Reading the proof in Bondal-Orlov reconstruction theorem (http://arxiv.org/pdf/alg-geom/9712029v1.pdf), I found the spectral sequence in the title

$E_2^{p,q}=\bigoplus_{k-j=q}\mathrm{Ext}^p(\mathcal{H}^j,\mathcal{H}^k)\Rightarrow \mathrm{Hom}^{p+q}(P,P)$

where $P$ is a point-like object and the $\mathcal{H}^j$'s are the cohomology sheaves of the complex $P$.

Is it a special case of some other spectral sequence? If not, where could I find a reference for it?

One thing I know is that all the cohomology sheaves $\mathcal{H}^j$ are supported at the same single point $x$.

Thank you


I was thinking whether it is possible to get it (or something similar, since the indexing conventions might be different between my other resource, i.e. Huybrecht's Fourier Mukai transforms in algebraic geometry) from the two spectral sequences \begin{equation} E^{p,q}_2=\mathrm{Hom}(A,\mathcal{H}^q(B)[p])\Rightarrow\mathrm{Hom}(A,B[p+q]) \end{equation} and \begin{equation} E^{p,q}_2=\mathrm{Hom}(\mathcal{H}^{-q}(A),B[p])\Rightarrow\mathrm{Hom}(A,B[p+q]) \end{equation} The rough idea was to consider \begin{equation} \mathrm{Hom}(\mathcal{H}^{-k},\mathcal{H}^j[p])\Rightarrow\mathrm{Hom}(P,\mathcal{H}^j[p+k]) \Rightarrow \mathrm{Hom}(P,P[p+j+k]) \end{equation} and then take the direct sum for $j+k=q$ and argue that \begin{equation} \bigoplus_{j+k=q}\mathrm{Hom}(\mathcal{H}^{-k},\mathcal{H}^j[p])\Rightarrow \mathrm{Hom}(P,P[p+q]) \end{equation}

I fear (and feel) it is not a correct approach (or at least it is not phrased correctly). Am I going in the right direction? If not, what could I do?


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