33
votes
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Why is Voevodsky's motivic homotopy theory 'the right' approach?
(Don't be afraid about the word "$\infty$-category" here: they're just a convenient framework to do homotopy theory in).
I'm going to try with a very naive answer, although I'm not sure I understand ...
- 16.5k
25
votes
Why is the motivic category defined over the site of smooth schemes only?
It's worth noting first that smooth schemes are essentially the smallest possible category from which one can define the motivic homotopy category: to make sense of $\mathbb A^1$-homotopy and of the ...
- 8,972
21
votes
Why is Voevodsky's motivic homotopy theory 'the right' approach?
I just want to point out, with regards to your second question, that the fact that the $\mathbb P^1$ is not equivalent to a simplicial complex homotopic to the sphere built out of affine spaces (or ...
- 148k
19
votes
$\zeta(-n)=2^{r_1}\frac{|K_{2n}(O)|}{|K_{2n+1}(O)|} R_K$ and replace $K$-theory with $\mathbb{S}$
Yes, you can do a lot of what you ask about in your question, and if you're willing to change the questions slightly, you can do even more. I have thought a lot about these kinds of questions. I'm ...
- 1,335
18
votes
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What's motivic about $\mathbb{A}^1$-homotopy theory? What's motivic about correspondences?
There is no need a priori to define these categories of motives starting from correspondences. The stable homotopy theory of schemes $SH$ may be characterized by a universal property saying that it is ...
- 13.6k
17
votes
Why is the motivic category defined over the site of smooth schemes only?
It makes sense to consider larger versions of the (unstable and stable) motivic homotopy categories built out of the site $Sch_S$ of all schemes over $S$ (say of finite type to avoid dealing with size ...
- 5,901
16
votes
Is there a geometric realization of $\mathbf{C}((t))$-varieties?
I know nothing about $\mathbf{A}^1$-homotopy. I will describe how one can get a functor from smooth varieties over $\mathbf{C}((t))$ to ${\rm Top}_{/\mathbf{S}^1}$ using log smooth proper models and ...
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15
votes
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Which motivic spectra are dualizable?
If $X$ is noetherian of dimension $>0$, there is always a compact object of $SH(X)$ which is not dualizable: e.g. $j_\sharp$ of the sphere spectrum where $j:U\to X$ is any dense open immersion with ...
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15
votes
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Who proved the motivic 6-functor formalism?
My understanding is that Scholze was citing Gallauer for the universal property of the 6FF of motivic spectra (an unpublished result as far as I know), not for the existence of the 6FF. This result ...
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14
votes
Accepted
Is algebraic $K$-theory a motivic spectrum?
Let me assume that $S$ is a regular Noetherian scheme (for example a field). Then algebraic K-theory is a motivic spectrum, and in fact it is represented by the $\mathbb{P}^1$-spectrum that is $BGL_\...
- 16.5k
13
votes
Voevodsky's Triangulated Categories of Motives and their Relationships
I won't embark on the difficult question of what one wants out of a category of motives, but I can make some comments on what might motivate the various choices of topologies.
Nisnevich (aka ...
- 251
12
votes
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Smallness of the category of schemes of finite type
You can prove essential smallness of the category of finite type $S$-schemes (without further assumptions), where $S$ is any scheme, as follows:
If $S$ is affine and we only consider affine finite ...
- 5,482
10
votes
Accepted
What are the advantages of various "models" for the motivic stable homotopy category
The injective model structure is monoidal (satisfies the pushout product axiom), see Hornbostel's paper "Localizations in motivic homotopy theory", Thm 1.9 and Lemma 1.10. The projective model ...
- 30.3k
9
votes
Could a motivic spectrum have a "zeta function"?
Etale cohomology factors through the stable $\mathbb A^1$ homotopy type. So you should simply consider your local zeta factor in terms of traces of frobenius on etale cohomology. Since smashing with $...
- 1,347
8
votes
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What is the best reference for motives?
I would add this as a comment, but I do not have enough reputation to do so.
While there are certainly more contemporary references, Voevodsky's "Triangulated category of motives over a field" is a ...
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8
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Is there a geometric realization of $\mathbf{C}((t))$-varieties?
