Synthetic spectra and the cellular motivic category
By Piotr Pstrągowski. Inventiones Mathematicae, 2018
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To an Adams-type homology theory we associate the notion of a synthetic spectrum; this is a product-preserving sheaf on the site of finite spectra with projective E -homology. We show that the $$\infty $$ ∞ -category of synthetic spectra based on E is in a precise sense a deformation of the $$\infty $$ ∞ -category of spectra into quasi-coherent sheaves over a certain algebraic stack, and show that this deformation encodes the $$E_{*}$$ E ∗ -based Adams spectral sequence. We describe a symmetric monoidal functor from the $$\infty $$ ∞ -category of cellular motivic spectra over $$\text {Spec}({\mathbb {C}})$$ Spec ( C ) into an even variant of synthetic spectra based on $$\textrm{MU}$$ MU and show that it induces an equivalence between the $$\infty $$ ∞ -categories of p -complete objects for all primes p . In particular, it follows that the p -complete cellular motivic category can be described purely in terms of chromatic homotopy theory.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/56637b791e12f3c571a0e34594cebcf084135563
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Citations in the corpus
- The special fiber of the motivic deformation of the stable homotopy category is algebraic (2018, background): “After the announcement of Theorem 1.1, alternative proofs of certain versions of Corollary 1.2 have appeared in work of Krause [35] and Pstra̧gowski [60].”