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On the relative Wall-Witt groups

Authors:	Yongjin Song

Institution:	(1) Department of Mathematics, Inha University, 402-751 Incheon, Korea

Abstract:	LetR* be a simplicial involutive ring. According to certain involutions onK(R) and_ε L R _∗, there are 1/2-local splittings $K(R_ ) \simeq K^S (R_ * ) \times K^a (R_ * )$ and $K(R_ * ) \simeq K^S (R_ * ) \times K^a (R_ * )$ . It is known 2] that_ε L _\ga ^α R _∗, the Wall-Witt group. SupposeI→R $I \to R \mathop \to \limits^f S$ S is a split extension of discrete involutive rings withI ²=0, andI is a freeS-bimodule. Then we have $K_n + 1(f) \otimes \mathbb{Q} \cong Prim_n \wedge ^ * M(I \otimes \mathbb{Q})$ and $K_n + 1(f) \otimes \mathbb{Q} \cong Prim_n \wedge ^ * M(I \otimes \mathbb{Q})$ . The trace map Tr: Prim_n∧*M(I ⊗ ℚ)→ $\bar W$ ₀ ρ _n;I ⊗ ℚ) is an isomorphism. We prove in Lemma 1 that the trace map Tr is ℤ/2-equivariant. In Theorem 2 we show that under a certain assumption the rational relative Wall-Witt group vanishes. Theorem 2 can be extended to a more general case (Theorem 3) by employing Goodwillie’s reduction technique 3]. This work was partially supported by KOSEF under Grant 923-0100-010-1.

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