Timeline for Reconstruction of commutative differential graded algebras
Current License: CC BY-SA 4.0
27 events
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| Oct 4, 2023 at 15:40 | history | edited | YkMz | CC BY-SA 4.0 |
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| Oct 4, 2023 at 11:46 | history | edited | YkMz | CC BY-SA 4.0 |
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| Oct 4, 2023 at 11:18 | comment | added | YkMz | Thank you! Is Lurie's Higher algebra the best reference for E1-center? Are there any other good references? | |
| Oct 4, 2023 at 8:13 | comment | added | Maxime Ramzi | Certainly yes - I'm not super familiar with dg literature but this cannot not be in there somewhere; in $\infty$-categorical language, the definition and description of the center is done in Higher Algebra (under the name "$E_1$-center"), and the description of its $\pi_0$ in the case where $A$ is discrete follows from the earlier comparison of the homotopy groups of mapping spectra with Ext-groups. (but this approach really only works when $A$ and $B$ are in degree $0$) | |
| Oct 4, 2023 at 2:35 | comment | added | YkMz | @MaximeRamzi Are there some good references to what you said? | |
| Oct 3, 2023 at 16:14 | comment | added | Maxime Ramzi | In $\infty$-categorical terms, it's simply the endomorphism ring of the identity functor of a given (say, stable) $\infty$-category. In dg-terms, it's something like Hochschild cohomology | |
| Oct 3, 2023 at 16:11 | comment | added | YkMz | Thank you. What is the definition of derived centers? | |
| Oct 3, 2023 at 16:04 | comment | added | Maxime Ramzi | In the case where $A,B$ are in degree $0$, this is true, because you can then recover $A$ as $\pi_0$ of the derived center of this category. Other than that, I don't know - I want to point out that there are other settings where this is false. As Marc points out, away from char 0, you can at best require $A,B$ to be quasi-iso as $E_1$-rings, and this becomes a reasonable question (although in char $0$ they are equivalent) - if you work $K(n)$- or $T(n)$-locally everywhere, there are counterexamples to even this version. | |
| Oct 3, 2023 at 15:34 | history | edited | YkMz | CC BY-SA 4.0 |
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| Oct 3, 2023 at 15:28 | history | edited | YkMz | CC BY-SA 4.0 |
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| Sep 22, 2023 at 2:22 | history | edited | YkMz | CC BY-SA 4.0 |
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| Sep 21, 2023 at 6:19 | history | edited | YkMz | CC BY-SA 4.0 |
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| Sep 21, 2023 at 6:12 | history | edited | YkMz | CC BY-SA 4.0 |
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| Sep 20, 2023 at 14:00 | comment | added | the L | You might find homepages.math.uic.edu/~bshipley/tilting7.pdf interesting, especially Section 5 | |
| Sep 20, 2023 at 7:17 | comment | added | Marc Hoyois | @DanPetersen Oh this is a surprise, thanks for the reference! | |
| Sep 20, 2023 at 3:49 | comment | added | Stahl | @QiaochuYuan Yes, you're right -- I see now that I missed the (c) assumption in the OP | |
| Sep 20, 2023 at 3:14 | comment | added | Qiaochu Yuan | @Stahl: the OP assumes commutativity, and ordinary commutative $k$-algebras are Morita equivalent iff they're isomorphic (because the module category knows the center of a ring). So this approach won't work for producing a counterexample. | |
| Sep 20, 2023 at 2:50 | history | edited | YkMz | CC BY-SA 4.0 |
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| Sep 20, 2023 at 2:46 | comment | added | YkMz | Thank you for the comment! So is it still possible that the question I am considering is correct? | |
| Sep 19, 2023 at 16:35 | comment | added | Dan Petersen | @MarcHoyois "and there can be several non-equivalent $E_\infty$-algebra structures on such $A$" --- this is actually false, when one works over a field of characteristic zero. This is Theorem A of arxiv.org/abs/1904.03585 . | |
| Sep 19, 2023 at 15:46 | history | edited | YkMz | CC BY-SA 4.0 |
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| Sep 19, 2023 at 13:01 | comment | added | YkMz | Thank you for the comments! Do you have any literature that elaborates on what you mentioned? In addition, can we expect to improve the situation by making any assumptions about $A$? | |
| Sep 19, 2023 at 9:40 | comment | added | Marc Hoyois | There is also a difference between dgas and cdgas. The dg-category $D_{dg}(A)$ depends only on $A$ as a dga/$E_1$-algebra, and there can be several non-equivalent $E_\infty$-algebra structures on such $A$. So the answer to the question is "no" for this reason. | |
| Sep 19, 2023 at 5:01 | comment | added | YkMz | Thank you for the comment and the many references! I understand the classical (derived) Morita theory. I want to know whether generalized results are well-known or not. | |
| Sep 19, 2023 at 3:16 | comment | added | Stahl | Putting together results from these references, I think that if $A$ and $B$ are Morita equivalent $k$-algebras, viewed as DG-$k$ algebras in degree $0,$ they should have quasi-equivalent DG-derived categories, but not necessarily be quasi-isomorphic as DG-$k$-algebras. | |
| Sep 19, 2023 at 3:05 | comment | added | Stahl | I believe the key phrase here is "(derived) Morita theory." There is a wide literature on this, for example, see here, here, here, or here, to start. It may also help to reference Cohn or Cisinski and Tabuada for a quick overview of model structures on DG categories, as well as other work of Tabuada on those model structures. | |
| Sep 19, 2023 at 2:04 | history | asked | YkMz | CC BY-SA 4.0 |