Ring Spectra Research Articles

Bundles of spectra and algebraic K-theory

A parametrized spectrum E is a family of spectra E_x continuously parametrized by the points x of a topological space X. We take the point of view that a parametrized spectrum is a bundle-theoretic geometric object. When R is a ring spectrum, we consider parametrized R-module spectra and show that they give cocycles for the cohomology theory determined by the algebraic K-theory K(R) of R in a manner analogous to the description of topological K-theory K^0(X) as the Grothendieck group of vector bundles over X. We prove a classification theorem for parametrized spectra, showing that parametrized spectra over X whose fibers are equivalent to a fixed R-module M are classified by homotopy classes of maps from X to the classifying space BAut_R(M) of the A_\infty space of R-module equivalences from M to M. In proving the classification theorem for parametrized spectra, we define of the notion of a principal G fibration where G is an A_\infty space and prove a similar classification theorem for principal G fibrations.

Near isotropic behavior of turbulent thermal convection

We investigate the anisotropy in turbulent convection in a 3D box using direct numerical simulation. We compute the anisotropic parameter $A = u_\perp^{2}/(2u_{\parallel}^{2})$, where $u_{\perp}$ and $u_{\parallel}$ are the components of velocity perpendicular and parallel to the buoyancy direction, the shell and ring spectra, and shell-to-shell energy transfers. We observe that the flow is nearly isotropic for the Prandtl number $\mathrm{Pr} \approx 1$, but the anisotropy increases with the Prandtl number. For $\mathrm{Pr}=\infty$, $A \approx 0.3$, thus anisotropy is not very significant even in extreme cases. We also observe that $u_{\parallel}$ feeds energy to $u_{\perp}$ via pressure. The computation of shell-to-shell energy transfers reveals that the energy transfer in turbulent convection is local and forward, similar to hydrodynamic turbulence. These results are consistent with the Kolmogorov's spectrum observed by Kumar et al.~[Phys. Rev. E {\bf 90}, 023016 (2014)] for turbulent convection.

The topological cyclic homology of the dual circle

We give a new proof of a result of Lazarev, that the dual of the circle S+1 in the category of spectra is equivalent to a strictly square-zero extension as an associative ring spectrum. As an application, we calculate the topological cyclic homology of DS1 and rule out a Koszul-dual reformulation of the Novikov conjecture.

Quasi-Bessel beams with longitudinally varying polarization state generated by employing spectrum engineering.

We report on the generation and control of quasi-Bessel beams having not only a uniform axial intensity but also a longitudinally varying polarization state. Based on the optimization routine of the spatial spectrum of the so-called Durnin ring, we generate quasi-Bessel beams possessing longitudinally variant axial intensity with linear or sinusoidal envelopes. By utilizing a Sagnac interferometer, we create and coaxially composite two orthogonally polarized beams with complementary axial intensities to form quasi-Bessel beams with uniform axial intensity but a longitudinally varying polarization state. Furthermore, we demonstrate the possibility and flexibility of manipulating the trajectory, speed, and period of polarization state transformation upon propagation. Our results may enable Bessel beams to be used in other applications including optical communications, material processing, and polarimetry.

Totally ordered sets and the prime spectra of rings

ABSTRACTLet T be a totally ordered set and let D(T) denotes the set of all cuts of T. We prove the existence of a discrete valuation domain Ov such that T is order isomorphic to two special subsets of Spec(Ov). We prove that if A is a ring (not necessarily commutative), whose prime spectrum is totally ordered and satisfies (K2), then there exists a totally ordered set U⊆Spec(A) such that the prime spectrum of A is order isomorphic to D(U). We also present equivalent conditions for a totally ordered set to be a Dedekind totally ordered set. At the end, we present an algebraic geometry point of view.

Affine weakly regular tensor triangulated categories

We prove that the Balmer spectrum of a tensor triangulated category is homeomorphic to the Zariski spectrum of its graded central ring, provided the triangulated category is generated by its tensor unit and the graded central ring is noetherian and regular in a weak sense. There follows a classification of all thick subcategories, and the result extends to the compactly generated setting to yield a classification of all localizing subcategories as well as the analog of the telescope conjecture. This generalizes results of Shamir for commutative ring spectra.

Prüfer algebraic spaces

This is the first in a series of two papers concerned with relative birational geometry of algebraic spaces. In this paper, we study Prufer spaces and Prufer pairs of algebraic spaces that generalize spectra of Prufer rings. As a particular case of Prufer spaces we introduce valuation algebraic spaces, and use them to establish valuative criteria of separatedness and properness that sharpen the standard criteria. In a sequel paper, we introduce a version of Riemann–Zariski spaces, and prove Nagata’s compactification theorem for algebraic spaces.

Multiplicative differential algebraic K-theory and applications

We construct a version of Beilinson's regulator as a map of sheaves of commutative ring spectra and use it to define a multiplicative variant of differential algebraic K-theory. We use this theory to give an interpretation of Bloch's construction of K_3-classes and the relation with dilogarithms. Furthermore, we provide a relation to Arakelov theory via the arithmetic degree of metrized line bundles, and we give a proof of the formality of the algebraic K-theory of number rings.

The strong ultrafilter topology on spaces of ideals

The patch/constructible refinement of the Zariski topology on the prime spectrum of a commutative ring is well known and well studied. Recently, Fontana and Loper gave an equivalent definition of this topology using ultrafilters. In this note we distinguish between two different types of ultrafilter convergence and use them to define two new topologies on the prime spectrum of a ring. We study various properties of these topologies. As applications we use the ultrafilters to classify all the compact subsets of a spectral space in the Zariski topology and we classify Grothendieck's retrocompact spaces again using ultrafilters.

Homotopy automorphisms of R-module bundles, and the K-theory of string topology

Let R be a ring spectrum and \( \mathcal {E}\rightarrow X\) an R-module bundle of rank n. Our main result is to identify the homotopy type of the group-like monoid of homotopy automorphisms of this bundle, \(hAut^R(\mathcal {E})\). This will generalize the result regarding R-line bundles proven by Cohen and Jones (Mex Bull Math, 2016). The main application is the calculation of the homotopy type of \(BGL_n(End ((\mathcal {L}))\) where \(\mathcal {L}\rightarrow X\) is any R-line bundle, and \(End^R(\mathcal {L})\) is the ring spectrum of endomorphisms. In the case when such a bundle is the fiberwise suspension spectrum of a principal bundle over a manifold, \(G \rightarrow P \rightarrow M\), this leads to a description of the K-theory of the string topology spectrum in terms of the mapping space from M to \(BGL \left( \Sigma ^\infty (G_+)\right) \).

Gauge theory and string topology

Given a principal bundle over a closed manifold, \(G \rightarrow P \rightarrow M\), let \(P^{Ad} \rightarrow M\) be the associated adjoint bundle. Gruher and Salvatore (Proc Lond Math Soc 96(3), 78106 2008) showed that the Thom spectrum \((P^\mathrm{Ad})^{-TM}\) is a ring spectrum whose corresponding product in homology is a Chas-Sullivan type string topology product. We refer to this spectrum as the “string topology spectrum of P”, \( \mathcal {S}(P)\). In the universal case when P is contractible, \(\mathcal {S}(P) \simeq LM^{-TM}\) where LM is the free loop space of the manifold. This ring spectrum was introduced by the authors in Cohen et al. (Math Annalen 324, 773–798 2002) as a homotopy theoretic realization of the Chas-Sullivan string topology of M. The main purpose of this paper is to introduce an action of the gauge group of the principal bundle, \(\mathcal {G}(P)\) on the string topology spectrum \(\mathcal {S}(P)\), and to study this action in detail. Indeed we study the entire group of units and the induced representation \(\mathcal {G}(P) \rightarrow GL_1(\mathcal {S}(P))\). We show that this group of units is the group of homotopy automorphisms of the fiberwise suspension spectrum of P. More generally we describe the homotopy type of the group of homotopy automorphisms of any E-line bundle for any ring spectrum E. We import some of the basic ideas of gauge theory, such as the action of the gauge group on the space of connections to the setting of E-line bundles over a manifold and do explicit calculations. We end by discussing a functorial perspective, which describes a sense in which the string topology spectrum \(\mathcal {S}(P)\) of a principal bundle is the “linearization” of the gauge group \(\mathcal {G}(P)\).

Grothendieck–Neeman duality and the Wirthmüller isomorphism

We clarify the relationship between Grothendieck duality à la Neeman and the Wirthmüller isomorphism à la Fausk–Hu–May. We exhibit an interesting pattern of symmetry in the existence of adjoint functors between compactly generated tensor-triangulated categories, which leads to a surprising trichotomy: there exist either exactly three adjoints, exactly five, or infinitely many. We highlight the importance of so-called relative dualizing objects and explain how they give rise to dualities on canonical subcategories. This yields a duality theory rich enough to capture the main features of Grothendieck duality in algebraic geometry, of generalized Pontryagin–Matlis duality à la Dwyer–Greenless–Iyengar in the theory of ring spectra, and of Brown–Comenetz duality à la Neeman in stable homotopy theory.

Higher homotopy excision and Blakers–Massey theorems for structured ring spectra

Working in the context of symmetric spectra, we prove higher homotopy excision and higher Blakers–Massey theorems, and their duals, for algebras and left modules over operads in the category of modules over a commutative ring spectrum (e.g., structured ring spectra).

