Ring Spectra Research Articles

A simplicial calculus for local intersection numbers at non-archimedian places on products of semi-stable curves

We analyse the subring of the Chow ring with support generated by the irreducible components of the special fibre of the Gross–Schoen desingularization of a d-fold self product of a semi-stable curve over the spectrum of a discrete valuation ring. For this purpose we develop a calculus which allows to determine intersection numbers in the special fibre explicitly. As input our simplicial calculus needs only combinatorial data of the special fibre. It yields a practical procedure for calculating even self intersections in the special fibre. The first ingredient of our simplicial calculus is a localization formula, which reduces the problem of calculating intersection numbers to a special situation. In order to illustrate how our simplicial calculus works, we calculate all intersection numbers between divisors with support in the special fibre in dimension three and four. The localization formula and the general idea were already presented for \(d=2\) in a paper of Zhang (Invent Math 179(1):1–73, 2010, §3). In our present work we achieve a generalisation to arbitrary d.

On the algebraicK-theory of higher categories

We prove that Waldhausen K-theory, when extended to a very general class of quasicategories, can be described as a Goodwillie differential. In particular, K-theory spaces admit canonical (connective) deloopings, and the K-theory functor enjoys a universal property. Using this, we give new, higher categorical proofs of both the additivity and fibration theorems of Waldhausen. As applications of this technology, we study the algebraic K-theory of associative ring spectra and spectral Deligne-Mumford stacks.

Engenera

Let R be an E_2 ring spectrum with zero odd dimensional homotopy groups. Every map of ring spectra MU to R is represented by a map of E_2 ring spectra. If 2 is invertible in pi_0(R), then every map of ring spectra MSO to R is represented by a map of E_2 ring spectra.

Universality of multiplicative infinite loop space machines

We establish a canonical and unique tensor product for commutative monoids and groups in an infinity-category C which generalizes the ordinary tensor product of abelian groups. Using this tensor product we show that E_n-(semi)ring objects in C give rise to E_n-ring spectrum objects in C. In the case that C is the infinity-category of spaces this produces a multiplicative infinite loop space machine which can be applied to the algebraic K-theory of rings and ring spectra. The main tool we use to establish these results is the theory of smashing localizations of presentable infinity-categories. In particular, we identify preadditive and additive infinity-categories as the local objects for certain smashing localizations. A central theme is the stability of algebraic structures under basechange; for example, we show Ring(D \otimes C) = Ring(D) \otimes C. Lastly, we also consider these algebraic structures from the perspective of Lawvere algebraic theories in infinity-categories.

Higher Hochschild cohomology of the Lubin–Tate ring spectrum

We give a method for computing factorization homology of $\oper{E}_n$-algebra using as an input an algebraic version of higher Hochschild homology due to Pirashvili. We then show how to compute higher Hochschild homology and cohomology when the algebra is \'etale in a sense that we make precise. As an application, we compute higher Hochschild cohomology of the Lubin-Tate ring spectrum.

Open Set Lattices of Subspaces of Spectrum Spaces

AbstractWe take a unified approach to study the open set lattices of various subspaces of the spectrum of a multiplicative lattice L. The main aim is to establish the order isomorphism between the open set lattice of the respective subspace and a sub-poset of L. The motivating result is the well known fact that the topology of the spectrum of a commutative ring R with identity is isomorphic to the lattice of all radical ideals of R. The main results are as follows: (i) for a given nonempty set S of prime elements of a multiplicative lattice L, we define the S-semiprime elements and prove that the open set lattice of the subspace S of Spec(L) is isomorphic to the lattice of all S-semiprime elements of L; (ii) if L is a continuous lattice, then the open set lattice of the prime spectrum of L is isomorphic to the lattice of all m-semiprime elements of L; (iii) we define the pure elements, a generalization of the notion of pure ideals in a multiplicative lattice and prove that for certain types of multiplicative lattices, the sub-poset of pure elements of L is isomorphic to the open set lattice of the subspace Max(L) consisting of all maximal elements of L.

Ausoni–Bökstedt duality for topological Hochschild homology

We consider the Gorenstein condition for topological Hochschild homology, and show that it holds remarkably often [7]. More precisely, if R is a commutative ring spectrum and R⟶k is a map to a field of characteristic p then, provided k is small as an R-module, THH(R;k) is Gorenstein in the sense of [11]. In particular, this holds if R is a (conventional) regular local ring with residue field k of characteristic p.Using only Bökstedt's calculation of THH(k), this gives a non-calculational proof of dualities visible in calculations of Bökstedt [9], Ausoni [3], Lindenstrauss and Madsen [17], Angeltveit and Rognes [2] and others.

