非线性轨迹
Nonlinear Traces
摘要：
将同伦代数中的迹理论与导代数几何中的层理论相结合，导出了一般的不动点和特征公式。在高等代数的背景下，可对偶对象的维数（或霍赫希尔德同调）的形式主义为诸如欧拉示性数、陈氏特征标和群表示的特征标等经典概念提供了一个统一的框架。此外，维度的简单功能性性质澄清了著名的恒等式，并将其扩展到新的上下文。我们观察到，这是有利的计算尺寸，迹和他们的函直接在对应范畴的非线性几何设置，在那里他们直接识别（衍生版本）闭路空间，不动点轨迹和循环映射，分别。这就产生了通用的非线性形式的Grothendieck-Riemann-Roch定理、Atiyah-Bott-Lefschetz迹公式和Frobenius-Weyl特征公式。一方面，我们可以通过应用层理论，例如由Gaitsgory和Rozenblyum发展的相干层和D-模理论，作为对应范畴之外的函子（在拓扑域论的精神中）来线性化。这恢复了熟悉的经典身份，在家庭和没有任何光滑性或横截性的假设。另一方面，形式主义也适用于更高的范畴设置没有被捕获在一个线性的框架，如群作用的类别上的字符。
英文摘要：
We combine the theory of traces in homotopical algebra with sheaf theory in derived algebraic geometry to deduce general fixed point and character formulas. The formalism of dimension (or Hochschild homology) of a dualizable object in the context of higher algebra provides a unifying framework for classical notions such as Euler characteristics, Chern characters, and characters of group representations. Moreover, the simple functoriality properties of dimensions clarify celebrated identities and extend them to new contexts. We observe that it is advantageous to calculate dimensions, traces and their functoriality directly in the nonlinear geometric setting of correspondence categories, where they are directly identified with (derived versions of) loop spaces, fixed point loci and loop maps, respectively. This results in universal nonlinear versions of Grothendieck-Riemann-Roch theorems, Atiyah-Bott-Lefschetz trace formulas, and Frobenius-Weyl character formulas. On the one hand, we can then linearize by applying sheaf theories, such as the theories of coherent sheaves and D-modules, developed by Gaitsgory and Rozenblyum, as functors out of correspondence categories (in the spirit of topological field theory). This recovers the familiar classical identities, in families and without any smoothness or transversality assumptions. On the other hand, the formalism also applies to higher categorical settings not captured within a linear framework, such as characters of group actions on categories.
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