Ring Spectra Research Articles

The cotangent complex and Thom spectra

The cotangent complex of a map of commutative rings is a central object in deformation theory. Since the 1990s, it has been generalized to the homotopical setting of E_infty -ring spectra in various ways. In this work we first establish, in the context of infty -categories and using Goodwillie’s calculus of functors, that various definitions of the cotangent complex of a map of E_infty -ring spectra that exist in the literature are equivalent. We then turn our attention to a specific example. Let R be an E_infty -ring spectrum and mathrm {Pic}(R) denote its Picard E_infty -group. Let Mf denote the Thom E_infty -R-algebra of a map of E_infty -groups f:Grightarrow mathrm {Pic}(R); examples of Mf are given by various flavors of cobordism spectra. We prove that the cotangent complex of Rrightarrow Mf is equivalent to the smash product of Mf and the connective spectrum associated to G.

Determinant map for the prestack of Tate objects

We construct a map from the prestack of Tate objects over a commutative ring $k$ to the stack of $\mathbb{G}_{\rm m}$-gerbes. The result is obtained by combining the determinant map from the stack of perfect complexes as proposed by Sch\"urg-To\"en-Vezzosi with a relative $S_{\bullet}$-construction for Tate objects as studied by Braunling-Groechenig-Wolfson. Along the way we prove a result about the K-theory of vector bundles over a connective $\mathbb{E}_{\infty}$-ring spectrum which is possibly of independent interest.

Multiplicative properties of integer valued polynomials over split-quaternions

We study localization properties and the prime spectrum of the integer-valued polynomial ring where (respectively ) is the algebra of split-quaternion over ℤ (respectively over ).

Strategies of porous network quinolone polymers: A comprehensive evaluation of their biological activity

Hydrazine polymers were synthesized under the condition of hydrazine hydrate/ethanol by a clever one-pot method, and then treated with methanesulfonyl chloride and acetyl chloride, respectively, two new quinolones were easily obtained. The basic structure characterization of the prepared quinolone polymers was established based on 1H NMR spectra of aromatic rings and branched chains. In terms of cell activity, MTT assay was used to study the anti-proliferation properties of HCT-116, A549, HepG-2, and McF-7 human cancer cell lines and normal cell lines L02 compared with positive control 5-FU.The distribution and nucleation of polymeric drugs in cells were further observed by means of cell imaging. Conventional Annexin V-FITC kit was subsequently used to detect cell cycle arrest with propidium iodide (PI)and to monitor apoptosis induction with flow cytometry. The results showed that both of the novel quinolone polymers had strong cytotoxic compounds to induce cell cycle stop and trigger apoptosis in the G2/M phase in different test cancer cells. In particular, polymer 1 was more effective.

The injective spectrum of a right noetherian ring

The injective spectrum is a topological space associated to a ring R, which agrees with the Zariski spectrum when R is commutative noetherian. We consider injective spectra of right noetherian rings (and locally noetherian Grothendieck categories) and establish some basic topological results and a functoriality result, as well as links between the topology and the Krull dimension of the ring (in the sense of Gabriel and Rentschler). Finally, we use these results to compute a number of examples.

An algebraic model for rational na\xefve-commutative ring SO(2)-spectra and equivariant elliptic cohomology

Equipping a non-equivariant topological text {E}_infty -operad with the trivial G-action gives an operad in G-spaces. For a G-spectrum, being an algebra over this operad does not provide any multiplicative norm maps on homotopy groups. Algebras over this operad are called naïve-commutative ring G-spectra. In this paper we take G=SO(2) and we show that commutative algebras in the algebraic model for rational SO(2)-spectra model rational naïve-commutative ring SO(2)-spectra. In particular, this applies to show that the SO(2)-equivariant cohomology associated to an elliptic curve C of Greenlees (Topology 44(6):1213–1279, 2005) is represented by an text {E}_infty -ring spectrum. Moreover, the category of modules over that text {E}_infty -ring spectrum is equivalent to the derived category of sheaves over the elliptic curve C with the Zariski torsion point topology.

The torsion in the cohomology of wild elliptic fibers

Given an elliptic fibration f:X→S over the spectrum of a complete discrete valuation ring with algebraically closed residue field, we use a Hochschild–Serre spectral sequence to express the torsion in R1f⁎OX as the first group cohomology H1(G,H0(S′,OS′)). Here, G is the Galois group of the maximal extension K′/K such that the normalization of X×SS′ induces an étale covering of X, where S′ is the normalization of S in K′. The case where S is a Dedekind scheme is easily reduced to the local case. Moreover, we generalize to higher-dimensional fibrations.

Local Gorenstein duality for cochains on spaces

We investigate when a commutative ring spectrum R satisfies a homotopical version of local Gorenstein duality, extending the notion previously studied by Greenlees. In order to do this, we prove an ascent theorem for local Gorenstein duality along morphisms of k-algebras. Our main examples are of the form R=C⁎(X;k), the ring spectrum of cochains on a space X for a field k. In particular, we establish local Gorenstein duality in characteristic p for p-compact groups and p-local finite groups as well as for k=Q and X a simply connected space which is Gorenstein in the sense of Dwyer, Greenlees, and Iyengar.

Discrete orderings in the real spectrum

We study discrete orderings in the real spectrum of a commutative ring by defining discrete prime cones and give an algebro-geometric meaning to some kind of diophantine problems over discretely ordered rings. Also for a discretely ordered ring M and a real closed field R containing M we prove a theorem on the distribution of the discrete orderings of M[X1,…,Xn] in Specr(R[X1,…,Xn]) in geometric terms. To be more precise, we prove that any ball B(α,r) in Specr(R[X1,…,Xn]) with center α and radius r (defined via Robson's metric) contains a discrete ordering of M[X1,…,Xn] whenever r is positive non-infinitesimal and α is at infinite distance from all hyperplanes over M.

On the electronic structure of benzene and borazine: an algebraic description

The spectrum of a hexagonal ring is analysed using concepts of group theory and a tight-binding model with first, second and third neighbours. The two doublets in the spectrum are explained with the C3 symmetry group together with time-reversal symmetry. Degeneracy lifts are induced by means of various mechanisms. Conjugation symmetry breaking is introduced via magnetic fields, while C3 breaking is studied with the introduction of defects, similar to the inclusion of fluorine atoms. Concrete applications to benzene and borazine are shown as an illustration of our description. Wave functions are described in connection with partial or full aromaticity. Electronic density currents are found for all cases. A detailed study of a supersymmetry in a 6-ring is presented and its consequences on electronic spectra are discussed.

On the Brun spectral sequence for topological Hochschild homology

We generalize a spectral sequence of Brun for the computation of topological Hochschild homology. The generalized version computes the $E$-homology of $THH(A;B)$, where $E$ is a ring spectrum, $A$ is a commutative $S$-algebra and $B$ is a connective commutative $A$-algebra. The input of the spectral sequence are the topological Hochschild homology groups of $B$ with coefficients in the $E$-homology groups of $B \wedge_A B$. The mod $p$ and $v_1$ topological Hochschild homology of connective complex $K$-theory has been computed by Ausoni and later again by Rognes, Sagave and Schlichtkrull. We present an alternative, short computation using the generalized Brun spectral sequence.

Completing perfect complexes

This note proposes a new method to complete a triangulated category, which is based on the notion of a Cauchy sequence. We apply this to categories of perfect complexes. It is shown that the bounded derived category of finitely presented modules over a right coherent ring is the completion of the category of perfect complexes. The result extends to non-affine noetherian schemes and gives rise to a direct construction of the singularity category. The parallel theory of completion for abelian categories is compatible with the completion of derived categories. There are three appendices. The first one by Tobias Barthel discusses the completion of perfect complexes for ring spectra. The second one by Tobias Barthel and Henning Krause refines for a separated noetherian scheme the description of the bounded derived category of coherent sheaves as a completion. The final appendix by Bernhard Keller introduces the concept of a morphic enhancement for triangulated categories and provides a foundation for completing a triangulated category.

