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 $K(G,\mathbb F\_\ell)$?



reduction of torsion modules
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.


