Timeline for What is a "block" in an abelian category?
Current License: CC BY-SA 2.5
5 events
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| Apr 28, 2010 at 22:06 | comment | added | Jim Humphreys | @Torsten: Your helpful added comment is in the spirit of Jantzen's discussion, using finiteness conditions. I'd like to avoid idempotents or centers, which can make it easier (and more interesting) in some special cases to find block decompositions and describe the blocks. Without indecomposables the whole notion of "block" does lose interest. Still I'm inclined to start with the most general language and then verify in special cases that the category is a direct sum of blocks even if they are hard to classify. I'd prefer not to solve that problem by ad hoc definition. | |
| Apr 24, 2010 at 13:21 | comment | added | Torsten Ekedahl | I do not see that what you say is true. Note that in the case of finite length categories two modules in different blocks have no common simple for their Jordan-Hölder factors so clearly there are no non-zero morphisms between them. | |
| Apr 24, 2010 at 9:27 | comment | added | JJH | In your definition, two indecomposable objects lie in the same block if there is no nonsplit extension; in the definition quoted by Noah, two indecomposable objects are in the same block if there is a nonzero morphism from the one to the other. It seems the two definitions are very different, so when and why they coincide? | |
| Apr 24, 2010 at 6:53 | history | edited | Torsten Ekedahl | CC BY-SA 2.5 |
Added elucidation of definition of blocks in the finite length case
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| Apr 24, 2010 at 5:33 | history | answered | Torsten Ekedahl | CC BY-SA 2.5 |