Reference for algebro-geometric definition of the $\cup$-product in (sheaf) cohomology.

Question

Can anyone give me a reference on how the $\cup$-product of sheaf cohomology is defined? I read somewhere that it has to do with the Yoneda pairing of Ext, but my naive approach did not work, because $H^i(X,\mathcal{F}) \cong \text{Ext}^i(\mathcal{O}_X,\mathcal{F})$ and $H^j(X,\mathcal{G}) \cong \text{Ext}^j(\mathcal{O}_X, \mathcal{G})$ do not pair up.

Answer 1

There is a really concrete definition without Yoneda product, so I write it just in case.

Pick an affine cover $\mathfrak U$ of $X$. An element of $H^i(X,F)$ can be though as a cocycle $(f_{I})$ where $f_I \in \Gamma(F, U_I)$ (I hope the abuse of notation is clear), and similarly for an element of $H^j(X,G)$. You can now simply define the product as $(f_I) \cup (g_J) = (f_I \otimes g_J)$ when you only consider the disjoint pairs of index $I,J$. This is by definition in $H^{i+j}(X, F \otimes G)$.

If you want to use the Yoneda product, there is a natural map $$ H^i(X,F) \otimes H^j(X,G) = Ext^i(\mathcal O_X,F) \otimes Ext^j(\mathcal O_X, G) \to Ext^i(\mathcal O_X, F) \otimes Ext^j(F, G \otimes F)$$ and you can apply the Yoneda product to the last term.

Remark : I think the two definitions coincide but I never see where it was written down formally.