Primary L-theory

Abstract

The various types of L-groups defined in Chaps. 26,27,30,32 (asymmetric, automorphism, endomorphism, isometric etc.) have primary analogues in which the endomorphism f is required to be such that p(f) = 0 for some polynomial p(z) in a prescribed class of polynomials. The primary L-groups feature in the L-theory analogues of the splitting theorems of Chap. 12

$${K_1}({s^{ - 1}}A[x]) = {K_1}(A[x]) \oplus End_0^s(A)$$

$${K_1}({\tilde S^{ - 1}}A[x]) = {K_1}(A) \oplus \tilde {End}_0^S(A)$$

$$End_0^s(A) = {K_0}(A) \oplus \tilde {End}_0^S(A)$$

which will will now be obtained, with S ⊂ A[x]a multiplicative subset with leading units and x ∈ S, and $\tilde S \subset A[x]$ the reverse multiplicative subset (12.15). The S-primary endomorphism L-groups LEnd ^* _S (A, є), $L\tilde {Edn}_S^*\left( {A, \in } \right)$ are defined for an involution-invariant S ⊂ A[x],such that

$$L_h^n({S^{ - 1}}A[x],\varepsilon ) = L_h^n(A[x],\varepsilon ) \oplus LEnd_S^n(A,\varepsilon )$$

$$L_h^n({{\tilde S}^{ - 1}}A[x],\varepsilon ) = L_h^n(A[x],\varepsilon ) \oplus L\mathop \sim \limits_{End_s^n} (A,\varepsilon ),$$

$$LEnd_S^n(A,\varepsilon ) = L_p^n(A,\varepsilon ) \oplus L\mathop \sim \limits_{End_s^n} (A,\varepsilon )$$

.