# Bousfield Classes and Ohkawa's Theorem

## Nagoya, Japan, August 28-30, 2015

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**BouCla 2015**

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Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 309)

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Conference proceedings
**BouCla 2015**

- 1 Citations
- 162 Downloads

Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 309)

This volume originated in the workshop held at Nagoya University, August 28–30, 2015, focusing on the surprising and mysterious Ohkawa's theorem: the Bousfield classes in the stable homotopy category *SH* form a set. An inspiring, extensive mathematical story can be narrated starting with Ohkawa's theorem, evolving naturally with a chain of motivational questions:

- Ohkawa's theorem states that the Bousfield classes of the stable homotopy category
*SH*surprisingly forms a set, which is still very mysterious. Are there any toy models where analogous Bousfield classes form a set with a clear meaning? - The fundamental theorem of Hopkins, Neeman, Thomason, and others states that the analogue of the Bousfield classes in the derived category of quasi-coherent sheaves
**D**_{qc}(*X*) form a set with a clear algebro-geometric description. However, Hopkins was actually motivated not by Ohkawa's theorem but by his own theorem with Smith in the triangulated subcategory*SH*, consisting of compact objects in^{c}*SH*. Now the following questions naturally occur: (1) Having theorems of Ohkawa and Hopkins-Smith in*SH*, are there analogues for the Morel-Voevodsky A^{1}-stable homotopy category*SH*(*k*), which subsumes*SH*when*k*is a subfield of**C**?, (2) Was it not natural for Hopkins to have considered**D**_{qc}(*X*)instead of^{c}**D**_{qc}(*X*)? However, whereas there is a conceptually simple algebro-geometrical interpretation**D**_{qc}(*X*)=^{c}**D**^{perf}(*X*), it is its close relative**D**^{b}_{coh}(*X*) that traditionally, ever since Oka and Cartan, has been intensively studied because of its rich geometric and physical information.

This book contains developments for the rest of the story and much more, including the chromatics homotopy theory, which the Hopkins–Smith theorem is based upon, and applications of Lurie's higher algebra, all by distinguished contributors.

stable homotopy theory Bousfield class Ohkawa's theorem motivic stable cohomology dualities of Tannakian type in memory of Tetsusuke Ohkawa 14Fxx 14Lxx 16Exx 18Dxx 18Fxx 19Exx 32Gxx 55Nxx 55Pxx

- DOI https://doi.org/10.1007/978-981-15-1588-0
- Copyright Information Springer Nature Singapore Pte Ltd. 2020
- Publisher Name Springer, Singapore
- eBook Packages Mathematics and Statistics
- Print ISBN 978-981-15-1587-3
- Online ISBN 978-981-15-1588-0
- Series Print ISSN 2194-1009
- Series Online ISSN 2194-1017
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