来自杂志 Advances in Mathematics 的文献。
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林海onrush
(2023-08-01 00:03):
#paper,doi.org/10.1016/j.aim.2023.109194,Equivariant algebraic K-theory, G-theory and
derived completions,论文研究的是群作用下的代数K理论的补全问题。主要内容可概括如下:
文章主要研究线性代数群作用在方案上的等ivariant代数K理论和G理论。目标是证明一个类似Atiyah-Segal在拓扑K理论中补全定理的结果。衍生补全的技术对此问题非常关键。Thomason在80年代就预测到需要一种同伦类的补全方法。本文使用第一作者2008年提出的衍生补全方法。Robert Thomason在建立与Atiyah-Segal对应的等变代数K理论的完备性定理时，发现了强限制性条件过于严格。他对等变代数G理论的情况提出了一个猜想，即对于线性代数群在概型上的作用，存在一个类似Atiyah和Segal的完备性定理，而不需要他之前证明的强限制性条件，这些条件也出现在原始的Atiyah-Segal定理中。
本文的主要目标是在尽可能广泛的背景下，利用导出完备性技术，对该猜想进行证明，并考虑几个应用。解决方案足够广泛，允许所有线性代数群的作用，无论它们是否连通，并作用于任何有限型域上的准投影概型，无论它们是否正则或投影。因此，可以考虑大类的变体的等变代数G理论，例如所有的齐次概型（由一个齐次环作用的情况）和所有球状概型（由一个约化群作用的情况）。通过限制为分裂齐次概型的作用，还可以考虑对代数空间的作用。此外，通常也不需要将基概型限制为域，但主要是为了简化部分阐述。这使得可以得到广泛的应用，其中一些被简要概述，并计划在将来详细探讨。实际上，我们在续篇中讨论了将结果扩展到等变同伦K理论以及各种Riemann-Roch定理。
通过将结果与先前已知的没有使用导出完备性的结果进行比较，可以看出如果不使用导出完备性，只能得到非常限制性的结果。
Abstract:
In the mid 1980s, while working on establishing completion theorems for equivariant Algebraic K-Theory similar to the well-known Atiyah-Segal completion theorem for equivariant topological K-theory, the late Robert Thomason found …
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In the mid 1980s, while working on establishing completion theorems for equivariant Algebraic K-Theory similar to the well-known Atiyah-Segal completion theorem for equivariant topological K-theory, the late Robert Thomason found the strong finiteness conditions that are required in such theorems to be too restrictive. Then he made a conjecture on the existence of a completion theorem in the sense of Atiyah and Segal for equivariant algebraic G-theory, for actions of linear algebraic groups on schemes that holds without any of the strong finiteness conditions that are required in such theorems proven by him, and also appearing in the original Atiyah-Segal theorem. The main goal of the present paper is to provide a proof of this conjecture in as broad a context as possible, making use of the technique of derived completion, and to consider several of the applications. Our solution is broad enough to allow actions by all linear algebraic groups, irrespective of whether they are connected or not, and acting on any quasi-projective scheme of finite type over a field, irrespective of whether they are regular or projective. This allows us therefore to consider the equivariant algebraic G-Theory of large classes of varieties like all toric varieties (for the action of a torus) and all spherical varieties (for the action of a reductive group). Restricting to actions by split tori, we are also able to consider actions on algebraic spaces. Moreover, the restriction that the base scheme be a field is also not required often, but is put in mainly to simplify some of our exposition. These enable us to obtain a wide range of applications, some of which are briefly sketched and which we plan to explore in detail in the future. In fact, we discuss an extension of our results to equivariant homotopy K-theory along with various Riemann-Roch theorems in a sequel. A comparison of our results with previously known results, none of which made use of derived completions, shows that without the use of derived completions one can only obtain results which are indeed very restrictive.
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