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Nov 18 |
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Reference request for cohomology of coverings
@Aleksey There are examples with higher cohomology when the fundamental group is just $\mathbb Z^n$. |
Nov 17 |
answered | Reference request for cohomology of coverings |
Nov 16 |
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Is there a category whose isomorphisms are precisely the simple homotopy equivalences?
It may be worth knowing that if you take the nerve of the simple homotopy equivalences (ie, abandoning interest non-equivalences, retreating from categories to groupoids), you get an interesting space, which is different from the nerve of the homotopy equivalences, and not just in its set of components. |
Nov 5 |
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Which mapping class group representations come from algebraic geometry?
Yes, it works. There are two difficulties. The first is that there are lots of choices; you just have to make all of them. That is: take the universal curve over the moduli stack of double covers of a genus $g$ curve ramified in $n=4g-4$ points. This maps to $M_g$, so its cohomology is a local system there. The second problem is the stackiness: when you take cohomology, it's like taking $\mathbb Z/2$ invariants, but the interesting cohomology all has action by $-1$. But you can just take the tensor square, or some other ad hoc option. |
Nov 5 |
answered | Which mapping class group representations come from algebraic geometry? |
Oct 27 |
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Are there nontrivial involutions of $S^7\times S^7$ with fixed point set homeo to $S^7$?
Yes, the method is surgery, so it has the drawbacks of surgery. We'd like to distinguish involutions by the isomorphism type of the quotient of the complement of a regular neighborhood of the fixed set, but that's hard so we mark so that we can apply surgery. And even if there are isomorphism once we drop the marking, that doesn't mean that there are automorphisms of $M$ that induce them. |
Oct 26 |
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Does every smooth, projective morphism to $\mathbb{C}P^1$ admit a section?
@Marty, that's a great idea. It suggests a generalization: that a smooth proper map to a curve contains a complete curve which maps to the target by an etale map. |
Oct 25 |
answered | Are there nontrivial involutions of $S^7\times S^7$ with fixed point set homeo to $S^7$? |
Oct 24 |
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Doing some homological algebra in triangulated categories
I am skeptical that you can iterate the construction in the third paragraph beyond $n=2$. |
Oct 23 |
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Does every smooth, projective morphism to $\mathbb{C}P^1$ admit a section?
Is there any reason for "projective" rather than "(algebraic) proper"? (Of course, for analytic proper maps, it's false.) |
Oct 12 |
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The quotient stack $[\mathbb{A}^n / \mathrm{GL}_n]$
This stack classifies vector bundles of rank $n$ together with $1$ global section. You see it just like you see that $A^1/G_m$ classifies line bundles with a global section. |
Oct 11 |
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Are there nontrivial involutions of $S^7\times S^7$ with fixed point set homeo to $S^7$?
This involution is conjugate to the swap by $(x,y)\mapsto (xy,y)$. |
Oct 7 |
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Why the Dold-Thom theorem?
@მამუკა ჯიბლაძე, I agree that is the right perspective, but you need a little bit more. $\mathbb Z[X]$ is a simplicial abelian group. There are two things you can do with it: turn it into a chain complex and take homology; or turn it into a space and take homotopy groups. You need to know that they correspond. This is sometimes included in Dold-Kan. |
Oct 5 |
accepted | Is homology finitely generated as an algebra? |
Oct 5 |
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Finite complexes whose homotopy groups are not “finitely generated”
Given the answers to my question, I've changed my mind and think the answer is no: that there are rational examples that are not finitely generated. But I don't have any. |
Oct 4 |
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Is homology finitely generated as an algebra?
@john Wow! I expected the opposite. I think you're right: the whole algebra is finitely generated as a module over the kernel of $d$, so the kernel of $d$ is finitely generated as an algebra. |
Oct 3 |
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Is homology finitely generated as an algebra?
The homological convention is that $d$ lowers degree, while the cohomological convention is that it raises degree. You can switch between them by negating degrees, but that switches whether a DGA is positive or negative. |
Oct 3 |
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Is homology finitely generated as an algebra?
Do you have citations for Bhatt and Halpern-Leistner? |
Oct 3 |
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Is homology finitely generated as an algebra?
Your last parenthetical sentence implies that $R$-modules are a reflective subcategory of $\mathbb Q[x,y]$-modules, thus an example of a smashing localization. This is generally true about open subschemes. This is familiar algebraic localization in codimension 1. Codimension higher than 2 isn't much different than 2. |
Oct 3 |
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Is homology finitely generated as an algebra?
This has the advantage over Akhil's example of being generated in positive degrees. It is positive with respect to the cohomological convention, but if you move $z$ to degree 4, it is positive with respect to the homological convention. . . . Can't you make a free one by $dz=abc$, with $a,b,c$ odd? |