Formal groups and stable homotopy of commutative rings

Abstract

We explain a new relationship between formal group laws and ring spectra in stable homotopy theory. We study a ring spectrum denoted DB which depends on a commutative ring B and is closely related to the topological André-Quillen homology of B . We present an explicit construction which to every 1-dimensional and commutative formal group law F over B associates a morphism of ring spectra F * : HZ −→ DB from the Eilenberg-MacLane ring spectrum of the integers. We show that formal group laws account for all such ring spectrum maps, and we identify the space of ring spectrum maps between HZ and DB . That description involves formal group law data and the homotopy units of the ring spectrum DB .

On lifting commutative dynamical systems

“…In addition to their elegance (most notably in [18]), formal groups continue to find powerful and sometimes surprising applications in number theory, algebraic geometry and topology (see [3,20,21], for example).…”

Absolute algebra and Segal's Γ-rings

“…where D n is defined by replacing the component at ∞ of D by a n {∞}. One then applies (23) to obtain (26). Finally, when λ = e a is also irrational one uses an increasing sequence a n → a such that e a n ∈ Z[1/p] together with (23) to obtain (i).…”

“…One then applies (23) to obtain (26). Finally, when λ = e a is also irrational one uses an increasing sequence a n → a such that e a n ∈ Z[1/p] together with (23) to obtain (i). (ii) For any q ∈ Q × , one has a canonical isomorphism O(D) → O(D + (q)) of sheaves of O Spec Z -modules over Spec Z which is defined by multiplication by q −1 .…”

“…Next, we review the definition of the assembly map in the simple case of Γ -sets: we refer to [21,23] for more details. Let N : Γ op −→ Sets * be a Γ -set, and Ñ : Sets * −→ Sets * its extension to a pointed endofunctor as in (31).…”