Atiyah duality for motivic spectra

We prove that Atiyah duality holds in the -category of non- A¹-invariant motivic spectra over arbitrary derived schemes: every smooth projective scheme is dualizable with dual given by the Thom spectrum of its negative tangent bundle. The Gysin maps recently constructed by L. Tang are a key ingredient in the proof. We then present several applications. First, we study A¹-colocalization, which transforms any module over the A¹-invariant sphere into an A¹-invariant motivic spectrum without changing its values on smooth projective schemes. This can be applied to all known p-adic cohomology theories and gives a new elementary approach to "logarithmic" or "tame" cohomology theories; it recovers for instance the logarithmic crystalline cohomology of strict normal crossings compactifications over perfect fields and shows that the latter is independent of the choice of compactification. Second, we prove a motivic Landweber exact functor theorem, associating a motivic spectrum to any graded formal group law classified by a flat map to the moduli stack of formal groups. Using this theorem, we compute the ring of P¹-stable cohomology operations on the algebraic K-theory of qcqs derived schemes, and we prove that rational motivic cohomology is an idempotent motivic spectrum.