We study pro excision in algebraic K-theory, following Suslin--Wodzicki,Cuntz--Quillen, Corti~nas, and Geisser--Hesselholt, as well as Artin--Rees andcontinuity properties of Andr'e--Quillen, Hochschild, and cyclic homology. Ourkey tool is to first establish the equivalence of various pro Tor vanishingconditions which appear in the literature. Using this we prove that all idealsof commutative, Noetherian rings are pro unital in a certain sense, and showthat such ideals satisfy pro excision in $K$-theory as well as in cyclic andtopological cyclic homology. In addition, our techniques yield a strong form ofthe pro Hochschild--Kostant--Rosenberg theorem, an extension to general baserings of the Cuntz--Quillen excision theorem in periodic cyclic homology, and ageneralisation of the Feu{i}gin--Tsygan theorem.
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机译：我们研究了Suslin-Wodzicki，Cuntz-Quillen，Corti 〜nas和Geisser-Hesselholt以及Artin-Rees和Andr'e-Quillen的连续性之后的代数K理论中的临切Hochschild和循环同源。我们的关键工具是首先确定文献中出现的各种亲Tor消失条件的等价性。利用这一点，我们证明了所有可交换的Noether环理想在某种意义上都是亲统一的，并且证明了这些理想在$ K $理论以及循环和拓扑循环同源性中都满足于割除。此外，我们的技术产生了亲Hochschild-Kostant-Rosenberg定理的强形式，扩展了Cuntz-Quillen切除定理在周期循环同源性中的一般基本环，并对Fe u { i} gin进行了一般化--Tsygan定理。
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