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We investigate the homological ideal $\mathfrak{J}_G^H$, the kernel of therestriction functors in compact Lie group equivariant Kasparov categories.Applying the relative homological algebra developed by Meyer and Nest, werelate the Atiyah-Segal completion theorem with the comparison of$\mathfrak{J}_G^H$ with the augmentation ideal of the representation ring.  In relation to it, we study on the Atiyah-Segal completion theorem forgroupoid equivariant $\mathrm{KK}$-theory, McClure's restriction map theorem,permanence property of the Baum-Connes conjecture under extensions of groupsand a class of $\mathfrak{J}_G$-injective objects coming from$\mathrm{C}^*$-dynamical systems, continuous Rokhlin property.

A categorical perspective on the Atiyah-Segal completion theorem in $\mathrm{KK}$-theory