Timeline for Almost acyclicity of the complex of configuration spaces of noncollinear points in projective plane over finite fields
Current License: CC BY-SA 4.0
11 events
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| Sep 8, 2022 at 8:30 | comment | added | XYC | @RichardStanley : Because if $F$ has infinite elements then for any cycle $z=\sum_in_i(x_0^i, \dots, x_p^i)\in C_p$ we can always find an element $x\in X$ such that $x$ is non-collinear to any two of $x_n^j$'s, and insert $x$ to all the base elements in $z$ at the first slot we have $z=d[\sum_in_i(x,x_0^i, \dots, x_p^i)]$. Because in $d[\sum_in_i(x,x_0^i, \dots, x_p^i)]$ all base elements with $x$ at the first slot cancels out since $z$ is a cycle. See Lemma 1 in Section 1 also Lemma 3.1 in Suslin's paper- $K_3$ of a field and the Bloch group. | |
| Aug 31, 2022 at 0:02 | comment | added | Richard Stanley | Why is the statement trivially right for infinite fields? | |
| S May 25, 2022 at 19:04 | history | bounty ended | CommunityBot | ||
| S May 25, 2022 at 19:04 | history | notice removed | CommunityBot | ||
| May 17, 2022 at 17:43 | comment | added | XYC | @DanPetersen: The one-dimention-less analogue of this question is the complex of injective words, so I think this version would also be right (at least when $n=1,2,3,4$). On the other hand if we replace finite fields by infinite fields this statement would be trivially right. BTW I know Goncharov's article but I never read it. | |
| S May 17, 2022 at 17:37 | history | bounty started | XYC | ||
| S May 17, 2022 at 17:37 | history | notice added | XYC | Canonical answer required | |
| May 11, 2022 at 13:53 | comment | added | David E Speyer | Interesting question! The reverse problem, where $(x_0, x_1, \dots, x_n)$ form a simplex if all the $x_i$ ARE collinear has been studied -- see mathscinet.ams.org/mathscinet-getitem?mr=772475 and mathscinet.ams.org/mathscinet-getitem?mr=3482438 . But I can't find anyone who is looked at this version. | |
| May 11, 2022 at 5:38 | comment | added | Dan Petersen | It would be helpful to say something about where this problem comes from. I assume you're aware of e.g. Goncharov's "Geometry of configurations, polylogarithms and motivic cohomology". | |
| May 11, 2022 at 4:55 | history | edited | XYC | CC BY-SA 4.0 |
added 88 characters in body
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| May 10, 2022 at 17:44 | history | asked | XYC | CC BY-SA 4.0 |