Related papers: Picard groups of higher real $K$-theory spectra at…

We compute the continuous cohomology of the Morava stabilizer group with coefficients in Morava $E$-theory, $H^*(\mathbb{G}_2, E_t)$, at $p=2$, for $0\leq t < 12$, using the Algebraic Duality Spectral Sequence. Furthermore, in that same…

As a consequence of the algebraicity of chromatic homotopy at large primes, we show that the Hopkins' Picard group of the $K(n)$-local category coincides with the algebraic one when $2p-2 > n^{2}+n$.

In this work, we derive the low index subgroups of the extended Hecke, Hecke and the Picard groups using tools in color symmetry theory. We also present the low index subgroups of the modular group.

For a finite abelian group $A$, we determine the Balmer spectrum of $\mathrm{Sp}_A^{\omega}$, the compact objects in genuine $A$-spectra. This generalizes the case $A=\mathbb{Z}/p\mathbb{Z}$ due to Balmer and Sanders \cite{Balmer-Sanders},…

The motivic version of the $c_1$-spherical cobordism spectrum is constructed. A connection of this spectrum with other motivic Thom spectra is established. Using this connection, we compute the $\mathbb{P}^1$-diagonal of the homotopy groups…

In this paper, we showed that the Stable Picard group of $A(n)$ for $n\geq 2$ is $\mathbb{Z}\oplus \mathbb{Z}$ by considering the endotrivial modules over $A(n)$. The proof relies on reductions from a Hopf algebra to its proper Hopf…

Traditionally, homotopy groups in $G$-equivariant stable homotopy theory have been graded over $\text{RO}(G)$, the real representation ring of $G$. It is arguably more natural to grade homotopical structures over the Picard group of the…

The genus spectrum of a finite group $G$ is a set of integers $g \geq 2$ such that $G$ acts on a closed orientable compact surface $\Sigma_g$ of genus $g$ preserving the orientation. In this paper we complete the study of spectrum sets of…

We calculate the integral homotopy groups of THH(l) at any prime and of THH(ko) at p=2, where l is the Adams summand of the connective complex p-local K-theory spectrum and ko is the connective real K-theory spectrum.

Let $G$ be a finite group. For a $G$-ring $A,$ let ${\rm Pic}^{\it G}({\it A})$ denote the equivariant Picard group of $A.$ We show that if $A$ is a finite type algebra over a field $k$ then ${\rm Pic}^{\it G}({\it A})$ is contracted in the…

We calculate the Picard group, over the integers, of the Hilbert scheme of smooth, irreducible, non-degenerate curves of degree $d$and genus $g \geq 4$ in ${\Bbb P}^r$, in the case when $d \geq 2g+1 $ and $r \leq d-g$. We express the…

We compute the $RO(G)$-graded equivariant algebraic $K$-groups of a finite field with an action by its Galois group $G$. Specifically, we show these $K$-groups split as the sum of an explicitly computable term and the well-studied…

We introduce the Morava-isotropic stable homotopy category and, more generally, the stable homotopy category of an extension $E/k$. These "local" versions of the Morel-Voevodsky stable ${\Bbb{A}}^1$-homotopy category $SH(k)$ are analogues…

We develop a framework for displaying the stable homotopy theory of the sphere, at least after localization at the second Morava K-theory K(2). At the prime 3, we write the spectrum L_{K(2)S^0 as the inverse limit of a tower of fibrations…

To any Adams-type spectrum $E$, Pstr\k{a}gowski produced a symmetric monoidal stable $\infty$-category $Syn_E$ whose objects are, in a sense, ''formal Adams spectral sequences''. $Syn_E$ comes equipped with a lax symmetric monoidal functor…

We explicitly compute the ellitpic points and isotropy groups for the action of the Picard modular group over the Gaussian integers on 2-dimensional complex hyperbolic space.

Let $n\geq 1$ be an integer, $p$, $q$ be distinct odd primes. Let ${G}$, $N$ be two groups of order $p^nq$ with their Sylow-$p$-subgroups being cyclic. We enumerate the Hopf-Galois structures on a Galois ${G}$-extension, with type $N$. This…

Let $K$ be a non-archimedean local field of residual characteristic $p\neq 2$. Let $G$ be a connected reductive group over $K$, let $\theta$ be an involution of $G$ over $K$, and let $H$ be the connected component of $\theta$-fixed subgroup…

Let p>3 be a prime, let ku be the connective complex K-theory spectrum, and let K(ku) be the algebraic K-theory spectrum of ku. We study the p-primary homotopy type of the spectrum K(ku) by computing its mod (p,v_1) homotopy groups. We show…

For a smooth geometrically integral variety $X$ over a field $k$ of characteristic 0, we introduce and investigate the extended Picard complex $UPic(X)$. It is a certain complex of Galois modules of length 2, whose zeroth cohomology is…