Let $\mathcal{F}$ and $\mathcal{G}$ be coherent analytic sheaves on $\mathbb{P}^n$. Let $\mathcal{F}_\bullet$ be a locally free resolution of $\mathcal{F}$. In Principles of Algebraic Geometry by Griffiths and Harris, $\text{Ext}^p(\mathcal{F},\mathcal{G})$ is defined as the hypercohomology of the complex $\mathcal{Hom}(\mathcal{F}_\bullet,\mathcal{G})$, i.e., the cohomology of the complex $\bigoplus_{p=k+\ell} C^k(\mathfrak{U},\mathcal{Hom}(\mathcal{F}_\ell,\mathcal{G}))$, see pages 705 and 446. Here $C^\bullet(\mathfrak{U},\mathcal{Hom}(\mathcal{F}_\ell,\mathcal{G}))$ denotes the Čech complex with respect to some affine open cover $\mathfrak{U}$ of $\mathbb{P}^n$.

If I understand correctly, the Yoneda pairing $$\text{Ext}^p(\mathcal{F},\mathcal{G}) \times \text{Ext}^q(\mathcal{G},\mathcal{H}) \rightarrow \text{Ext}^{p+q}(\mathcal{F},\mathcal{H})$$ should then be induced by the cup product in Čech cohomology. However, I fail to see precisely how this works out.

Edit: To clarify: What I am primarily interested in is how the Yoneda pairing can be expressed in terms of the concrete representatives when $\text{Ext}$ is realized as hypercohomology. In [HL, Section 10.1.1], there is an explicit cup product on hypercohomology, which then is said to induce a product $Ext^i(F^\bullet,G^\bullet)\otimes Ext^j(E^\bullet,F^\bullet) \to Ext^{i+j}(E^\bullet,G^\bullet)$. It is then also stated that "If we interpret $Ext^i(E^\bullet,F^\bullet)$ as $Hom_{\mathcal D}(E^\bullet,F^\bullet[i])$, where $\mathcal{D}$ is the derived category of quasi-coherent sheaves, then the cup product for Ext-groups is simply given by composition." To me, it is not clear neither how the cup product induces the product on $\text{Ext}$, nor why this product coincides with composition in the derived category. Any references to where this is discussed in more detail, or hints on how to prove this would be welcome.

[HL] Huybrechts, Lehn: The Geometry of Moduli Spaces of Sheaves

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    $\begingroup$ I think these questions become clearer if you go full derived and see $\mathrm{Ext}^p(\mathcal{F},\mathcal{G})$ as the cohomology of the internal hom in the derived category of quasi-coherent analytic sheaves. $\endgroup$ Mar 11, 2020 at 11:34

1 Answer 1


Suppose Č(U)→Hom(Q(F),G)[p] and Č(V)→Hom(Q(G),H)[q] represent elements in Ext^p(F,G) and Ext^q(G,H). Here Č(U) denotes the Čech chains for an open cover U and Q(-) is the functor that computes appropriate replacements (often cofibrant), e.g., locally free resolutions.

Choose a common refinement W of U and V and restrict the above maps along the morphisms Č(W)→Č(U) and Č(W)→Č(V). Tensor the two resulting maps together and restrict along the diagonal map, obtaining a map Č(W)→Hom(Q(F),G)⊗Hom(Q(G),H)[p+q]

The remaining problem is the mismatch between G and Q(G). The canonical map Q(G)→G goes in the wrong direction, and so does the induced map Hom(Q(F),Q(G))→Hom(Q(F),G). This prevents us from composing things in the most obvious way.

If one is not interested in specific models for representatives, one can observe that Hom(Q(F),Q(G))→Hom(Q(F),G) becomes an isomorphism in the derived category, so can be tensored with Hom(Q(G),H) and then inverted and composed with the other map, yielding a map Č(W)→Hom(Q(F),H)[p+q], as desired.

Otherwise, working in (say) the projective or flat model structure one can explicitly lift the cofibration 0→Č(U) against the acyclic fibration Hom(Q(F),Q(G))[p]→Hom(Q(F),G)[p], obtaining a new representative Č(U)→Hom(Q(F),Q(G))[p], and then apply the second paragraph above to get the desired map Č(W)→Hom(Q(F),H)[p+q].


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