The Stacks project

Remark 19.13.11. Let $\mathcal{A}$ be a Grothendieck abelian category. Let $M, K$ be objects of $D(\mathcal{A})$. For any choice of complex $M^\bullet $ representing $M$ we can use the filtration $F^ pM^\bullet = \tau _{\leq -p}M^\bullet $ and the discussion in Remark 19.13.8 to get a spectral sequence with

\[ E_1^{p, q} = \mathop{\mathrm{Ext}}\nolimits ^{2p + q}(H^ p(M), K) \]

This spectral sequence is independent of the choice of complex $M^\bullet $ representing $M$. After renumbering $p = -j$ and $q = i + 2j$ we find a spectral sequence $(E'_ r, d'_ r)_{r \geq 2}$ with $d'_ r$ of bidegree $(r, -r + 1)$, with

\[ (E'_2)^{i, j} = \mathop{\mathrm{Ext}}\nolimits ^ i(H^{-j}(M), K) \]

If $M \in D^-(\mathcal{A})$ and $K \in D^+(\mathcal{A})$ then $E_ r$ and $E'_ r$ are bounded and converge to $\mathop{\mathrm{Ext}}\nolimits ^{p + q}(M, K)$. If we use the filtration $F^ pM^\bullet = \sigma _{\geq p}M^\bullet $ then we get

\[ E_1^{p, q} = \mathop{\mathrm{Ext}}\nolimits ^ q(M^{-p}, K) \]

If $K \in D^+(\mathcal{A})$ and $M^\bullet $ is bounded above, then this spectral sequence is bounded and converges to $\mathop{\mathrm{Ext}}\nolimits ^{p + q}(M, K)$.

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