In this post, we will address Ext groups (see 5.86 here) and the notion of the local-to-global spectral sequence. Some kinds of sheaves can be thought of as modules over rings of algebraic functions (and we have talked about them previously in Quill over here), namely (quasi)-coherent sheaves. Let us work with coherent sheaves. Given two sheaves $\mathcal{G}, \mathcal{H}$, we can define either global $Ext^n_X(\mathcal{G}, \mathcal{H})$, which is abelian group for each $n \in \mathbb{Z}$, where $X$ is a (regular) scheme or local $\underline{Ext}^n_{\mathcal{O}_X}(\mathcal{G}, \mathcal{H})$ which are not groups but sheaves of abelian groups.

Just like a projective resolution of a module, we have a locally free resolution of a sheaf. Given a coherent sheaf $\mathcal{G}$, we write a resolution by locally-free sheaves (bundles)

$$0 \to \mathcal{E}_n \to \mathcal{E}_{n-1} \to \cdots \to \mathcal{E}_0 \to \mathcal{G} \to 0$$

which is exact for some $n$. Then we apply $\underline{Hom}(-,\mathcal{H})$ to the locally-free resolution $$0 \to \underline{Hom}(\mathcal{E}_0, \mathcal{H}) \to \underline{Hom}(\mathcal{E}_1, \mathcal{H}) \to \cdots \to \underline{Hom}(\mathcal{E}_n, \mathcal{H}) \to 0$$ and take the cohomology of the following resolution so $$\underline{Ext}^n_{\mathcal{O}_X} \cong H^n (\underline{Hom}(\mathcal{E}_{\bullet}, \mathcal{H}))$$ this is exactly the local Ext sheaves $\underline{Ext}^n_{\mathcal{O}_X}$. Even though the locally-free resolution of the sheaf is not unique, the local Ext sheaves can be uniquely written.

Now, there is a very interesting spectral sequence which relates the global Ext and local Ext (sheaves), which is the local-to-global spectral sequence (in cohomological version) $$E^{p,q}_2 = H^p \left( X, \underline{Ext}^q_{\mathcal{O}_X} (\mathcal{G},\mathcal{H}) \right) \Rightarrow Ext^{p+q}_X (\mathcal{G}, \mathcal{H})$$ and this is an example of Grothendieck spectral sequence!

(*) The open-string spectra between D-branes can be computed using the Ext group. See here.

Postscript: 1) Anveshanā enters into its second year of publication, and the issue for January 2026 is up online to read here. 2) The Prelude to Schemes (PtoS) project was started a few months ago, and regular updates follow on this website. After homological algebra, I am writing notes for the Algebraic Geometry section.