Related papers: Strict units of commutative ring spectra

We describe a special instance of the Goerss-Hopkins obstruction theory, due to Senger, for calculating the moduli of $E_\infty$ ring spectra with given mod-$p$ homology. In particular, for the $2$-primary Brown-Peterson spectrum we give a…

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…

Our results are of three types. First we describe a general procedure of adjoining polynomial variables to $A_\infty$-ring spectra whose coefficient rings satisfy certain restrictions.A host of examples of such spectra is provided by…

We use geometric fixed points to describe the homotopy theory of genuine equivariant commutative ring spectra after inverting the group order. The main innovation is the use of the extra structure provided by the Hill-Hopkins-Ravenel norms…

We develop a new geometric method of understanding principal G-Higgs bundles through their spectral data, for G a real form of a complex Lie group. In particular, we consider the case of G a split real form, as well as G = SL(2,R), U(p,p),…

We combine Lurie's generalization of the Hopkins-Miller theorem with work of Zink-Lau on displays to give a functorial construction of even-periodic commutative ring spectra, concentrated in chromatic layers 2 and above, associated to…

We compute the connective spectra of maps from $\mathbb{Z}$ to the Picard spectra of the spherical Witt vectors associated with perfect rings of characteristic $p$. As an application, we determine the connective spectrum of maps from…

We establish basic results on subrings of finite commutative rings and closely related rings. Among other applications we calculate the number of maximal subrings of a finite commutative local ring.

Let GL_1(R) be the units of a commutative ring spectrum R. In this paper we identify the composition BGL_1(R)->K(R)->THH(R)->\Omega^{\infty}(R), where K(R) is the algebraic K-theory and THH(R) the topological Hochschild homology of R. As a…

In recent work, Darmon, Pozzi and Vonk explicitly construct a modular form whose spectral coefficients are $p$-adic logarithms of Gross-Stark units and Stark-Heegner points. Here we describe how this construction gives rise to a practical…

We consider the change-of-rings spectral sequence as it applies to Hochschild cohomology, obtaining a description of the differentials on the first page which relates it to the multiplicative stucture on cohomology. Using this information,…

Certifying the positivity of trigonometric polynomials is of first importance for design problems in discrete-time signal processing. It is well known from the Riesz-Fej\'ez spectral factorization theorem that any trigonometric univariate…

We investigate sums of exceptional units in a quaternion ring $H(R)$ over a finite commutative ring $R$. We prove that in order to find the number of representations of an element in $H(R)$ as a sum of $k$ exceptional units for some integer…

We extend a classical fact about deformations of groups of units of commutative rings to $\mathbb{E}_{\infty}$-ring spectra, and we use this result to provide a map of spectra generalizing the ordinary logarithmic derivative induced by an…

Let $R$ be a commutative ring with identity. A unit $u$ of $R$ is called exceptional if $1-u$ is also a unit. When $R$ is a finite commutative ring, we determine the additive and multiplicative structures of its exceptional units; and then…

We construct hyper-homology spectral sequences of Z-graded and ROG-graded Mackey functors for Ext and Tor over G-equivariant S-algebras (A-infty ring spectra) for finite groups G. These specialize to universal coefficient and Kunneth…

We compare derived categories of the category of strict polynomial functors over a finite field and the category of ordinary endofunctors on the category of vector spaces. We introduce two intermediate categories: the category of…

Molecular line intensity calculations are not a straightforward task. We present a description of the basics for including molecular linesin synthetic spectra, and of the input data needed. We aim both at describing ways in which molecular…

We give a higher-algebraic interpretation of complex orientations of ring spectra as "$\mathbb{E}_2$ strictifications" of the identity element. We show that higher strictifications do not exist for most ring spectra of interest in chromatic…

The notion of highly structured ring spectra of prime characteristic is made precise and is studied via the versal examples S//p for prime numbers p. These can be realized as Thom spectra, and therefore relate to other Thom spectra such as…