通过Rees代数得到的抽象指标定理
An Abstract Index Theorem via Rees Algebras
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中文总结 AI 辅助
该研究基于滤过微分分次代数的Rees构造，发展纯代数框架得到抽象指标定理，将Getzler重标度技术与环空间陈示性类局域化公式结合，建立了相关代数结构的迹等式。
中文摘要 AI 辅助
我们基于滤过微分分次代数（FDGA）的Rees构造，发展了一套纯代数的指标型定理框架。除经典Rees模外，我们引入了适用于分析论证的光滑变体$C^\nu_{\boldsymbol{\tau}}A$，并研究迹及其逐点、按系数的延拓到Rees代数。主要结果（推论3.11）为抽象指标定理：给定带迹的兼容滤过微分分次结合代数组$A,B,G$，其中有态射$\boldsymbol{\tau}\boldsymbol{\tau}:A\to B$、作用$\rho:G\boldsymbol{\tau}A\to A$、态射$i:G\to B$，以及一个分次中心元素，则有$\text{tr}_B(e^{f_0+f_1})=\text{tr}_{\boldsymbol{\tau}\text{gr}A}(e^\boldsymbol{\tau}h)$，其中左侧为$B$中的迹，右侧为$A$的Laurent级数相伴分次$\boldsymbol{\tau}\text{gr}A$中的迹。这里$f_0+f_1$是曲率型元素，即对某奇次元素$\beta$，形如$d_B\beta+\beta^2$的元素，而$\boldsymbol{\tau}$和$h$是$\boldsymbol{\tau}\text{gr}A$中的特定元素。该形式体系以Getzler重标度技术及Ludewig与Yi推导的环空间陈示性类的局域化公式为模型。
英文摘要
We develop a purely algebraic framework for index-type theorems based on the Rees construction for filtered differential graded algebras (FDGAs). Alongside the classical Rees module we introduce a smooth variant $C^ω_{\mathcal{R}}A$, adapted to analytic arguments, and we study traces and their pointwise and coefficient-wise extensions to Rees algebras. The main result (Corollary 3.11) is an abstract index theorem: given a compatible datum of filtered differential graded associative algebras with traces $A$, $B$, $G$ with a morphism $ϕ\colon A\to B$, an action $ρ\colon G\otimes A\to A$ and a morphism $i\colon G\to B$, together with a graded-central element, one has $\mathrm{tr}_B(e^{f_0+f_1}) = \mathrm{tr}_{\widehat{\mathrm{gr}}A}(e^γh)$, where the left-hand side is the trace in $B$ and the right-hand side the trace in the Laurent series associated graded $\widehat{\mathrm{gr}}A$ of $A$. Here $f_0+f_1$ is a curvature-type element, i.e., an element of the form $d_Bβ+β^2$ for some odd-degree element $β$, while $γ$ and $h$ are certain elements in $\widehat{\mathrm{gr}}A$. The formalism is modelled on the Getzler rescaling technique and on the derivation of the localization formula for the loop space Chern character by Ludewig and Yi.
