Moduli stack of oriented formal groups and cellular motivic spectra over $\mathbf C$

Abstract

We exhibit a relationship between motivic homotopy theory and spectral algebraic geometry, based on the motivic τ -deformation picture of Gheorghe, Isaksen, Wang, Xu. More precisely, we identify cellular motivic spectra over C with ind-coherent sheaves (in a slighly non-standard sense) on a certain spectral stack τ≥0(M or FG ). The latter is the connective cover of the non-connective spectral stack M or FG , the moduli stack of oriented formal groups, which we have introduced previously and studied in connection with chromatic homotopy theory. We also provide a geometric origin on the level of stacks for the observed τ -deformation behavior on the level of sheaves, based on a notion of extended effective Cartier divisors in spectral algebraic geometry.