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15.67 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

\[ \mathop{\mathrm{Ext}}\nolimits ^ n_ R(L, K) = \mathop{\mathrm{Hom}}\nolimits _{D(R)}(L, K[n]) \]

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

15.67.0.1
\begin{equation} \label{more-algebra-equation-first-ss-ext} E_2^{i, j} = \mathop{\mathrm{Ext}}\nolimits _ R^ i(M, H^ j(K)) \Rightarrow \mathop{\mathrm{Ext}}\nolimits _ R^{i + j}(M, K) \end{equation}

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

15.67.0.2
\begin{equation} \label{more-algebra-equation-second-ss-ext} E_1^{i, j} = \mathop{\mathrm{Ext}}\nolimits _ R^ j(M, K^ i) \Rightarrow \mathop{\mathrm{Ext}}\nolimits _ R^{i + j}(M, K) \end{equation}

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

Comments (3)

Comment #4834 by Weixiao Lu on

It seems that this section has not been finished yet?

Comment #5133 by on

Everybody is free to contribute https://stacks.math.columbia.edu/contribute

Comment #5141 by on

@#4834: OK, I have added a reference to how to construct these spectral sequences.

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