Timeline for Projective objects in chain complexes of an abelian category: Further question
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Nov 12, 2023 at 1:02 | comment | added | David White | Lastly, you wrote to view $C_*$ as a graded $\mathbb{Z}$-module, and I don't know what you mean by this. Usually when we say M is a graded R module then R is a graded ring: en.wikipedia.org/wiki/Graded_ring | |
| Nov 12, 2023 at 1:02 | comment | added | David White | It seems you are very set on the idea that every chain complex is equivalent to a graded $R$-module, and that $\bigoplus$ in the context of graded $R$-modules places a finiteness restriction. This is just not right. Show me where in Weibel the result you think is true is proven. If you can't, then go read Weibel's definition 1.1.1 and tell me why it rules out the example I gave you of a chain complex that has every $C_n$ nonzero. | |
| Nov 11, 2023 at 13:46 | comment | added | David White | I disagree with your previous two comments. This has nothing to do with Freyd-Mitchell. You wrote "If a chain complex 𝐶∙ is a $\mathbb{Z}$-graded 𝑅−module, then a typical element is non-zero only in finitely many degrees whether 𝐶∙ is unbounded or not" and I disagree. Consider the chain complex that is $C_n = \mathbb{Z}$ for all $n$, and $d_n = 0$ for all $n$. This is perfectly allowed by Weibel's Definition 1.1.1 | |
| Nov 10, 2023 at 1:25 | comment | added | locally trivial | Ah. So the direct sum $C_\bullet = \bigoplus_{n\in \mathbb{Z}} P(n)$ still contains the element $c = (..., c_{-1}, c_0, c_1, ... )$ which is non-zero in infinitely many degrees. Very good! Now we're clear! All good, thanks! | |
| Nov 10, 2023 at 0:40 | comment | added | David White | The answer to your first question is "yes." There's no finiteness restriction on a general chain complex $C_\bullet$. It's ok if it has all $C_i$ nonzero. | |
| Nov 9, 2023 at 23:11 | vote | accept | locally trivial | ||
| Nov 9, 2023 at 20:42 | comment | added | locally trivial | Does the chain complex $C_\bullet = \{C_i, \partial_i)$ contain the "element" $c = \{c_i \text{ }|\text{ } c_i \in C_i \text{ for each }i\in \mathbb{Z} \}$, or does it only contain the 'pure' elements $c$ such that $c\in C_i$ for some $i\in \mathbb{Z}$? (Do I fundamentally misunderstand what the elements of a chain complex are?) i.e., what is a typical element of $C = \{C_i, \partial_i\} = \bigoplus_{n\in \mathbb{Z}} P(n)$ | |
| Nov 9, 2023 at 20:18 | history | answered | David White | CC BY-SA 4.0 |