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Let $\mathcal{C}$ be a site on the category of schemes for some Grothendieck topology, $X$ a scheme, and let $F,G$ be two sheaves of abelian groups on $\mathcal{C}$. Do we have cup-product as follows? $$H^i(X,F)\otimes H^j(X,G)\rightarrow H^{i+j}(X,F\otimes_{\mathbb{Z}}G)$$ or even the relative version $$R^if_*F\otimes R^jf_*G\rightarrow R^{i+j}f_*(F\otimes_{\mathbb{Z}}G)$$ along a map $f:X\rightarrow Y$ of schemes.

For etale cohomology, it seems that Godement resolution is needed in the construction, see [Etale cohomology theory, the paragraph before Prop. 7,4,10] by Fu Lei, see also Appendix B of http://math.stanford.edu/~conrad/Weil2seminar/Notes/L12-13.pdf.

But in a general site (e.g. fppf site), we may not have Godement resolution. I checked the chapter "cohomology of sites" of Stack-Project, unfortunately, cup -product is in the list of topics that "should be discussed in this chapter, and have not yet been written"

It would be very helpful, if someone knows a reference.

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Yes. This is treated in detail in Section 8.4 of Jardine's book “Local homotopy theory”. See also the introduction to Chapter 8 there for a historical comment on cup products and Godement resolutions.

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  • $\begingroup$ The construction there is for simplicial sheaf. I guess I can consider a sheaf a a constant simplicial sheaf, right? $\endgroup$
    – Heer
    Commented Jun 12, 2019 at 13:11
  • $\begingroup$ Thanks for the reference. The content of the book is far out of my area, I hope more accessible (for me) answer could appear. $\endgroup$
    – Heer
    Commented Jun 12, 2019 at 13:13
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    $\begingroup$ @Heer: A cohomology class c∈H^i(X,F) is simply a derived map in the category of presheaves of chain complexes Z[X]→F[i]. This translates to a derived map of simplicial presheaves X→K(F,i). See ncatlab.org/nlab/show/abelian+sheaf+cohomology for a detailed description of the correspondence between simplicial presheaves and the older language of sheaf resolutions. $\endgroup$
    – Dmitri Pavlov
    Commented Jun 12, 2019 at 13:45
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    $\begingroup$ As Jardine says in the beginning of chapter 8, one approach to defining cup-products is via Verdier's hypercovering theorem. The Stacks project has a presentation of the hypercovering theorem which you might find useful (stacks.math.columbia.edu/tag/01FX). $\endgroup$
    – Jesse Silliman
    Commented Jun 12, 2019 at 15:40

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