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Let $X$ be a scheme (we can assume $X$ is smooth over a field $k$). Let $\mathcal F_1$ and $\mathcal F_2$ be two sheaves of abelian groups on $X$.

Is it always true that $H^i_{\text{flat}}(X, \mathcal F_1 \oplus \mathcal F_2) = H^i_{\text{flat}}(X, \mathcal F_1) \oplus H^i_{\text{flat}}(X, \mathcal F_2)$? If not in general, is it true in particular that $H^i_{\text{flat}}(X, \mu_{n} \times \mu_n) = H^i_{flat}(X, \mu_n ) \oplus H^i_{\text{flat}}(X, \mu_n )$, or if we assume extra conditions on $X$ (like quasi-compactness etc. …)?

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Am I missing something? It seems like this should be true for any cohomology functor:

We have inclusion maps $i_1:F_1 \rightarrow F_1 \oplus F_2, i_2:F_2\rightarrow F_1 \oplus F_2$ and projection maps $\pi_1: F_1 \oplus F_2 \rightarrow F_1, \pi_2:F_1 \oplus F_2 \rightarrow F_2$ such that $\pi_i \circ i_i = id$.

Considering the exact sequence $0 \rightarrow F_1 \rightarrow F_1 \oplus F_2 \rightarrow F_2 \rightarrow 0$,

we have from the long exact sequence in cohomology the exact sequence

$H^i(F_1) \rightarrow H^i(F_1 \oplus F_2) \rightarrow H^i(F_2)$,

where the first map is induced by $i_1$ and the second map by $\pi_2$. But by functoriality, we have $\pi^*_i \circ i^*_i = id$ so the map induced by $i_1$ must be an injection and the map induced by $\pi_2$ must be a surjection. So we actually have an exact sequence

$0 \rightarrow H^i(F_1) \rightarrow H^i(F_1 \oplus F_2) \rightarrow H^i(F_2) \rightarrow 0$

But again by functoriality, this splits, so the middle is the direct sum of the two ends.

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