A connectedness theorem over the spectrum of a formal power series ring

We study the connectedness of the non-klt locus over the spectrum of a formal power series ring. In dimension 3, we prove the existence and normality of the smallest lc center, and apply it to the ACC for minimal log discrepancies greater than 1 on smooth 3-folds.

TheER(n)-cohomology ofBO(q) and real Johnson–Wilson orientations for vector bundles

Using the Bockstein spectral sequence developed previously by the authors, we compute the ring ER(n)^*(BO(q)) explicitly. We then use this calculation to show that the ring spectrum MO[2^{n+1}] is ER(n)-orientable (but not ER(n+1)-orientable), where MO[2^{n+1}] is defined as the Thom spectrum for the self map of BO given by multiplication by 2^{n+1}.

The impact of vibrational Raman scattering of air on DOAS measurements of atmospheric trace gases

Abstract. In remote sensing applications, such as differential optical absorption spectroscopy (DOAS), atmospheric scattering processes need to be considered. After inelastic scattering on N2 and O2 molecules, the scattered photons occur as additional intensity at a different wavelength, effectively leading to "filling-in" of both solar Fraunhofer lines and absorptions of atmospheric constituents, if the inelastic scattering happens after the absorption. Measured spectra in passive DOAS applications are typically corrected for rotational Raman scattering (RRS), also called Ring effect, which represents the main contribution to inelastic scattering. Inelastic scattering can also occur in liquid water, and its influence on DOAS measurements has been observed over clear ocean water. In contrast to that, vibrational Raman scattering (VRS) of N2 and O2 has often been thought to be negligible, but it also contributes. Consequences of VRS are red-shifted Fraunhofer structures in scattered light spectra and filling-in of Fraunhofer lines, additional to RRS. At 393 nm, the spectral shift is 25 and 40 nm for VRS of O2 and N2, respectively. We describe how to calculate VRS correction spectra according to the Ring spectrum. We use the VRS correction spectra in the spectral range of 420–440 nm to determine the relative magnitude of the cross-sections of VRS of O2 and N2 and RRS of air. The effect of VRS is shown for the first time in spectral evaluations of Multi-Axis DOAS data from the SOPRAN M91 campaign and the MAD-CAT MAX-DOAS intercomparison campaign. The measurements yield in agreement with calculated scattering cross-sections that the observed VRS(N2) cross-section at 393 nm amounts to 2.3 ± 0.4 % of the cross-section of RRS at 433 nm under tropospheric conditions. The contribution of VRS(O2) is also found to be in agreement with calculated scattering cross-sections. It is concluded, that this phenomenon has to be included in the spectral evaluation of weak absorbers as it reduces the measurement error significantly and can cause apparent differential optical depth of up to 3 ×10−4. Its influence on the spectral retrieval of IO, glyoxal, water vapour and NO2 in the blue wavelength range is evaluated for M91. For measurements with a large Ring signal a significant and systematic bias of NO2 dSCDs (differential slant column densities) up to (−3.8 ± 0.4) × 1014 molec cm−2 is observed if this effect is not considered. The effect is typically negligible for DOAS fits with an RMS (root mean square) larger than 4 × 10−4.

Completed power operations for MoravaE–theory

We construct and study an algebraic theory which closely approximates the theory of power operations for Morava E-theory, extending previous work of Charles Rezk in a way that takes completions into account. These algebraic structures are made explicit in the case of K-theory. Methodologically, we emphasize the utility of flat modules in this context, and prove a general version of Lazard's flatness criterion for module spectra over associative ring spectra.

An extension of a theorem of Hartshorne

We extend a classical theorem of Hartshorne concerning the connectedness of the punctured spectrum of a local ring by analyzing the homology groups of a simplicial complex associated with the minimal primes of a local ring.

Operadic multiplications in equivariant spectra, norms, and transfers

We study homotopy-coherent commutative multiplicative structures on equivariant spaces and spectra. We define N∞ operads, equivariant generalizations of E∞ operads. Algebras in equivariant spectra over an N∞ operad model homotopically commutative equivariant ring spectra that only admit certain collections of Hill–Hopkins–Ravenel norms, determined by the operad. Analogously, algebras in equivariant spaces over an N∞ operad provide explicit constructions of certain transfers. This characterization yields a conceptual explanation of the structure of equivariant infinite loop spaces.To explain the relationship between norms, transfers, and N∞ operads, we discuss the general features of these operads, linking their properties to families of finite sets with group actions and analyzing their behavior under norms and geometric fixed points. A surprising consequence of our study is that in stark contract to the classical setting, equivariantly the little disks and linear isometries operads for a general incomplete universe U need not determine the same algebras.Our work is motivated by the need to provide a framework to describe the flavors of commutativity seen in recent work of the second author and Hopkins on localization of equivariant commutative ring spectra.

