kG), and that if kG is homotopy-commutative then t(kG) will also be homotopycommutative (see [2, Proposition 3.5]). It is therefore natural to ask whether an equivariant E∞-ring structure on kG induces an equivariant E∞-ring structure on t(kG) (we will recall the definition in a moment). We offer both positive and negative answers to this question. On the positive side, we show that t(kG) inherits a structure which is somewhat weaker than an equivariant E∞-ring structure, but which should be adequate for most practical purposes. To explain this, let us recall from [3, Example VII.1.4] that to each G-universe U is associated an equivariant operad L(U). Let us fix a complete G universe U and let V denote the trivial G-universe U. An equivariant E∞-ring structure is defined to be an action of an equivariant operad equivalent to L(U) (see [3, Definitions VII.2.1 and VII.1.2 and Remark VII.1.3]). Let us define an E′ ∞-ring structure to be an action of an equivariant operad equivalent to L(V ); since G acts trivially on L(V ) we can rephrase this by saying that an E′ ∞-ring structure is an action of a nonequivariant E∞ operad through G-maps. Since there is a map of operads L(V ) → L(U), an equivariant E∞ structure specializes to an E′ ∞ structure. On the other hand, Remark VII.2.5 of [3] shows that if kG is an E ′ ∞-ring spectrum then the fixed point spectra (kG) H have (nonequivariant) E∞-ring structures which are consistent as H varies; this is likely to be the point most relevant for applications. Our positive result is: