A Theory of Base Motives

When A is a commutative local ring with residue field k, the derived tensor product −⊗ L A k : D(A) → (k −Mod) lifts to a functor taking values in a category of modules over the ‘Tate cohomology’ RHom∗A(k, k), which is the universal enveloping algebra of a certain Lie algebra. Under reasonable conditions this lift satisfies a spectral sequence of Adams (or Bockstein) type. In a suitable category of ring-spectra, replacing A → k by A(∗) → S or TC(S) → S yields interesting Hopf objects, with Lie algebras free after tensoring with Q, analogous to those of motivic groups studied recently by Deligne, Connes and Marcolli, and others. [This is a sequel to and continuation of a talk at last summer’s conference in Bonn honoring Haynes Miller [23]. I owe many mathematicians thanks for helpful conversations and encouragement, but want to single out John Rognes particularly, and thank him as well for organizing this wonderful conference.]