K理论与同调

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[1] arXiv:2508.03477 (交叉列表自 math.KT) [中文pdf, pdf, html, 其他]

标题： 计算分列精确代数$KK$-理论中的$K$-同调$K$-理论乘积 显示英文标题

标题： Computing the $K$-homology $K$-theory product in splitexact algebraic $KK$-theory

Bernhard Burgstaller

主题： K理论与同调 (math.KT) ; 算子代数 (math.OA)

显式公式表明，可以计算分层一元素$z \in KK^G(A,{\bf C})$与分裂精确代数$KK^G$-理论中的任何元素$w \in KK^G({\bf C},B)$的乘积$z \cdot w$，或者$KK^G$-理论中的$C^*$-代数，具有非常特殊的$G$-作用。 我们还通过验证从代数分拆精确 $KK$-理论到 $kk$-理论的函子的存在性，使这些乘积对线性分拆半精确 $kk$-理论可访问。 显示英文摘要

Explicit formulas are indicated that compute the product $z \cdot w$ of a level-one element $z \in KK^G(A,{\bf C})$ and any element $w \in KK^G({\bf C},B)$ in splitexact algebraic $KK^G$-theory, or $KK^G$-theory for $C^*$-algebras, with very special $G$-actions. We also make such products accessible to linear-split half-exact $kk$-theory by verifying the existence of a functor from algebraic splitexact $KK$-theory to $kk$-theory.

[2] arXiv:2508.03621 (交叉列表自 math.KT) [中文pdf, pdf, html, 其他]

标题： 一个真实的$G$-谱对于剪切和粘贴$K$-理论的$G$-流形 显示英文标题

标题： A genuine $G$-spectrum for the cut-and-paste $K$-theory of $G$-manifolds

Maxine Calle, David Chan

评论： 15页，欢迎提出意见！

主题： K理论与同调 (math.KT) ; 代数拓扑 (math.AT)

最近的工作已将剪切相容性$K$-理论应用于研究流形的经典剪切和粘贴($SK$)不变量。 本文证明了这样一个猜想，即等变$SK$-流形的平方$K$-理论作为真正的$G$-谱的空间固定点出现。 我们的方法利用了谱Mackey函子的框架，作为真正$G$-谱的模型，我们主要的技术结果是使用平方$K$-理论构建谱Mackey函子的一般过程。 显示英文摘要

Recent work has applied scissors congruence $K$-theory to study classical cut-and-paste ($SK$) invariants of manifolds. This paper proves the conjecture that the squares $K$-theory of equivariant $SK$-manifolds arises as the fixed points of a genuine $G$-spectrum. Our method utilizes the framework of spectral Mackey functors as models for genuine $G$-spectra, and our main technical result is a general procedure for constructing spectral Mackey functors using squares $K$-theory.

替换提交 (展示 2 之 2 条目 )

[3] arXiv:2304.00527 (替换) [中文pdf, pdf, html, 其他]

标题： G-维数对于交换DG环上的DG模 显示英文标题

标题： G-dimensions for DG-modules over commutative DG-rings

Jiangsheng Hu, Xiaoyan Yang, Rongmin Zhu

评论： 20页，将发表于《爱丁堡数学学会会刊》

主题： 交换代数 (math.AC) ; K理论与同调 (math.KT) ; 环与代数 (math.RA)

我们定义并研究了在非正分次交换诺特DG环$A$上的DG模的G-维数概念。通过应用Minamoto引入的DG投影分解的DG版本[以色列数学杂志 245 (2021) 409-454]，给出了DG模的G-维数有限性的某些条件。此外，证明了G-维数的有限性刻画了$A$的局部Gorenstein性质。应用分为三个方向。第一个是建立G-维数与 &\mathcal{A}& 的小有限维数之间的联系。第二个是通过最大局部-Cohen-Macaulay DG模类与DG模的一个特殊G类之间的关系来刻画Cohen-Macaulay和Gorenstein DG环。第三个是将经典的Buchweitz-Happel定理及其逆从交换诺特局部环扩展到交换诺特局部DG环的设置中。我们的方法与经典交换环的方法有些不同。 显示英文摘要