Framed transfers and motivic fundamental classes

We relate the recognition principle for infinite $\mathbf P^1$-loop spaces to the theory of motivic fundamental classes of D\'eglise, Jin, and Khan. We first compare two kinds of transfers that are naturally defined on cohomology theories represented by motivic spectra: the framed transfers given by the recognition principle, which arise from Voevodsky's computation of the Nisnevish sheaf associated with $\mathbf A^n/(\mathbf A^n-0)$, and the Gysin transfers defined via Verdier's deformation to the normal cone. We then introduce the category of finite E-correspondences for E a motivic ring spectrum, generalizing Voevodsky's category of finite correspondences and Calm\`es and Fasel's category of finite Milnor-Witt correspondences. Using the formalism of fundamental classes, we show that the natural functor from the category of framed correspondences to the category of E-module spectra factors through the category of finite E-correspondences.

REAL K-COHOMOLOGY OF COMPLEX PROJECTIVE SPACES

In this paper, we determine the structure of KO-cohomology of complex projective space P l and its product space P l × P m as algebras over the coefficient ring KO ∗ . We also give a description of the map KO ∗ ( P l+m ) → KO ∗ ( P l × P m ) induced by the map that classifies the tensor product of the canonical line bundles and show that its image is not contained in the image of the cross product KO ∗ ( P l ) ⊗KO∗ KO ∗ ( P m ) → KO ∗ ( P l × P m ) to see that non-existence of the formal group structure on KO ∗ ( P ∞ ). Introduction A commutative ring spectrum E is said to be complex oriented if an element x of the reduced E-cohomology of the infinite dimensional complex projective space CP ∞ is given such that x maps to a generator of the reduced E-cohomology of 1-dimensional complex projective space CP 1 ((2)). We call such an element x a complex orientation of E. On the other hand, if E-homology E∗E of E is a flat over the coefficient ring E∗, E∗E has a structure of a Hopf algebroid and E-homology theory takes values in the category of E∗E-comodule, in other words, the category of representations of the groupoid represented by the affine groupoid scheme represented by E∗E ((1)). If E is a complex oriented ring spectrum, the E-cohomology of the complex projective space is just a truncated polynomial algebra over E∗ and it is shown that E-homology E∗E of E is a flat over E∗. Moreover the product structure of CP ∞ gives a one dimensional formal group law over E ∗ ((5)) which closely relates with the structure of the Hopf algebroid ((2)). The complex K-theory is one of the most basic examples of complex oriented cohomology theories. However, KO-spectrum representing the real K-theory is one of a few well-known examples of spectra E without any complex orientation such that E∗E is flat over E∗ ((2), (7)). In fact, we see that KO-spectrum does not have any complex orientation by showing that the Atiyah-Hirzebruch spectral sequence converging to KO ∗ (CP l ) has a non-trivial

Higher Orbifolds and Deligne-Mumford Stacks as Structured Infinity-Topoi

We develop a universal framework to study smooth higher orbifolds on the one hand and higher Deligne-Mumford stacks (as well as their derived and spectral variants) on the other, and use this framework to obtain a completely categorical description of which stacks arise as the functor of points of such objects. We choose to model higher orbifolds and Deligne-Mumford stacks as infinity-topoi equipped with a structure sheaf, thus naturally generalizing the work of Lurie, but our approach applies not only to different settings of algebraic geometry such as classical algebraic geometry, derived algebraic geometry, and the algebraic geometry of commutative ring spectra as in Lurie's work, but also to differential topology, complex geometry, the theory of supermanifolds, derived manifolds etc., where it produces a theory of higher generalized orbifolds appropriate for these settings. This universal framework yields new insights into the general theory of Deligne-Mumford stacks and orbifolds, including a representability criterion which gives a categorical characterization of such generalized Deligne-Mumford stacks. This specializes to a new categorical description of classical Deligne-Mumford stacks, a result sketched in previous work of the author, which extends to derived and spectral Deligne-Mumford stacks as well.

