Rational algebraic 𝐾-theory of certain truncated polynomial rings

Abstract

Abstract.In this paper we derive a formula for rationalized algebraic A'-theory of certain overlings of rings of integers in number fields. Truncated polynomial algebras are examples. Our method is homological calculation which is facilitated by some basic rational homotopy theory and interpreted in terms of the cyclic homology theory of algebras invented by Alain Connes.The object of this paper is to compute, in terms of A. Connes' cyclic homology and the rational algebraic /^-theory of a ring of integers 0 in a number field the dual numbers over 0. Using a classical Lie algebra homology calculation based on invariant theory, but no cyclic homology, he found that for the dual numbers dimg^ = d, or = 0, if q is odd, or even. In §2 of this paper we sketch a computation in cyclic homology relevant to the case A = 0[T]/(T" + 1) following the steps of a computation of the rational algebraic ^-theory of the space CP" shown to us by T. Goodwillie. It turns out that dim V = n ■ d, or =0, depending on whether or not q is odd, or even.The proof of this theorem, together with the proof of the theorem of Loday and Quillen relating Lie algebra homology and cyclic homology, actually gives when Z = 0 a chain of natural isomorphisms linking the rationalized relative A"-group Kq+X(A -^Z) = 7r9(fibre(BGL + (/l) -» BGL(Z)+)) ® Q

On the algebraic K-theory of truncated polynomial algebras in several variables

“…This generalizes results of Soulé [14] and Staffeldt [15] in the single variable case. Note that the difference between K * (A) for k = Z and k = Q is torsion, so Theorem 1.4 also gives the Poincaré series of K * (Q[x 1 , .…”

On theK-theory of truncated polynomial algebras over the integers

“…Here T is the multiplicative group of complex numbers of modulus 1, C r ⊂ T is the finite subgroup of the indicated order, λ is a finitedimensional complex T-representation, and S λ is the one-point compactification of λ. Since the groups K q (Z[x]/(x m ), (x)) and TR r q −λ (Z) are finitely generated by [13,14] and Lemma 1.3, respectively, these earlier results amount to a long exact sequence…”