I don't know about $\mathbb A^1$-homotopy categories. Let me try to do something that I think will work just for the category of quasiprojective varieties.
The key lemma is this:
There is a ...
- 148k
8
votes
Accepted
What is the etale sheafification of the (unramified) Milnor-Witt $K$-theory
We have a short exact sequence of sheaves of abelian groups (see e.g. Morel's book on $\mathbb{A}^1$-algebraic topology):
$$
0\to\mathbf{I}^{n+1}\to\mathbf{K}^{\rm MW}_n\to \mathbf{K}^{\rm M}_n\to 0,
...
- 17.4k
8
votes
Accepted
Inverting objects in a symmetric monoidal category
To be clear, this claim refers to a very specific construction of $\mathcal{C}[X^{-1}]$, where you copy the construction of the localization of the ring and defines it as the colimits of:
$$ \mathcal{...
- 42.4k
7
votes
Smallness of the category of schemes of finite type
Asking about smallness instead of essential smallness is a distraction; smallness is not invariant under equivalence of categories, so the answer potentially depends delicately on how you construct ...
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7
votes
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When is the Thom spectrum of a virtual vector bundle effective?
Yes.
A bit more generally, if $\xi$ is a perfect complex of rank $\geq 0$, then $Th(\xi)$ is effective (even very effective): the question is Nisnevich-local on $X$ and $\xi$ is locally a complex of ...
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7
votes
Are there non trivial maps from $H\mathbb{Z}$ to $MGL$?
NO such a map does not exist.
Thanks to Eric Peterson for making me realize that the argument carries through even if the map is not a map of algebras.
By rigidity,you can only consider the case ...
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7
votes
Accepted
Is the $\mathbb A^1$ homotopy type of a punctured curve independent of the choice of puncture?
If I understand it correctly, this follows from the results of Severitt's master's thesis. (I am not an expert on this, so it is possible I misread one of the statements.)
Indeed, Lemma 9.1.1 shows ...
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6
votes
Motivic cohomology is universal with respect to what (co)homology theories?
I think a proof that motivic Borel-Moore homology is a universal Borel-Moore homology theory could be deduced from work of Bloch, Geisser-Levine, and Voevodsky, at least for varieties over fields. I ...
- 592
6
votes
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Basic questions on spectra
Here is a slightly more fleshed out version of the comment above.
First, the claim that the collection $\mathcal{C} = \{ \Sigma^{p,q} U \mid U \in Sm/S, p,q \in \mathbb{Z} \}$ is a collection of ...
- 3,785
6
votes
Accepted
$\mathbf{A}^1$-invariance of Brauer groups and $H^2_{\mathrm{et}}(-;\mathbb{G}_m)$
(For $i=0$, the map $H_{\mathrm{et}}^{0}(\operatorname{Spec} A,\mathbb{G}_{m}) \to H_{\mathrm{et}}^{0}(\operatorname{Spec} A[t],\mathbb{G}_{m})$ is an isomorphism if and only if $A$ is reduced.)
For $...
- 2,017
5
votes
What is the best reference for motives?
A wealth of information is contained in:
Uwe Jannsen, Steven L. Kleiman, Jean Pierre Serre, “Proceedings of Symposia in Pure Mathematics, Vol 55, Parts 1 and 2”. Amer Mathematical Society (February 1, ...
- 5,494
5
votes
Accepted
Model category structure on spectra
Is there a model structure on Spt(S), having SH(S) as homotopy category, such that every object is fibrant? If so, could you provide a reference?
No, if the given model category of spectra Spt(S) (...
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5
votes
Model category structure on spectra
Since you tagged this as a reference request, let me give you some relevant references. The first injective model structure for motivic spectra that I am aware of is Jardine's paper Motivic Symmetric ...
- 30.3k
5
votes
A question about the vanishing of motivic cohomology in negative Tate twist
Here's an easy way to see this (which is more or less an elaboration of Mikhail's answer):
Suppose $i > 0$ and $n\in \mathbb Z$. We have the Thom isomorphism
$$
H^n(X, \mathbb Z(-i)) \cong H^{n+2i}...
- 8,972
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