Timeline for Quasi-isomorphism and homotopy equivalence between diagram of chain complex
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 11, 2019 at 17:32 | comment | added | HenrikRüping | you also run into completeness issues for this question. If you consider the filtered $k$-modules $k[x]$ and $k[[x]]$ (formal power series) (both filtered by powers of the ideal $(x)$), you get a map of filtered modules that induces an isomorphism on each subquotient and thus it also induces isomorphisms on the associated graded modules, but the two modules are not isomorphic. Viewing these modules as chain complexes concentrated in one degree gives to non-homotopy equivalent chain complexes, and a filtration preserving map between them which induces an iso on the associated graded. | |
| Nov 10, 2019 at 6:49 | comment | added | J.D.Chern | @R. van Dobben de Bruyn, Then then question is : can we build two chain complex of $k[x]$-module and a map $f$ between them, that $f$ induce quasi-isomorphism on associated graded pieces of $(x)$-adic filtration and $f$ itself is not a homotopy equivalence. May be this kind of example is easy? | |
| Nov 10, 2019 at 6:48 | comment | added | J.D.Chern | @R. van Dobben de Bruyn, thank you for your comment, I think there are also a small gap in your counterexample. I believe it is easy to give two chain of $k[x]$-module that is quasi-isomorphic but not homotopy equivalence, but if we want it to be a counterexample of my question, the associated graded pieces of (x)-adic filtration should be quasi-isomorphic. | |
| Nov 10, 2019 at 4:47 | comment | added | R. van Dobben de Bruyn | I expect there should be similar counterexamples for easier index categories, possibly already if $\mathscr J = (\bullet \to \bullet)$, but you'd need to think a little harder. | |
| Nov 10, 2019 at 4:46 | comment | added | R. van Dobben de Bruyn | If by 'diagram of chain complexes' you mean a chain complex in a functor category $[\mathscr J,\mathbf{Vec}_k]$ for some small category $\mathscr J$, then I suspect the answer is negative. For example for $\mathscr J$ the monoid $\mathbf N$ viewed as a one-object category, the category $[\mathscr J,\mathbf{Vec}_k]$ is equivalent to $\mathbf{Mod}_{k[x]}$. If you take the $(x)$-adic filtration, then the associated graded pieces are $k$-vector spaces, for which homotopy equivalence coincides with quasi-isomorphism. The same is not true for $k[x]$-modules, so this should give counterexamples. | |
| Nov 10, 2019 at 4:10 | review | First posts | |||
| Nov 10, 2019 at 7:00 | |||||
| Nov 10, 2019 at 4:05 | history | asked | J.D.Chern | CC BY-SA 4.0 |