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We define the Chow t-structure on the ∞-category of motivic spectra SH(k) over an arbitrary base field k. We identify the heart of this t-structure SH(k)c♡ when the exponential characteristic of k is inverted. Restricting to the cellular subcategory, we identify the Chow heart SH(k)cell,c♡ as the category of even graded MU2∗MU-comodules. Furthermore, we show that the ∞-category of modules over the Chow truncated sphere spectrum 1c=0 is algebraic. Our results generalize the ones in Gheorghe–Wang–Xu in three aspects: to integral results; to all base fields other than just C; to the entire ∞-category of motivic spectra SH(k), rather than a subcategory containing only certain cellular objects. We also discuss a strategy for computing motivic stable homotopy groups of (p-completed) spheres over an arbitrary base field k using the Postnikov–Whitehead tower associated to the Chow t-structure and the motivic Adams spectral sequences over k.
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The stable Galois correspondence for real closed fields
In previous work [7], the authors constructed and studied a lift of the Galois correspondence to stable homotopy categories. In particular, if L/k is a finite Galois extension of fields with Galois group G, there is a functor c∗L/k : SHG → SHk from the G-equivariant stable homotopy category to the stable motivic homotopy category over k such that c∗L/k(G/H+) = Spec(LH)+. The main theorem of [7] says that when k is a real closed field and L = k[i], the restriction of c∗L/k to the η-complete subcategory is full and faithful. Here we “uncomplete” this theorem so that it applies to c∗L/k itself. Our main tools are Bachmann’s theorem on the (2,η)- periodic stable motivic homotopy category and an isomorphism range for the map πRS → πC2 S induced by C2-equivariant Betti realization.
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- Award ID(s):
- 1709302
- PAR ID:
- 10067698
- Date Published:
- Journal Name:
- Contemporary mathematics - American Mathematical Society
- Volume:
- 707
- ISSN:
- 0271-4132
- Page Range / eLocation ID:
- 1-9
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1090/conm/707/14250
