The tangent complex of K-theory

Benjamin Hennion
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Abstract

We prove that the tangent complex of K-theory, in terms of (abelian) deformation problems over a characteristic 0 field k, is cyclic homology (over k). This equivalence is compatible with the $\lambda$-operations. In particular, the relative algebraic K-theory functor fully determines the absolute cyclic homology over any field k of characteristic 0. We also show that the Loday-Quillen-Tsygan generalized trace comes as the tangent morphism of the canonical map $BGL_\infty \to K$. The proof builds on results of Goodwillie, using Wodzicki's excision for cyclic homology and formal deformation theory a la Lurie-Pridham.
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k理论的正切复合体
我们证明了在特征0场k上的(阿贝尔)变形问题中,k理论的切复是循环同调的(在k上)。这个等价与$\lambda$ -运算相容。特别是,相对代数k -理论函子完全确定了特征为0的任意域k上的绝对循环同调。我们还证明Loday-Quillen-Tsygan广义迹是规范映射$BGL_\infty \to K$的切态射。这个证明建立在Goodwillie的结果之上,使用了Wodzicki对循环同调的剔除和Lurie-Pridham的形式变形理论。
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