For a finite group $G$, we define a constant spectral $G$-Mackey functor as a genuine $G$-spectrum in which the restrictions between categorical fixed points are equivalences. We further identify the full subcategory of constant spectral $G$-Mackey functors with the category of modules over a certain $\mathbb{E}_1$-ring spectrum $R_G$. When $G=C_{p^n}$ is a cyclic $p$-group, we provide an explicit construction of the $\mathbb{E}_1$-ring spectrum $R_{C_{p^n}}$. The proof uses a reconstruction theorem for genuine $C_{p^n}$-spectra, which might be of independent interest. As an application, we compute the $\mathbb{Z}$-graded $C_{p^n}$-equivariant mod $p$ Steenrod algebra.
We establish a global equivariant refinement of the theorem of Snaith that presents the complex $K$-theory spectrum as a localization of the suspension spectrum of $\mathbb C P^\infty$. We prove that the morphism of ultra-commutative global ring spectra from $\Sigma^\infty_+ B_{\text{gl}} U(1)$ that classifies the taut...
We prove an equivariant Landweber exact functor theorem for abelian compact Lie groups $G$. For a graded module $N$ over the $G$-equivariant Lazard ring $L_G$, we give necessary and sufficient algebraic conditions on $N$ for the functor \[ X\longmapsto (MU_G)_*(X)\otimes_{L_G} N \] to define a homology theory on the ca...
Let $(A,\mathfrak{m})$ be a Cohen-Macaulay local ring of dimension $d$, residue field $k$ and let $I$ be an $\mathfrak{m}$-primary ideal. Assume $k$ is infinite. Let $N$ be a perfect $A$-module of dimension $t \geq 1$. Let $M$ be a MCM $A$-module. The function $n \rightarrow \ell(\text{Tor}^A_j(M, N/I^{n+1}N))$ is of p...
Let $C$ be a smooth projective curve of genus $g$, and let $T_{n,L}$ denote the tautological line bundle on its $n$-th symmetric product $C_n$. We prove effective vanishing criteria for Koszul cohomology $K_{p,q}(C_n,T_{n,B};T_{n,L})$, extending two classical theorems of Mark Green. The main ingredient is a weight-one...
Let $G$ be a finite group. In this note, we construct an element ${\sigma}^G$ in the Burnside ring of $G$ over $\mathbb{Q}$, which gives a rational number $m(G)$ that is the sum of the inverses of the element orders in $G$, by using the deflation map ${\mathrm{Def}}^G_{G/N}$ induced by the $(G/N,G)$-biset $G/N$ for a n...
Let $T$ be a finite group and let $kB_T$ denote the shifted Burnside biset functor over a commutative ring $k$. We study the essential algebra $\widehat{kB_T}(G)$ of $kB_T$ at an arbitrary finite group $G$. We first establish criteria determining when a covering subgroup of $G\times G\times T$ factors, with respect to...
Olcay Coşkun, Ruslan Muslumov· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
