Talk:Compact object (mathematics)

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Thanks, Wundzer for your recent edits including examples of categories without compact objects! (And overall, for your many great edits on my watchlist!) Here, I find the wording of the consecutive sentences concerning D_qc(X) a bit confusing. Do you want to state the following?:

• D_qc(X) is generally not compactly generated, even if X is qcqs.
• For X=BG_a, the category has only zero as a compact object. (Here, say for k=F_p, it would be nice to give the elementary explanation, wouldn't it?
• More generally, D_qc(X) has only zero as a compact object if X=BG for G as in the statement below.

Jakob.scholbach (talk) 08:08, 28 August 2020 (UTC)

Hi Jakob.scholbach, thanks for the comments. I re-ordered the statement and reworded some bits slightly. I think these edits made it much more coherent, but let me know if there are still any issues. Also, I'm not sure how they get the equivalence between the category of $\mathbb {G} _{a}$-modules and the category of locally small modules over the ring $R$ on page 7. I suspect this has something to do with representations of unipotent groups over finite fields, but I'm not familiar with this area. Do you have any direct references or direction for pursuing a citation of this material? Wundzer (talk) 19:18, 28 August 2020 (UTC)

Thanks; I have in turn made some minor reformulations. Yes, I think the basic issue is that the trivial Z/p-representation (with Z/p-coefficients) is not compact in the derived category, basically since group cohomology for cyclic groups is periodic. I unfortunately don't have a reference which rephrases this classical fact as saying that the trivial rep is not compact in such cases. Some related discussion is here, say, but I don't have an official reference right now. (On the positive side of things, in the paper by Ben-Zvi et. al that we already cite they explain why in char. 0, these problems disappear.) Jakob.scholbach (talk) 14:40, 31 August 2020 (UTC)