Regularity of Structured Ring Spectra and Localization in K-theory

We introduce a new notion of regularity for structured ring spectra, and we prove, in the presence of this condition, the existence of new Blumberg–Mandell-type localization sequences for algebraic K-theory. In particular, we confirm a conjecture of Ausoni–Rognes on the behavior of K(KO) as well as a natural generalization of this conjecture to K(Tmf). An early success of Quillen’s algebraic K-theory was his localization sequence [Qui73, §5]. This sequence shows that algebraic K-theory enjoys a strong excision property that permits one to cut out certain subvarieties. For example, suppose R a regular noetherian ring, and suppose x ∈ R an element such that the quotient R/x is also regular. Then the localization sequence takes the form of a long exact sequence · · · Kn(R/x) Kn(R) Kn(R[x]) Kn−1(R/x) · · · . The aim of this short paper is to identify a regularity property for structured ring spectra that allows one to prove a natural analogue of Quillen’s theorem in this context. To obtain Quillen’s sequence, one may begin with a localization sequence · · · Kn(Nil(R,x)) Gn(R) Gn(R[x]) Kn−1(Nil(R,x)) · · · where Nil(R,x) is the category of finitely generated, x-nilpotent R-modules. This sequence is exact irrespective of any regularity hypotheses on R and x; for example, it is an instance of the Fibration Theorem of Waldhausen [Wal85, Th. 1.6.4]. The regularity condition actually enters twice to convert this sequence into the sequence above. First, one deduces isomorphisms G∗(R/x) ∼= K∗(R/x), G∗(R) ∼= K∗(R), and G∗(R[x]) ∼= K∗(R[x]) from corresponding equivalences of derived categories. Second, one uses Quillen’s Dévissage Theorem [Qui73, Th. 4] to identify K∗(Nil(R,x)) and G∗(R/x). What’s remarkable about Quillen’s Dévissage Theorem is that it provides an equivalence of K-theories that is not induced by an equivalence of derived categories. Results of this kind are rare commodities, but the work of Blumberg and Mandell [BM08] provides another Dévissage Theorem: they show that for any connective E1 ring Λ, the algebraic K-theory of the category of Λ-modules with finitely generated homotopy and finite Postnikov towers is the same as the algebraic Ktheory of π0Λ. A version of the Blumberg–Mandell Dévissage Theorem can, for example, be deduced from the Theorem of the Heart for Waldhausen K-theory [Bar], which states that the inclusion A ♥ ⊂ A of the heart of a bounded t-structure on a stable ∞-category A induces an isomorphism K∗(A ♥) ∼= K∗(A ).