11
votes
3answers
541 views

Applications of homotopy purity theorem of Morel-Voevodsky

One of the most important theorems in motivic homotopy theory is the homotopy purity theorem of Morel-Voevodsky which says that the motivic Thom space of the normal bundle $\mathcal N_{Z/X}$ of a ...
3
votes
0answers
291 views

Two questions on motivic homotopy theory

I am a beginner in motivic homotopy theory and motivic cohomology and have just begun reading books by Dundas et al. and Mazza-Voevodsky-Weibel. I have two basic questions: Why is the question of ...
10
votes
1answer
444 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 ...
6
votes
0answers
148 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 ...
3
votes
1answer
231 views

$T$-stable vs. $S^1$-stable motivic homotopy category: which sorts of transfers are available?

I would like to have some sort of transfers in motivic stable homotopy categories in order to adapt Voevodsky's split standard triple argument to cohomology theories that can be factorized through ...
7
votes
1answer
313 views

Ring structure for the motivic spectrum/complex that represents singular cohomology?

As the discussion here Is singular cohomology representable by a (Voevodsky's) motivic complex? shows, the singular cohomology of (smooth) complex varieties is represented by a motivic complex ...
7
votes
1answer
298 views

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 ...
8
votes
1answer
311 views

Is the injective model structure on symmetric spectra Bousfield localizable?

I am interested in injective model structures on both symmetric spectra as exposed in Hovey/Shipley/Smith and motivic symmetric spectra as in Jardine's article. Both authors take a model structure on ...