I ask this question because of the following fact:
Let $R$ be a ring and $0 \to M_1 \to M_2 \to M_3 \xrightarrow{\alpha} M_4 \to 0$ be an exact sequence of $R$-modules. Suppose we know $Ext_R(M_1,R)$, $Ext_R(M_2,R)$, $Ext_R(M_3,R)$.
Then we can assemble the long exact sequences associated to the sequences $0 \to M_1 \to M_2 \to \ker \alpha \to 0\text{ and }0 \to \ker \alpha \to M_3 \xrightarrow{\alpha} M_4 \to 0$ to find $Ext^i_R(M_4,R)$.
Question: Is there is a standard spectral sequence that outputs the associated graded of $Ext^i_R(M_4,R)$, or degenerates to something that outputs $assgrd(Ext^i_R(M_4,R))$.
I am pretty familiar with the Serre spectral sequence, so feel free to be heavy on spectral sequence notation.