On a nilpotence conjecture of J. P. May

We prove a conjecture of J.P. May concerning the nilpotence of elements in ring spectra with power operations, i.e., $H_\infty$-ring spectra. Using an explicit nilpotence bound on the torsion elements in $K(n)$-local $H_\infty$-algebras over $E_n$, we reduce the conjecture to the nilpotence theorem of Devinatz, Hopkins, and Smith. As corollaries we obtain nilpotence results in various bordism rings including $M\mathit{Spin}_*$ and $M\mathit{String}_*$, results about the behavior of the Adams spectral sequence for $E_\infty$-ring spectra, and the non-existence of $E_\infty$-ring structures on certain complex oriented ring spectra.

A thick subcategory theorem for modules over certain ring spectra

We classify thick subcategories of the $\infty$-categories of perfect modules over ring spectra which arise as functions on even periodic derived stacks satisfying affineness and regularity conditions. For example, we show that the thick subcategories of perfect modules over $\mathrm{TMF}$ are in natural bijection with the subsets of the underlying space of the moduli stack of elliptic curves which are closed under specialization.

Landau level quantization and almost flat modes in three-dimensional semimetals with nodal ring spectra

We investigate Landau level structures of semimetals with nodal ring dispersions. When the magnetic field is applied parallel to the plane in which the ring lies, there exist almost nondispersive Landau levels at the Fermi level (${E}_{F}=0$) as a function of the momentum along the field direction inside the ring. We show that the Landau levels at each momentum along the field direction can be described by the Hamiltonian for the graphene bilayer with fictitious interlayer couplings under a tilted magnetic field. Near the center of the ring where the in-terlayer coupling is negligible, we have Dirac Landau levels which explain the appearance of the zero modes. Although the interlayer hopping amplitudes become finite at higher momenta, the splitting of zero modes is exponentially small and they remain almost flat due to the finite artificial in-plane component of the magnetic field. The emergence of the density of states peak at the Fermi level would be a hallmark of the ring dispersion.

COX RINGS OF MINIMAL RESOLUTIONS OF SURFACE QUOTIENT SINGULARITIES

AbstractWe investigate Cox rings of minimal resolutions of surface quotient singularities and provide two descriptions of these rings. The first one is the equation for the spectrum of a Cox ring, which is a hypersurface in an affine space. The second is the set of generators of the Cox ring viewed as a subring of the coordinate ring of a product of a torus and another surface quotient singularity. In addition, we obtain an explicit description of the minimal resolution as a divisor in a toric variety.

Decomposition Matrices are Generically Trivial

We establish a genericity property in the representation theory of a flat family of finite-dimensional algebras in the sense of Cline-Parshal-Scott. More precisely, we show that the decomposition matrices as introduced by Geck and Rouquier of an algebra which is free of finite dimension over a noetherian integral domain and which splits over the fraction field of this ring are generically trivial, i.e., they are trivial in an open neighborhood of the generic point of the spectrum of the base ring. This generalizes a classical result by Brauer in modular representation theory of finite groups. We furthermore show that this is true precisely on an open set in case all fibers of the algebra split. In this way we get a stratification of the base scheme such that decomposition maps are trivial on each stratum. Moreover, this defines a new discriminant of such algebras which generalizes Schur elements of simple modules for symmetric split semisimple algebras. We provide some extensions to the theory of decomposition maps allowing us to work without the usual normality assumption on the base ring.

Power Operations and Coactions in Highly Commutative Homology Theories

Power operations in the homology of infinite loop spaces, and H_\infty or E_\infty ring spectra have a long history in Algebraic Topology. In the case of ordinary mod p homology for a prime p , the power operations of Kudo, Araki, Dyer and Lashof interact with Steenrod operations via the Nishida relations, but for many purposes this leads to complicated calculations once iterated applications of these functions are required. On the other hand, the homology coaction turns out to provide tractable formulae better suited to exploiting multiplicative structure. We show how to derive suitable formulae for the interaction between power operations and homology coactions in a wide class of examples; our approach makes crucial use of modern frameworks for spectra with well behaved smash products. In the case of mod p homology, our formulae extend those of Bisson and Joyal to odd primes. We also show how to exploit our results in sample calculations, and produce some apparently new formulae for the Dyer-Lashof action on the dual Steenrod algebra.

Affineness and chromatic homotopy theory

Given an algebraic stack $X$, one may compare the derived category of quasi-coherent sheaves on $X$ with the category of dg-modules over the dg-ring of functions on $X$. We study the analogous question in stable homotopy theory, for derived stacks that arise via realizations of diagrams of Landweber-exact homology theories. We identify a condition (quasi-affineness of the map to the moduli stack of formal groups) under which the two categories are equivalent, and study applications to topological modular forms. In particular, we provide new examples of Galois extensions of ring spectra and vanishing results about Tate spectra.