Spectra of units for periodic ring spectra and group completion of graded E∞ spaces

We construct a new spectrum of units for a commutative symmetric ring spectrum that detects the difference between a periodic ring spectrum and its connective cover. It is augmented over the sphere spectrum. The homotopy cofiber of its augmentation map is a non-connected delooping of the usual spectrum of units whose bottom homotopy group detects periodicity. Our approach builds on the graded variant of E-infinity spaces introduced in joint work with Christian Schlichtkrull. We construct a group completion model structure for graded E-infinity spaces and use it to exhibit our spectrum of units functor as right adjoint on the level of homotopy categories. The resulting group completion functor is an essential tool for studying ring spectra with graded logarithmic structures.

Higher homotopy commutativity of small ring spectra

Let X be a 2-local ring spectrum of type n which is homotopy commutative up to a finite level of coherence (homotopy associativity is not assumed). We prove that, if \(f:\Sigma ^kX\rightarrow X\) is a \(v_n\) self-map, there exists an N such that, for all \(m>0\), the cofiber \(C(f^{mN})\) of \(f^{mN}\) has a ring spectrum structure extending that on X which is homotopy commutative up to the same level of coherence. A strong analogue of this result holds at odd primes. We also discuss the relevance of this result to the problem of constructing explicit \(v_{n+1}\) self-maps and provide a framework for extending our results to higher multiplicative structures including higher homotopy associativity.

Mathcal {E}_infty ring spectra and elements of Hopf invariant\xa01

The 2-primary Hopf invariant 1 elements in the stable homotopy groups of spheres form the most accessible family of elements. In this paper, we explore some properties of the mathcal {E}_infty ring spectra obtained from certain iterated mapping cones by applying the free algebra functor. In fact, these are equivalent to Thom spectra over infinite loop spaces related to the classifying spaces Bmathrm {SO},,Bmathrm {Spin},,Bmathrm {String}. We show that the homology of these Thom spectra are all extended comodule algebras of the form mathcal {A}_*square _{mathcal {A}(r)_*}P_* over the dual Steenrod algebra mathcal {A}_* with mathcal {A}_*square _{mathcal {A}(r)_*}mathbb {F}_2 as an algebra retract. This suggests that these spectra might be wedges of module spectra over the ring spectra Hmathbb {Z}, kmathrm {O} or mathrm {tmf}; however, apart from the first case, we have no concrete results on this.

Topological Hochschild homology and the cyclic bar construction in symmetric spectra

The cyclic bar construction in symmetric spectra and Bökstedt’s original construction are two possible ways to define the topological Hochschild homology of a symmetric ring spectrum. In this short note we explain how to correct an error in Shipley’s original comparison of these two approaches.

A spectrum-level Hodge filtration on topological Hochschild homology

We define a functorial spectrum-level filtration on the topological Hochschild homology of any commutative ring spectrum R, and more generally the factorization homology $$R \otimes X$$ for any space X, echoing algebraic constructions of Loday and Pirashvili. We give a geometric description of this filtration, investigate its multiplicative properties, and show that it breaks $${\text {THH}}$$ up into common eigenspectra of the Adams operations.

Abstract representation theory of Dynkin quivers of type A

We study the representation theory of Dynkin quivers of type A in abstract stable homotopy theories, including those associated to fields, rings, schemes, differential-graded algebras, and ring spectra. Reflection functors, (partial) Coxeter functors, and Serre functors are defined in this generality and these equivalences are shown to be induced by universal tilting modules, certain explicitly constructed spectral bimodules. In fact, these universal tilting modules are spectral refinements of classical tilting complexes. As a consequence we obtain split epimorphisms from the spectral Picard groupoid to derived Picard groupoids over arbitrary fields.These results are consequences of a more general calculus of spectral bimodules and admissible morphisms of stable derivators. As further applications of this calculus we obtain examples of universal tilting modules which are new even in the context of representations over a field. This includes Yoneda bimodules on mesh categories which encode all the other universal tilting modules and which lead to a spectral Serre duality result.Finally, using abstract representation theory of linearly oriented An-quivers, we construct canonical higher triangulations in stable derivators and hence, a posteriori, in stable model categories and stable ∞-categories.

Galois descent for real spectra

We prove analogs of faithfully flat descent and Galois descent for categories of modules over \(E_{\infty }\)-ring spectra using the \(\infty \)-categorical Barr-Beck theorem proved by Lurie. In particular, faithful G-Galois extensions are shown to be of effective descent for modules. Using this we study the category of ER(n)-modules, where ER(n) is the \(\mathbb {Z}/2\)-fixed points under complex conjugation of a generalized Johnson-Wilson spectrum E(n). In particular, we show that ER(n)-modules is equivalent to \(\mathbb {Z}\)/2-equivariant E(n)-modules as stable \(\infty \)-categories.