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Register a new user$K$-theory of locally compact modules over orders
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Adam-Christiaan van Roosmalen
Publication date
2020
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English
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We present a quick approach to computing the $K$-theory of the category of locally compact modules over any order in a semisimple $mathbb{Q}$-algebra. We obtain the $K$-theory by first quotienting out the compact modules and subsequently the vector modules. Our proof exploits the fact that the pair (vector modules plus compact modules, discrete modules) becomes a torsion theory after we quotient out the finite modules. Treating these quotients as exact categories is possible due to a recent localization formalism.
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The K-theory of a functor may be viewed as a relative version of the K-theory of a ring. In the case of a Galois extension of a number field F/L with rings of integers A/B respectively, this K-theory of the norm functor is an extension of a subgroup of the ideal class group Cl(A) by the 0-Tate cohomology group with coefficients in A*. The Mayer-Vietoris exact sequence enables us to describe quite explicitly this extension which is related to the coinvariants of Cl(A) under the action of the Galois group. We apply these ideas to find results in Number Theory, which are known for some of them with different methods.
Let $H$ be a compact subgroup of a locally compact group $G$ and let $m$ be the normalized $G$-invariant measure on homogeneous space $G/H$ associated with Weils formula. Let $varphi$ be a Young function satisfying $Delta_2$-condition. We introduce the notion of left module action of $L^1(G/H, m)$ on the Orlicz spaces $L^varphi(G/H, m).$ We also introduce a Banach left $L^1(G/H, m)$-submodule of $L^varphi(G/H, m).$
We show that if $X$ is a toric scheme over a regular commutative ring $k$ then the direct limit of the $K$-groups of $X$ taken over any infinite sequence of nontrivial dilations is homotopy invariant. This theorem was previously known for regular commutative rings containing a field. The affine case of our result was conjectured by Gubeladze. We prove analogous results when $k$ is replaced by an appropriate $K$-regular, not necessarily commutative $k$-algebra.
We show that the K_{2i}(Z[x]/(x^m),(x)) is finite of order (mi)!(i!)^{m-2} and that K_{2i+1}(Z[x]/(x^m),(x)) is free abelian of rank m-1. This is accomplished by showing that the equivariant homotopy groups of the topological Hochschild spectrum THH(Z) are finite, in odd degrees, and free abelian, in even degrees, and by evaluating their orders and ranks, respectively.
A long standing problem, which has its roots in low-dimensional homotopy theory, is to classify all finite groups $G$ for which the integral group ring $mathbb{Z}G$ has stably free cancellation (SFC). We extend results of R. G. Swan by giving a condition for SFC and use this to show that $mathbb{Z}G$ has SFC provided at most one copy of the quaternions $mathbb{H}$ occurs in the Wedderburn decomposition of the real group ring $mathbb{R}G$. This generalises the Eichler condition in the case of integral group rings.
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