Bousfield Classes and Ohkawa's Theorem

Zusatztext

<p>This volume originated in the workshop held at Nagoya University, August 2830, 2015, focusing on the surprising and mysterious Ohkawa's theorem: the Bousfield classes in the stable homotopy category<i>SH</i> form a set. An inspiring, extensive mathematical story can be narrated starting with Ohkawa's theorem, evolving naturally with a chain of motivational questions:</p><p></p><ul><li> Ohkawa's theorem states that the Bousfield classes of the stable homotopy category<i>SH</i> 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?</li><li>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<b>D</b><sub>qc</sub>(<i>X</i>) 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 Smithin the triangulated subcategory<i>SH<sup>c</sup></i>, consisting of compact objects in<i>SH</i>. Now the following questions naturally occur: (1) Having theorems of Ohkawa and Hopkins-Smith in<i>SH</i>, are there analogues for the Morel-Voevodsky A<sup>1</sup>-stable homotopy category<i>SH</i>(<i>k</i>), which subsumes<i>SH</i> when<i>k</i> is a subfield of<b>C</b>?, (2) Was it not natural for Hopkins to have considered<b>D</b><sub>qc</sub>(<i>X</i>)<i><sup>c</sup></i> instead of<b>D</b><sub>qc</sub>(<i>X</i>)? However, whereas there is a conceptually simple algebro-geometrical interpretation<b>D</b><sub>qc</sub>(<i>X</i>)<i><sup>c</sup></i> =<b>D</b><sup>perf</sup>(<i>X</i>), it is its close relative<b>D</b><i><sup>b</sup></i><sub>coh</sub>(<i>X</i>) that traditionally, ever since Oka and Cartan, has been intensively studied because of its rich geometric and physical information.</li></ul><p></p><p></p><p>This book contains developments for the rest of the storyand much more, including the chromatics homotopy theory, which the HopkinsSmith theorem is based upon, and applications of Lurie's higher algebra, all by distinguished contributors.<br></p>

Weitere Details