reduction of torsion modules

Question

Let $G$ be a profinite group.

Let $K(G,\mathbb Z_\ell)$ be the Grothendieck group of the derived category of finitely generated $\mathbb Z_\ell$-modules with continuous $G$-action.

Let $K(G,\mathbb F_\ell)$ be the Grothendieck group of the derived category of finitely generated $\mathbb F_\ell$-modules with continuous $G$-action.

The map $[T]\mapsto [T\otimes_{\mathbb Z_\ell}^L\mathbb F_\ell]$ defines a group homomorphism $K(G,\mathbb Z_\ell)\to K(G,\mathbb F_\ell)$ (called the reduction map).

Question: If $T$ is a $\mathbb Z_\ell$-modules with continuous $G$-action such that $T$ is killed by some power of $\ell$, do we have $[T\otimes^L_{\mathbb Z_\ell}\mathbb F_\ell]=0$ in $K(G,\mathbb F_\ell)$?

Answer 1

It is true. Take any finitely generated module $M$ and filter it by powers of $\ell$, namely consider the filtration $M_i = \ell^iM$. It suffice to prove the claim for the associated graded $\oplus_i \ell^i M / \ell^{i+1}M$. But all the graded pieces are $\ell$-torsion and for them $M / \ell M \cong M \cong M[\ell]$ so that $M / \ell M \cong Tor_0(M,\mathbb{F}_\ell) \cong Tor_1(M,\mathbb{F}_\ell) \cong M[\ell]$ and the Euler characteristic of the derived tensor product vanishes.