Since the last updateon July 1st we have added

- an introductory chapter on algebraic stacks,
- a short chapter on Brauer groups of fields,
- a chapter on cohomology of sheaves on algebraic spaces,
- a chapter on adequate modules on schemes as discussed in this and this post,
- a chapter on sheaves on stacks, following the layout suggested in this post,
- a chapter on cohomology on algebraic stacks which contains a discussion of functoriality for quasi-coherent sheaves on algebraic stacks including higher direct images for quasi-compact and quasi-separated morphisms.

Let’s discuss the last topic a bit. We use *locally quasi-coherent* sheaves (sheaves that we called “quasi quasi-coherent” in this post) as an essential technical tool to prove the results. We also think about parasitic modules, which was a hint in an email of Martin Olsson. It turns out that the category of quasi-coherent modules is the quotient of the category of locally quasi-coherent modules satisfying the flat base change condition by the subcategory of parasitic ones. Then one can proceed as discussed in the post on adequate modules. This is not precisely how the results are stated, since the description of the category of a quasi-coherent sheaves as a quotient category isn’t needed. The main result at the moment is Proposition Tag 077A.

Enjoy!

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