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It's definitely known that the derived category of ${\cal M}_{FG}$ and the stable homotopy category are not equivalent. This is an instance of

The Mahowald Uncertainty Principle: Any spectral sequence converging to the homotopy groups of spheres with an $E_2$-term that can be named using homological algebra will be infinitely far from the actual answer.

(The naming is due to Ravenel; this quote is from Paul Goerss' "The Adams-Novikov Spectral Sequence and the Homotopy Groups of Spheres".) There is often a feeling that stable homotopy theory always deviates from algebra as soon as is possible.

As Neil said, the Adams-Novikov spectral sequence starts with morphisms in the derived category and computes stable homotopy groups of spheres. Every place where this spectral sequence does not degenerate indicates a point where the stable homotopy category deviates from the derived categoryalgebraic approximation. This includes the following phenomena.

• Hidden additive extensions, such as the hidden additive extension making $\pi_3^s$ into $\mathbb{Z}/24$ rather than $\mathbb{Z}/12 \times \mathbb{Z}/2$.

• Hidden multiplicative extensions. In the (2-local) stable homotopy groups there are elements $\eta \in \pi_1^s$, $\nu \in \pi_3^s$, and $\sigma \in \pi_7^s$. My recollection is that such that $\eta^2 \sigma = \nu^3$ on the $E_2$-term, but Toda showed that this relationship doesn't hold on-the-nose in stable homotopy groups of spheres.

• Differentials. For any prime $p$, there is always a nontrivial differential in the Adams-Novikov spectral sequence, and the first differential is called the Toda differential.

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It's definitely known that the derived category of ${\cal M}_{FG}$ and the stable homotopy category are not equivalent. This is an instance of

The Mahowald Uncertainty Principle: Any spectral sequence converging to the homotopy groups of spheres with an $E_2$-term that can be named using homological algebra will be infinitely far from the actual answer.

(The naming is due to Ravenel; this quote is from Paul Goerss' "The Adams-Novikov Spectral Sequence and the Homotopy Groups of Spheres".) There is often a feeling that stable homotopy theory always deviates from algebra as soon as is possible.

As Neil said, the Adams-Novikov spectral sequence starts with morphisms in the derived category and computes stable homotopy groups of spheres. Every place where this spectral sequence does not degenerate indicates a point where the stable homotopy category deviates from the derived category. This includes the following phenomena.

• Hidden additive extensions, such as the hidden additive extension making $\pi_3^s$ into $\mathbb{Z}/24$ rather than $\mathbb{Z}/12 \times \mathbb{Z}/2$.

• Hidden multiplicative extensions. In the (2-local) stable homotopy groups there are elements $\eta \in \pi_1^s$, $\nu \in \pi_3^s$, and $\sigma \in \pi_7^s$. My recollection is that such that $\eta^2 \sigma = \nu^3$ on the $E_2$-term, but Toda showed that this relationship doesn't hold on-the-nose in stable homotopy groups of spheres.

• Differentials. For any prime $p$, there is always a nontrivial differential in the Adams-Novikov spectral sequence, and the first differential is called the Toda differential.