2016/08/31 by Jeremiah Heller, Kyle Ormsby
Mathematics · #Algebraic Geometry and Number Theory #Field (mathematics) #Geometric and Algebraic Topology #Homotopy #Homotopy and Cohomology in Algebraic Topology #Homotopy category #Homotopy group #Spectrum (functional analysis) #Tensor (intrinsic definition) #math.AT #math.KT #msc:14F42 #msc:18E30 #msc:19D45 #msc:55P42 #n-connected
paper · pdf · doi:10.2140/gt.2018.22.2187
published as Geom. Topol. 22 (2018) 2187-2218 · Pre-proofs version accepted for publication in Geometry & Topology
openalex created_date 2016/08/23 · arxiv created 2017/08/29 · openalex publication_date 2018/04/05 · arxiv updated 2018/04/18 · openalex updated_date 2026/08/05
Let [math] be a field of characteristic different from [math] . We establish surjectivity of Balmer’s comparison map\n¶\n<math display="block">\n<mrow>\n<msup>\n<mrow>\n<mi>ρ</mi>\n</mrow>\n<mrow>\n<mo class="MathClass-bin">∙</mo>\n</mrow>\n</msup>\n<mo class="MathClass-punc">:</mo>\n<mspace class="thinspace" width="0.3em"/>\n<mo class="qopname">Spc</mo>\n<mrow>\n<mo class="MathClass-open">(</mo>\n<mrow>\n<msup>\n<mrow>\n<mo class="qopname">SH</mo>\n</mrow>\n<mrow>\n<msup>\n<mrow>\n<mi mathvariant="double-struck">A</mi>\n</mrow>\n<mrow>\n<mn>1</mn>\n</mrow>\n</msup>\n<mspace class="thinspace" width="0.3em"/>\n</mrow>\n</msup>\n<msup>\n<mrow>\n<mrow>\n<mo class="MathClass-open">(</mo>\n<mrow>\n<mi>F</mi>\n</mrow>\n<mo class="MathClass-close">)</mo>\n</mrow>\n</mrow>\n<mrow>\n<mi>c</mi>\n</mrow>\n</msup>\n</mrow>\n<mo class="MathClass-close">)</mo>\n</mrow>\n<mo class="MathClass-rel">→</mo>\n<msup>\n<mrow>\n<mo class="qopname"> Spec</mo>\n</mrow>\n<mrow>\n<mi>h</mi>\n</mrow>\n</msup>\n<mrow>\n<mo class="MathClass-open">(</mo>\n<mrow>\n<msubsup>\n<mrow>\n<mi>K</mi>\n</mrow>\n<mrow>\n<mo class="MathClass-bin">∗</mo>\n</mrow>\n<mrow>\n<mi>M</mi>\n<mi>W</mi>\n</mrow>\n</msubsup>\n<mrow>\n<mo class="MathClass-open">(</mo>\n<mrow>\n<mi>F</mi>\n</mrow>\n<mo class="MathClass-close">)</mo>\n</mrow>\n</mrow>\n<mo class="MathClass-close">)</mo>\n</mrow>\n</mrow>\n</math>\n¶ from the tensor triangular spectrum of the homotopy category of compact motivic spectra to the homogeneous Zariski spectrum of Milnor–Witt [math] –theory. We also comment on the tensor triangular geometry of compact cellular motivic spectra, producing in particular novel field spectra in this category. We conclude with a list of questions about the structure of the tensor triangular spectrum of the stable motivic homotopy category.