## 15.66 Spectral sequences for Ext

In this section we collect various spectral sequences that come up when considering the Ext functors. For any pair of objects $L$, $K$ of the derived category $D(R)$ of a ring $R$ we denote

according to our general conventions in Derived Categories, Section 13.27.

For $M$ an $R$-module and $K \in D^+(R)$ there is a spectral sequence

and if $K$ is represented by the bounded below complex $K^\bullet $ of $R$-modules there is a spectral sequence

These spectral sequences come from applying Derived Categories, Lemma 13.21.3 to the functor $\mathop{\mathrm{Hom}}\nolimits _ R(M, -)$.

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