Towards topological Hochschild homology of Johnson–Wilson spectra

We offer a complete description of $THH(E(2))$ under the assumption that the Johnson-Wilson spectrum $E(2)$ at a chosen odd prime carries an $E_\infty$-structure. We also place $THH(E(2))$ in a cofiber sequence $E(2) \rightarrow THH(E(2))\rightarrow \overline{THH}(E(2))$ and describe $\overline{THH}(E(2))$ under the assumption that $E(2)$ is an $E_3$-ring spectrum. We state general results about the $K(i)$-local behaviour of $THH(E(n))$ for all $n$ and $0 \leq i \leq n$. In particular, we compute $K(i)_*THH(E(n))$.

Morita theory and singularity categories

We propose an analogue of the bounded derived category for an augmented ring spectrum, defined in terms of a notion of Noether normalization. In many cases we show this category is independent of the chosen normalization. Based on this, we define the singularity and cosingularity categories measuring the failure of regularity and coregularity and prove they are Koszul dual in the style of the BGG correspondence. Examples of interest include Koszul algebras and Ginzburg DG-algebras, C⁎(BG) for finite groups (or for compact Lie groups with orientable adjoint representation), cochains in rational homotopy theory and various examples from chromatic homotopy theory.

K-theoretic Tate–Poitou duality and the fiber of the cyclotomic trace

Let $$p\in {\mathbb {Z}}$$ be an odd prime. We prove a spectral version of Tate–Poitou duality for the algebraic K-theory spectra of number rings with p inverted. This identifies the homotopy type of the fiber of the cyclotomic trace $$K({\mathcal {O}}_{F})^{\scriptscriptstyle \wedge }_{p}\mathchoice{\longrightarrow }{\rightarrow }{\rightarrow }{\rightarrow }TC({\mathcal {O}}_{F})^{\scriptscriptstyle \wedge }_{p}$$ after taking a suitably connective cover. As an application, we identify the homotopy type at odd primes of the homotopy fiber of the cyclotomic trace for the sphere spectrum in terms of the algebraic K-theory of $${\mathbb {Z}}$$ .

Injective spectra and torsion theories

This is one of two papers on the injective spectrum of a right noetherian ring. In [4], we defined the injective spectrum as a topological space associated to a ring (or, more generally, a Grothendieck category), which generalises the Zariski spectrum. We established some results about the topology and its links with Krull dimension, and computed a number of examples.In the present paper, which can largely be read independently of the first, we extend these results by defining a sheaf of rings on the injective spectrum and considering sheaves of modules over this structure sheaf and their relation to modules over the original ring. We then explore links with the spectrum of prime torsion theories developed by Golan [2] and use this torsion-theoretic viewpoint to prove further results about the topology.

Descent in algebraic $K$-theory and a conjecture of Ausoni–Rognes

Let A \to B be a G -Galois extension of rings, or more generally of \mathbb{E}_\infty -ring spectra in the sense of Rognes. A basic question in algebraic K -theory asks how close the map K(A) \to K(B)^{hG} is to being an equivalence, i.e., how close algebraic K -theory is to satisfying Galois descent. An elementary argument with the transfer shows that this equivalence is true rationally in most cases of interest. Motivated by the classical descent theorem of Thomason, one also expects such a result after periodic localization. We formulate and prove a general result which enables one to promote rational descent statements as above into descent statements after periodic localization. This reduces the localized descent problem to establishing an elementary condition on K_0(-)\otimes \mathbb{Q} . As applications, we prove various descent results in the periodically localized K -theory, TC , THH , etc. of structured ring spectra, and verify several cases of a conjecture of Ausoni and Rognes.