Timeline for Verdier duality under more general conditions
Current License: CC BY-SA 4.0
19 events
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| Mar 11, 2023 at 20:53 | answer | added | Peter Scholze | timeline score: 5 | |
| Aug 17, 2021 at 16:21 | vote | accept | Gabriel | ||
| Aug 17, 2021 at 11:10 | comment | added | Dan Petersen | As you say, $Rf_!$ preserves direct sums iff it is a left adjoint. But as I tried to suggest in my answer, I don't think it is a left adjoint in general. My expectation is that the version of $Rf_!$ for $\infty$-sheaves is a left adjoint and satisfies a version of Verdier duality, whereas Spaltenstein's $Rf_!$ is the composition of three functors: inclusion of hypersheaves into $\infty$-sheaves (right adjoint), $Rf_!$ for $\infty$-sheaves (left adjoint), then hypersheafification (left adjoint). | |
| Aug 17, 2021 at 10:53 | history | edited | Gabriel | CC BY-SA 4.0 |
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| Aug 17, 2021 at 10:51 | comment | added | Gabriel | @FernandoMuro that's where my proposed idea comes from. I studied this paper and wanted to apply its techniques to Verdier duality. | |
| Aug 17, 2021 at 10:51 | comment | added | Gabriel | @DanPetersen Well... if we know that Verdier duality holds (i.e., if $f$ satisfies the condition (*)), then $\mathsf{R}f_!$ has to preserve infinite direct sums. If we could prove this fact without using its adjoint, then we win. But I agree with you that my argument doesn't work. (I had in mind that direct sums of $K$-injectives were $K$-injective.) | |
| Aug 17, 2021 at 9:42 | comment | added | Fernando Muro | This might be of some help: google.com/url?sa=t&source=web&rct=j&url=https://… | |
| Aug 17, 2021 at 2:04 | comment | added | Dan Petersen | I don't believe that $Rf_!$ commutes with infinite direct sum in general, and I don't see why it would follow from direct sums in $D(X)$ being degreewise. Wouldn't you need something like a class of complexes which can be used to compute $Rf_!$ and which is closed under direct sums? Infinite direct sums of $K$-injectives are not $K$-injective in general. | |
| Aug 16, 2021 at 22:15 | history | edited | Gabriel | CC BY-SA 4.0 |
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| Aug 16, 2021 at 21:52 | history | edited | Gabriel | CC BY-SA 4.0 |
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| Aug 16, 2021 at 21:38 | history | edited | Gabriel | CC BY-SA 4.0 |
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| Aug 16, 2021 at 21:25 | history | edited | Gabriel | CC BY-SA 4.0 |
wrote a proposed proof
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| Aug 16, 2021 at 17:40 | answer | added | Dan Petersen | timeline score: 9 | |
| Aug 16, 2021 at 12:16 | answer | added | Vivek Shende | timeline score: 3 | |
| S Aug 13, 2021 at 20:04 | history | bounty ended | CommunityBot | ||
| S Aug 13, 2021 at 20:04 | history | notice removed | CommunityBot | ||
| S Aug 5, 2021 at 18:58 | history | bounty started | Gabriel | ||
| S Aug 5, 2021 at 18:58 | history | notice added | Gabriel | Draw attention | |
| Aug 3, 2021 at 18:19 | history | asked | Gabriel | CC BY-SA 4.0 |