We define and study a notion of G-dimension for DG-modules over a non-positively graded commutative noetherian DG-ring $A$. Some criteria for the finiteness of the G-dimension of a DG-module are given by applying a DG-version of projective resolution introduced by Minamoto [Israel J. Math. 245 (2021) 409-454]. Moreover, it is proved that the finiteness of G-dimension characterizes the local Gorenstein property of $A$. Applications go in three directions. The first is to establish the connection between G-dimensions and the little finitistic dimensions of &\mathcal{A}&. The second is to characterize Cohen-Macaulay and Gorenstein DG-rings by the relations between the class of maximal local-Cohen-Macaulay DG-modules and a special G-class of DG-modules. The third is to extend the classical Buchwtweiz-Happel Theorem and its inverse from commutative noetherian local rings to the setting of commutative noetherian local DG-rings.Our method is somewhat different from classical commutative ring.

[4] arXiv:2411.03257 (替换) [中文pdf, pdf, html, 其他]

标题： 谱弗洛尔理论和切向结构 显示英文标题

标题： Spectral Floer theory and tangential structures

Noah Porcelli, Ivan Smith

评论： 欢迎提出意见！v3：接受的版本，包含了审稿人的意见和建议

主题： 辛几何 (math.SG) ; 代数拓扑 (math.AT) ; K理论与同调 (math.KT)

在\cite{PS}中，对于一个稳定框架的黎曼流形$X$，我们定义了一个球面谱上的 Donaldson-Fukaya 范畴$\mathcal{F}(X;\mathbb{S})$，并发展了一种将拟同构从$\mathcal{F}(X;\mathbb{Z})$提升到$\mathcal{F}(X;\mathbb{S})$的障碍理论。 这里，我们为任何“分级切向对”$\Theta \to \Phi$定义了一个谱Donaldson-Fukaya范畴，这些空间位于$BO \to BU$上，其对象是切丛的分类映射提升到$\Theta \to \Phi$的拉格朗日子流形$L\to X$。 之前的案例对应于$\Theta = \Phi = \{\mathrm{pt}\}$。 我们将我们的障碍理论扩展到这种情形。 调整$\Theta$和$\Phi$的选择的灵活性增加了可以消除障碍的情况范围，这在相应的 bordism 理论中对 Lagrangian 嵌入的 bordism 类有应用$\Omega^{(\Theta,\Phi),\circ}_*$。我们包含了一个关于在环谱$R$上（精确）谱 Floer 理论何时存在的自含讨论，这可能具有独立的兴趣。 显示英文摘要

In \cite{PS}, for a stably framed Liouville manifold $X$ we defined a Donaldson-Fukaya category $\mathcal{F}(X;\mathbb{S})$ over the sphere spectrum, and developed an obstruction theory for lifting quasi-isomorphisms from $\mathcal{F}(X;\mathbb{Z})$ to $\mathcal{F}(X;\mathbb{S})$. Here, we define a spectral Donaldson-Fukaya category for any `graded tangential pair' $\Theta \to \Phi$ of spaces living over $BO \to BU$, whose objects are Lagrangians $L\to X$ for which the classifying maps of their tangent bundles lift to $\Theta \to \Phi$. The previous case corresponded to $\Theta = \Phi = \{\mathrm{pt}\}$. We extend our obstruction theory to this setting. The flexibility to `tune' the choice of $\Theta$ and $\Phi$ increases the range of cases in which one can kill the obstructions, with applications to bordism classes of Lagrangian embeddings in the corresponding bordism theory $\Omega^{(\Theta,\Phi),\circ}_*$. We include a self-contained discussion of when (exact) spectral Floer theory over a ring spectrum $R$ should exist, which may be of independent interest.