Let $K$ be a $p$-adic local field; $G = \mathop{\mathrm{Gal}}(\overline K | K)$; $\tau \in \{\text{HT}, \text{dR}, \text{crys}\}$, $B_\tau$ the corresponding period ring; $\mathop{\mathrm{Rep}}_{\mathbf Q_p}^\tau(G) \subseteq \mathop{\mathrm{Rep}}_{\mathbf Q_p}(G)$ the full subcategory of $B_\tau$-admissible representations. Is it true that $\mathbf K_0(\mathop{\mathrm{Rep}}_{\mathbf Q_p}(G)) = \mathbf K_0(\mathop{\mathrm{Rep}}_{\mathbf Q_p}^\tau(G))$?
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1$\begingroup$ Isn't the Grothendieck group of $K_0(Rep(G))$ just the free abelian group on the irreducible representations? Since there exist irreducible reps which are not $\tau$-admissible, these groups are different. $\endgroup$– David LoefflerCommented Oct 20, 2021 at 6:31
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