Talk:A¹ homotopy theory
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Article improvements
[edit]- Discuss fiber functors in the relevant article as it relates here
- Clean up denoting between unstable and A^1 local unstable theories
- Add cohomology theories such as K-theory, (pg 381 of Morel notes http://users.ictp.it/~pub_off/lectures/lns015/Morel/Morel.pdf and pg 313 of Thomason's higher K-theory of schemes paper https://www.gwern.net/docs/math/1990-thomason.pdf)
- Mention how K-theory works in Zariski, but not etale topology here
- Example 3.1.6 in morel notes
- Example 3.1.7 gives references to infinite grassmanian and BGL, giving algebraic k-theory the expected representatives
- Example 3.1.10
- Theorem 3.3.4 on Thom spaces
- The two stable categories
T-spectra
[edit]- Define the homotopy spheres (remark 5.1.5) + pg 417-418
- Rost motives (Rost cycles)
- Theorems 6.3.3 and onwards on Milnor K-theory
Wundzer (talk) 16:58, 4 February 2021 (UTC)
Adding material on sphere spectra
[edit]There should be a discussion for the stable motivic homotopy spheres. This should include the definition of the spheres and the related vanishing theorems. Moreover, there should be ample motivation behind their definition/construction so it's clear why they are defined the way they are.
- Extract symmetric monoidal structure from chapter 4 and state it for the spectra in chapter 5
- This way the spheres can be defined in chapter 5 using T-spectra
- Discuss A^1 localization
- Give diagram showing the isomorphism P^1/A^1 \cong T
- pg 417 gives the definition, note the negative smash products come from the symmetric monoidal structure
- Also note the unit is the zero sphere
Additional spectra
[edit]This is in ch 5
- Eilenberg-Maclane spectra
- Motivic cohomology spectra
Some other references/resources
[edit]- Stable connectivity over a base - https://arxiv.org/abs/1911.05014
- Vanishing in stable motivic homotopy sheaves - https://arxiv.org/abs/1704.04744
- A1-Algebraic Topology over a Field - 6.43 on page 167 gives vanishing result in the unstable situation - 978-3-642-29513-3
Other
[edit]- https://arxiv.org/abs/1002.5007 contains a collapsing spectral sequence
- Voevodsky's Seattle Lectures: K-theory and Motivic Cohomology - https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/Seatle_lectures_notes_by_Weibel_published.pdf
pg 403 of Morel notes gives a Postnikov tower for t-structures, this should be included in the Postnikov article