Timeline for Monoidal Dold–Kan correspondence for non-connected CDGA
Current License: CC BY-SA 4.0
16 events
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| Dec 8, 2022 at 20:21 | vote | accept | Grisha Taroyan | ||
| Dec 8, 2022 at 18:56 | history | edited | David White | CC BY-SA 4.0 |
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| Dec 8, 2022 at 18:56 | answer | added | David White | timeline score: 2 | |
| Nov 7, 2022 at 18:27 | comment | added | Grisha Taroyan | Hi David. Actually, I was able to find an ad hoc construction that allows passing between connected and non-connected algebras (basically taking a quotient by an augmentation ideal in degree 0), but it would be super-interesting to see a more high-tech answer:) | |
| Nov 7, 2022 at 15:32 | comment | added | David White | Hi Grisha. Sorry it has taken me so long to write an answer. We are hiring and it's been a ton of on-campus interview candidates, eating up pretty much all my free time. I haven't forgotten your question. | |
| Oct 31, 2022 at 17:43 | history | edited | Grisha Taroyan | CC BY-SA 4.0 |
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| Oct 31, 2022 at 17:42 | history | edited | LSpice | CC BY-SA 4.0 |
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| Oct 31, 2022 at 17:40 | history | edited | Grisha Taroyan | CC BY-SA 4.0 |
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| Oct 31, 2022 at 17:25 | comment | added | Ben Wieland | Connective means vanishing in negative degrees. Connected means connective and one dimensional in degree 0. It is rarely used, but came first. | |
| Oct 31, 2022 at 16:26 | comment | added | David White | Hi Grisha. Ok. I got it. I will write up an answer tonight. There is no obstruction. Everything works in characteristic zero. The paper I linked above proves it but I can explain in an answer here. | |
| Oct 31, 2022 at 15:33 | comment | added | Grisha Taroyan | @DavidWhite No, I meant non-connective in the sense that they have stuff (more than just the ground field) in dimension 0 and nothing below zero. Should I have called them non-reduced then? | |
| Oct 31, 2022 at 14:57 | comment | added | David White | Right but in your question you wrote that you wanted it for nonconnective cdgas. Was that a typo? Non connective usually means it's allowed to have stuff in dim below zero | |
| Oct 31, 2022 at 13:54 | comment | added | Grisha Taroyan | @DavidWhite I believe that Quillen stated that the equivalence works over any field of characteristic 0 (after all we only need a retraction of the tensor algebra onto the symmetric algebra). I think that Theorem 2.5 from here math.uchicago.edu/~amathew/doldkan.pdf states the correspondence between non-negatively graded chain complexes and simplicial abelian groups. Perhaps, I don't understand what is your definition of a connective chain complex. In my question, I assume that connective CDGA are ones that have the ground field in dimension $0$ and have nothing in dimensions below $0.$ | |
| Oct 30, 2022 at 16:14 | comment | added | David White | It sounds like you are proposing two generalizations at once. One generalization, to CDGAs over a field of characteristic zero, is no problem, e.g., see Corollary 5.4.1 in arxiv.org/abs/1606.01803. The other generalization, to the non-connective setting, is less clear. Classically, Dold-Kan is between simplicial abelian groups and connective chain complexes. If you move to non-connective chain complexes, what do you have on the other side? | |
| Oct 30, 2022 at 14:35 | history | edited | Grisha Taroyan |
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| Oct 30, 2022 at 14:29 | history | asked | Grisha Taroyan | CC BY-SA 4.0 |