Related papers: Invertible $K(2)$-Local $E$-Modules in $C_4$-Spect…

In \cite{SCH1} Schuster proved that $mod$ 2 Morava $K$-theory $K(s)^*(BG)$ is evenly generated for all groups $G$ of order 32. There exist 51 non-isomorphic groups of order 32. In \cite{H}, these groups are numbered by $1, \cdots ,51$. For…

Recently we introduced a class $\widetilde{\mathcal{Z}}$ of $2$-local finite spectra and showed that all spectra $Z\in \widetilde{\mathcal{Z}}$ admit a $v_2$-self-map of periodicity $1$. The aim of this article is to compute the…

We compute the algebraic Picard group of the category of $K(n)$-local spectra, for all heights $n$ and all primes $p$. In particular, we show that it is always finitely generated over $\mathbb{Z}_p$ and, whenever $n \geq 2$, is of rank $2$,…

Picard 2-categories are symmetric monoidal 2-categories with invertible 0-, 1-, and 2-cells. The classifying space of a Picard 2-category $\mathcal{D}$ is an infinite loop space, the zeroth space of the $K$-theory spectrum $K\mathcal{D}$.…

We compute the Picard group of the moduli space $U'$ of semistable vector bundles of rank $n$ and degree $d$ on an irreducible nodal curve $Y$ and show that $U'$ is locally factorial. We determine the canonical line bundles of $U'$ and…

In this paper we use the approach introduced in an earlier paper by Goerss, Henn, Mahowald and Rezk in order to analyze the homotopy groups of L_{K(2)}V(0), the mod-3 Moore spectrum V(0) localized with respect to Morava K-theory K(2). These…

We describe a deformation of the $\infty$-category of Borel $G$-spectra for a finite group $G$. This provides a new presentation of the $a$-complete real Artin--Tate motivic stable homotopy category when $G=C_2$ and gives a new…

We calculate the homotopy type of $L_1L_{K(2)}S^0$ and $L_{K(1)}L_{K(2)}S^0$ at the prime 2, where $L_{K(n)}$ is localization with respect to Morava $K$-theory and $L_1$ localization with respect to $2$-local $K$ theory. In $L_1L_{K(2)}S^0$…

Let $E_n$ be Morava $E$-theory and let $G \subset G_n$ be a finite subgroup of $G_n$, the extended Morava stabilizer group. Let $E_{n}^{tG}$ be the Tate spectrum, defined as the cofiber of the norm map $N:(E_n)_{hG} \to E_n^{hG}$. We use…

We present a new technique for analyzing the v_0-Bockstein spectral sequence studied by Shimomura and Yabe. Employing this technique, we derive a conceptually simpler presentation of the homotopy groups of the E(2)-local sphere for p > 3.…

The compact simply connected Riemannian 4-symmetric spaces were classified by J.A. Jim{\'{e}}nez according to type of the Lie algebras. As homogeneous manifolds, these spaces are of the form $G/H$, where $G$ is a connected compact simple…

In this paper, we investigate the rigidity of the stable comodule category of a specific class of Hopf algebroids known as finite Adams, shedding light on its Picard group. Then we establish a reduction process through base changes,…

We construct a well-behaved stable category of modules for a large class of infinite groups. We then consider its Picard group, which is the group of invertible (or endotrivial) modules. We show how this group can be calculated when the…

We construct spectral sequences for computing the cohomology of automorphism groups of formal groups with complex multiplication by a $p$-adic number ring. We then compute the cohomology of the group of automorphisms of a height four formal…

The compact simply connected Riemannian 4-symmetric spaces were classified by J.A. Jim{\'{e}}nez according to type of the Lie algebras. As homogeneous manifolds, these spaces are of the form $G/H$, where $G$ is a connected compact simple…

Let $A$ be a Cartan matrix and $G(A)$ be the Kac-Moody group associated to Cartan matrix $A$. In this paper, we discuss the computation of the rank $i_k$ of homotopy group $\pi_k(G(A))$. For a large class of Kac-Moody groups, we construct…

We have shown that the n-th Morava K-theory K^*(X) for a CW-spectrum X with action of Morava stabilizer group G_n can be recovered from the system of some height-(n+1) cohomology groups E^*(Z) with G_{n+1}-action indexed by finite…

For a motivic spectrum $E\in \mathcal{SH}(k)$, let $\Gamma(E)$ denote the global sections spectrum, where $E$ is viewed as a sheaf of spectra on $\mathrm{Sm}_k$. Voevodsky's slice filtration determines a spectral sequence converging to the…

We introduce the stable module $\infty$-category for groups of type $\Phi$ as an enhancement of the stable category defined by N. Mazza and P. Symonds. For groups of type $\Phi$ which act on a tree, we show that the stable module…

We construct the Continuous Wavelet Transform (CWT) on the homogeneous space (Cartan domain) D_4=SO(4,2)/(SO(4)\times SO(2)) of the conformal group SO(4,2) (locally isomorphic to SU(2,2)) in 1+3 dimensions. The manifold D_4 can be mapped…