Timeline for How does the Balmer spectrum fail to describe the algebraic geometry of categories of non-compact objects?
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Nov 4, 2022 at 19:27 | vote | accept | Doron Grossman-Naples | ||
| Nov 3, 2022 at 22:29 | answer | added | Doron Grossman-Naples | timeline score: 1 | |
| Nov 2, 2022 at 5:40 | history | edited | Doron Grossman-Naples | CC BY-SA 4.0 |
thick vs localizing subcategories
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| Nov 1, 2022 at 22:03 | comment | added | Denis T | Because colimits behave badly in unbounded category, most results using checking conditions on stalks in some way either become wrong, or start requiring workarounds. | |
| Nov 1, 2022 at 21:57 | comment | added | Denis T | I'd just say that the answer depends on whether you'd like to think about a complex of $\Bbb Z/4$'s with differential being multiplication by 2 as a zero object, or not. (more or less, whether you think about the category of acyclic complexes as a generalized Serre class, or as an approximation to possible "Bousfield class"-like structure). Balmer spectrum is useful to define a notion generalising support of cohomology for perfect complexes on qcqs scheme; some useful localisations of unbounded complexes fail in being either or both "qc" or "qs". Usual homotopy category fails in second regard. | |
| Nov 1, 2022 at 21:55 | comment | added | Maxime Ramzi | (note that Remark 2.18 in the linked survey of Balmer's indicates that smallness plays a role, as well as his mention of infinite coproducts) | |
| Nov 1, 2022 at 21:52 | comment | added | Maxime Ramzi | Isn't the problem more about smallness than compactness ? If you hand me an abstract category $C$, compactness is more or less meaningless as I can always view the objects of $C$ as compact in $Ind(C)$. I think smallness might be a bigger problem with respect to some of the fundamental results of tt-geometry. Also, if you're working in the "big" case, thick $\otimes$-ideals are typically not the "correct" thing to look at, you usually want to also assume that your ideals are closed under arbitrary coproducts | |
| Nov 1, 2022 at 21:44 | history | edited | Doron Grossman-Naples | CC BY-SA 4.0 |
clarified title
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| Nov 1, 2022 at 21:13 | history | asked | Doron Grossman-Naples | CC BY-SA 4.0 |