Spanier–Whitehead K-duality for C∗-algebras

Classical Spanier–Whitehead duality was introduced for the stable homotopy category of finite CW complexes. Here we provide a comprehensive treatment of a noncommutative version, termed Spanier–Whitehead [Formula: see text]-duality, which is defined on the category of [Formula: see text]-algebras whose [Formula: see text]-theory is finitely generated and that satisfy the UCT, with morphisms the [Formula: see text]-groups. We explore what happens when these assumptions are relaxed in various ways. In particular, we consider the relationship between Paschke duality and Spanier–Whitehead [Formula: see text]-duality.

ENDOMORPHISMS OF EXOTIC MODELS

We calculate the endomorphism dga of Franke's exotic algebraic model for the K-local stable homotopy category at odd primes. We unravel its original abstract structure to give explicit generators, differentials and products.

Motivic and real étale stable homotopy theory

Let$S$be a Noetherian scheme of finite dimension and denote by$\unicode[STIX]{x1D70C}\in [\unicode[STIX]{x1D7D9},\mathbb{G}_{m}]_{\mathbf{SH}(S)}$the (additive inverse of the) morphism corresponding to$-1\in {\mathcal{O}}^{\times }(S)$. Here$\mathbf{SH}(S)$denotes the motivic stable homotopy category. We show that the category obtained by inverting$\unicode[STIX]{x1D70C}$in$\mathbf{SH}(S)$is canonically equivalent to the (simplicial) local stable homotopy category of the site$S_{\text{r}\acute{\text{e}}\text{t}}$, by which we mean thesmallreal étale site of$S$, comprised of étale schemes over$S$with the real étale topology. One immediate application is that$\mathbf{SH}(\mathbb{R})[\unicode[STIX]{x1D70C}^{-1}]$is equivalent to the classical stable homotopy category. In particular this computes all the stable homotopy sheaves of the$\unicode[STIX]{x1D70C}$-local sphere (over$\mathbb{R}$). As further applications we show that$D_{\mathbb{A}^{1}}(k,\mathbb{Z}[1/2])^{-}\simeq \mathbf{DM}_{W}(k)[1/2]$(improving a result of Ananyevskiy–Levine–Panin), reprove Röndigs’ result that$\text{}\underline{\unicode[STIX]{x1D70B}}_{i}(\unicode[STIX]{x1D7D9}[1/\unicode[STIX]{x1D702},1/2])=0$for$i=1,2$and establish some new rigidity results.

A monoidal algebraic model for rational SO(2)-spectra

The category of rational SO(2)–equivariant spectra admits an algebraic model. That is, there is an abelian category ${\mathcal A}$(SO(2)) whose derived category is equivalent to the homotopy category of rational SO(2)–equivariant spectra. An important question is: does this algebraic model capture the smash product of spectra?The category ${\mathcal A}$(SO(2)) is known as Greenlees' standard model, it is an abelian category that has no projective objects and is constructed from modules over a non–Noetherian ring. As a consequence, the standard techniques for constructing a monoidal model structure cannot be applied. In this paper a monoidal model structure on ${\mathcal A}$(SO(2)) is constructed and the derived tensor product on the homotopy category is shown to be compatible with the smash product of spectra. The method used is related to techniques developed by the author in earlier joint work with Roitzheim. That work constructed a monoidal model structure on Franke's exotic model for the K(p)–local stable homotopy category.A monoidal Quillen equivalence to a simpler monoidal model category R•-mod that has explicit generating sets is also given. Having monoidal model structures on ${\mathcal A}$(SO(2)) and R•-mod removes a serious obstruction to constructing a series of monoidal Quillen equivalences between the algebraic model and rational SO(2)–equivariant spectra.

Noncommutative stable homotopy and stable infinity categories

The noncommutative stable homotopy category NSH is a triangulated category that is the universal receptacle for triangulated homology theories on separable C*-algebras. We show that the triangulated category NSH is topological as defined by Schwede using the formalism of (stable) infinity categories. More precisely, we construct a stable presentable infinity category of noncommutative spectra and show that NSHop sits inside its homotopy category as a full triangulated subcategory, from which the above result can be deduced. We also introduce a presentable infinity category of noncommutative pointed spaces that subsumes C*-algebras and define the noncommutative stable (co)homotopy groups of such noncommutative spaces generalizing earlier definitions for separable C*-algebras. The triangulated homotopy category of noncommutative spectra admits (co)products and satisfies Brown representability. These properties enable us to analyze neatly the behavior of the noncommutative stable (co)homotopy groups with respect to certain (co)limits. Along the way we obtain infinity categorical models for some well-known bivariant homology theories like KK-theory, E-theory, and connective E-theory via suitable (co)localizations. The stable infinity category of noncommutative spectra can also be used to produce new examples of generalized (co)homology theories for noncommutative spaces.