Equivariant Algebraic K-Theory and Derived completions II: the case of Equivariant Homotopy K-Theory and Equivariant K-Theory
arxiv(2024)
摘要
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. In an
earlier work by the first two authors, we solved this conjecture by providing a
derived completion theorem for equivariant G-theory. In the present paper, we
provide a similar derived completion theorem for the homotopy Algebraic
K-theory of equivariant perfect complexes, on schemes that need not be regular.
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 normal
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 Homotopy Algebraic K-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). With finite coefficients invertible in the
base fields, we are also able to obtain such derived completion theorems for
equivariant algebraic K-theory but with respect to actions of diagonalizable
group schemes. These enable us to obtain a wide range of applications, several
of which are also explored.
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