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Let $\mathcal{F}^{\bullet}$ and $\mathcal{G}^{\bullet}$ be complexes of coherent sheaves on a variety $X$. There is a spectral sequence $$E_2^{p,q}=\mathcal{Ext}^p(\mathcal{H}^q(\mathcal{F}^{\bullet}),\mathcal{G}^{\bullet}) \Longrightarrow \mathcal{Ext}^{p-q}(\mathcal{F}^{\bullet},\mathcal{G}^{\bullet})$$

Are the second differentials in this sequence $$d_2^{p,q}:\mathcal{Ext}^p(\mathcal{H}^q(\mathcal{F}^{\bullet}),\mathcal{G}^{\bullet}) \rightarrow \mathcal{Ext}^{p+2}(\mathcal{H}^{q-1}(\mathcal{F}^{\bullet}),\mathcal{G}^{\bullet})$$

or

$$d_2^{p,q}:\mathcal{Ext}^p(\mathcal{H}^q(\mathcal{F}^{\bullet}),\mathcal{G}^{\bullet}) \rightarrow \mathcal{Ext}^{p+2}(\mathcal{H}^{q+1}(\mathcal{F}^{\bullet}),\mathcal{G}^{\bullet})$$

Thanks in advance for any insight.

Luc
  • 751
  • Without looking into the details of this particular spectral sequence, I can say that indexing is generally chosen so that the differential is always of total degree $\pm 1$ (depending on if you are in homological or cohomological setup) and so if either answer is correct, it is the first one (as the second raises degree by 3). – Aaron Apr 04 '14 at 23:36
  • Dear @Luc, Could you explain the notation? For example it looks like sheaf ext but it's righthand argument is a complex. And what is $\mathcal{H}^q(\mathcal{F}^\bullet)$? – Keenan Kidwell Apr 04 '14 at 23:43
  • $\mathcal{H}^p(\mathcal{F}^{\bullet})$ is the p-th cohomology of the complex $\mathcal{F}^{\bullet}$. $\mathcal{Ext}^p(\mathcal{F}^{\bullet},\mathcal{G}^{\bullet})=R^p\mathcal{Hom}(\mathcal{F}^{\bullet},\mathcal{G}^{\bullet})$ This is Grothendieck's spectral sequence for the composition of the identity functor and $\mathcal{Hom}$ (see for instance pg. 76 in Huybrechts' "Fourier-Mukai transforms in Algebraic Geometry"). – Luc Apr 04 '14 at 23:55

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