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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 $[T\otimes_{\mathbb Z_\ell}\mathbb F_\ell]=0$ in $K(G,\mathbb F_\ell)$?

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 $K(G,\mathbb F_\ell)$?

Let $G$ be a profinite group.

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

Let $K(G,\mathbb F_\ell)$ be the Grothendieck group of the derived category of $\mathbb F_\ell$-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 $[T\otimes_{\mathbb Z_\ell}\mathbb F_\ell]=0$ in $K(G,\mathbb F_\ell)$?

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 $[T\otimes_{\mathbb Z_\ell}\mathbb F_\ell]=0$ in $K(G,\mathbb F_\ell)$?

Let $G$ be a profinite group.

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

Let $K(G,\mathbb F_\ell)$ be the Grothendieck group of the derived category of $\mathbb F_\ell$-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 $K(G,\mathbb F_\ell)$?

Let $G$ be a profinite group.

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

Let $K(G,\mathbb F_\ell)$ be the Grothendieck group of the derived category of $\mathbb F_\ell$-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 $[T\otimes_{\mathbb Z_\ell}\mathbb F_\ell]=0$ in $K(G,\mathbb F_\ell)$?