Based on Balmer's tensor triangular Chow group [2], we propose (Milnor)K-theoretic Chow groups of derived categories of schemes. These Milnor K-theoretic Chow groups recover the classical ones [6] for smooth projective varieties and can detect nilpotent, while the classical ones can't do. As an application, we extend Soul\'e's variant of Bloch-Quilled identification from smooth projective varieties to their trivial infinitesimal thickenings. This answers affirmatively a question by Green-Griffiths for trivial deformations, see Question 1.1 below.

中文翻译：

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扩展 Soulé 的 Bloch-Quillen 识别变体

基于 Balmer 的张量三角形 Chow 群 [2]，我们提出了 (Milnor)K 理论 Chow 群的派生类别方案。这些 Milnor K-theoretic Chow 群恢复了经典群 [6] 的平滑射影变体，并且可以检测幂零，而经典群则不能。作为一个应用，我们将 Soul\'e 的 Bloch-Quilled 识别变体从平滑的射影变体扩展到它们微不足道的无穷小增厚。这肯定地回答了 Green-Griffiths 提出的关于微不足道的变形的问题，请参见下面的问题 1.1。