Questions tagged [motivic-homotopy]

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Why is the motivic category defined over the site of smooth schemes only?

Fix a base scheme $S$. Stable and unstable motivic categories over $S$ are defined as certain categories of higher stacks on the Nisnevich site $Sm_S$ of smooth schemes over $S$. Why smooth? As a ...
Tim Campion's user avatar
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23 votes
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What is the status of the Friedlander-Milnor conjecture today?

For the purposes of this question, the Friedlander-Milnor (FM) conjecture asserts an equality of the group homology for algebraic groups, and their discretizations in the following sense: Conjecture ...
user49544's user avatar
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26 votes
1 answer
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Voevodsky's counterexample to the existence of a motivic t-structure

I have been trying to unravel some of the known relationships between various ideas on mixed motives. I find the literature quite hard to follow -"from experts, for experts". Voevodsky in "...
plm's user avatar
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16 votes
1 answer
801 views

Representability of Weil Cohomology Theories in Stable Motivic Homotopy Theory

My understanding is that one purpose of stable motivic homotopy theory is to emulate classical stable homotopy theory. In particular, we would like Weil cohomology theories to be representable by ...
user94118's user avatar
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3 votes
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Connecting Quillen functors between motivic homotopy categories (of different "types"): references?

For a perfect base field $k$ there exists the following collection of "motivic homotopy" categories related to it: (a) the homotopy category of simplicial presheaves (from smooth $k$-varieties); here ...
Mikhail Bondarko's user avatar
30 votes
1 answer
4k views

Reasons for the use of Nisnevich topology in motivic homotopy theory

The objects of interest in motivic homotopy theory are "spaces"-which are simplicial sheaves of sets on the big Nisnevich site $Sm/k$ of smooth schemes of finite type over a field $k$. I understand ...
Carl's user avatar
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11 votes
2 answers
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The vanishing of $MGL^{2n+i,n}(X)$; do spectra of smooth projective varieties generate $SH_{l}$?

I have two questions related to the stable motivic homotopy categories of Morel-Voevodsky. The first is probably simple; I wonder what is known on the second one. For the algebraic cobordism theory $...
Mikhail Bondarko's user avatar
8 votes
1 answer
484 views

Inverting objects in a symmetric monoidal category

In Voevodsky’s ICM address: https://www.uio.no/studier/emner/matnat/math/MAT9580/v18/documents/voevodsky-a1-homotopy-theory-icm-1998.pdf In theorem 4.3 it is claimed that given a symmetric monoidal ...
user374433's user avatar
7 votes
1 answer
782 views

Etale and Algebraic K-theory with rational coefficients

We know that the Quillen-Lichtenbaum conjecture, now proved by Rost, Voevodsky, and Weibel, says that for smooth finite type $k$-schemes $X$, etale and algebraic $K$-theory with finite coefficient $\...
user97282's user avatar
7 votes
2 answers
394 views

Properness of the category of modules over a spectrum (that represents algebraic cobordism or motivic cohomology)

The abstract form of the question: let $C$ be a closed proper stable model category, $R$ is a ring object in it. Which conditions ensure that the category $R-mod$ is also proper? Since weak ...
Mikhail Bondarko's user avatar
7 votes
1 answer
532 views

Is $MGL$ an $H\mathbb{Z}$-algebra?

Let $\mathrm{MGL}$ be the $\mathbb{P}^1$-ring spectrum over a field $k$ representing algebraic cobordism. Suppose, for simplicity, that $k$ is of characteristic 0. Let $H\mathbb{Z}$ be the motivic ...
Anandam Banerjee's user avatar
6 votes
0 answers
229 views

Nice references for injective model structures and Quillen functors between motivic homotopy categories

It seems that for the injective model structures for the categories simplicial presheaves and simplicial Nisnevich sheaves (on the category of smooth varieties over a perfect field $k$) there exist ...
Mikhail Bondarko's user avatar