Timeline for What is a "block" in an abelian category?
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
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| Apr 28, 2010 at 22:18 | vote | accept | Jim Humphreys | ||
| Apr 28, 2010 at 22:18 | comment | added | Jim Humphreys | @Noah: I prefer the generality of the Comes-Ostrik viewpoint, as explained in my comment to Torsten. My question was motivated by the unsolved problem of determining blocks for the parabolic subcategories of the BGG category, if "block" is defined in a general way. It's tempting to solve such a problem just by giving a definition to fit the situation. Given the finiteness properties in the BGG case one wants a parabolic subcategory to be a direct sum of indecomposable subcategories fitting a general notion of block. Describing those subcategories may be tricky. | |
| Apr 25, 2010 at 13:21 | comment | added | Jim Humphreys | @Victor: Thanks for the clarification. It's good to have a notion of block in a general situation, but always with the understanding that (for instance) finiteness conditions on the category may lead to better results. For me it's been confusing to encounter inconsistent uses of the label "block", with some meanings looser than others. | |
| Apr 24, 2010 at 16:09 | comment | added | Victor Ostrik | Jim, we gave this definition for the same reason you are asking this question: different people mean different things when they are talking about blocks. The characteristic 0 assumption is irrelevant for this definition (but it is very important for other parts of our paper). Our category in question was not abelian, so we had no option to talk about simple objects. And I totally agree with Torsten: this definition is not very reasonable for categories where Krull-Schmidt theorem fails. | |
| Apr 24, 2010 at 13:17 | comment | added | Jim Humphreys | It's hard to say what is a "standard reference", which partly motivates my question. The approach of Comes-Ostrik seems reasonable, leaving aside their underlying field of characteristic 0. They emphasize indecomposable objects and morphisms rather than simple objects and extensions; both versions amount to the same thing in familiar examples, but their version may be more flexible. | |
| Apr 24, 2010 at 0:17 | history | answered | Noah Snyder | CC BY-SA 2.5 |