The Algebras $\mathcal{O}^{\prime} {\rm k}$ and k

Abstract

13.1.1. We preserve the setup of the previous chapter. Given v ∈ N[I]^a, we may regard ${\rm{V}} \mapsto K({Q_V})$ as a functor on the category $V_v^a$ with values in the category of O -modules. An isomorphism V ≅ V in $V_v^a$ induces an isomorphism Ev ≅ Ev compatible with the a-actions; this induces an isomorphism Qv ≅ Qv which induces an isomorphism K(Qv) ≅ K(Qv ) that is actually independent of the choice of the isomorphism V ≅ V by the equivariance properties of the complexes considered. Hence we may take the direct limit $\lim\limits_{\to_{\rm{v}}}$ K(Qv) over the category $V_v^a$. This direct limit is denoted by O kv By the previous discussion, the natural homomorphism K(Qv)→ O kv is an isomorphism for any V ∈ $V_v^a$.