Timeline for When do chain complexes decompose as a direct sum?
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 23, 2010 at 7:03 | vote | accept | Dylan Wilson | ||
| Jul 23, 2010 at 7:03 | comment | added | Dylan Wilson | Thank you! Excuse my inanity; I'm learning things sort of backwards here- looking at Balmer's paper before understanding all the constructions that inspired it :). Anyway, I don't think I'll be getting criteria more useful than the ones you gave, so I'll accept your answer. Thanks again! | |
| Jul 22, 2010 at 21:16 | comment | added | Greg Stevenson | There are two ways to view the homological support on $K^b(R\text{-}\mathrm{proj})$. As you mention one is given by Balmer's support on the spectrum of prime tensor ideals. For $A$ in $K^b(R\text{-}\mathrm{proj})$ we can also consider $\mathrm{supp}_R A = \cup_i \mathrm{supp}_R H^i(A)$ where we are just taking supports as $R$-modules, i.e., the set of primes $\mathfrak{p}$ such that $H^i(A)_{\mathfrak{p}} \neq 0$. These two notions agree, by the universality property of Balmer's support and the fact that homological support classifies thick subcategories. | |
| Jul 22, 2010 at 15:45 | comment | added | Dylan Wilson | Can you quickly remind me what homological support is? Is it just the collection of prime ideals of $K^b(R-proj)$ not containing an element, or is it something different? And I have Neeman's book, you don't need to hunt down the proof- I think it's rather straightforward. | |
| Jul 22, 2010 at 0:04 | history | answered | Greg Stevenson | CC BY-SA 2.5 |