arXiv subjects
Publications and source records attributed to Sanath Devalapurkar.
We produce refinements of the known multiplicative structures on the Brown--Peterson spectrum $BP$, its truncated variants $BP\langle n \rangle$, Ravenel's spectra $X(n)$, and evenly graded polynomial rings over the sphere spectrum. Consequently, topological Hochschild homology relative to these rings inherits a circle action.
Given a principally polarized abelian variety $A$ of dimension $g$ over an algebraically closed field $k$ of characteristic $p$, the $p$ torsion $A[p]$ is a finite flat $p$-torsion group scheme of rank $p^{2g}$. There are exactly $2^g$ possible group schemes that can occur as some such $A[p]$. In this paper, we study which group schemes can occur as $J[p]$, where $J$ is the Jacobian of a hyperelliptic curve defined over $\mathbb{F}_p$. We do this by computing explicit formulae for the action of Frobenius and its dual on the de Rham cohomology of a hyperelliptic curve with respect to a given basis. A theorem of Oda's in [Oda69] allows us to relate these actions to the $p$-torsion structure of the Jacobian. Using these formulae and the computer algebra system Magma, we affirmatively resolve questions of Glass and Pries in [Cor05] on whether certain group schemes of rank $p^8$ and $p^{10}$ can occur as $J[p]$ of a hyperelliptic curve of genus $4$ and $5$ respectively.
We define the notion of Cartesian 2-fibrations, and prove a weak analogue of straightening. Using Barwick's notion of operator categories and the notion of a Cartesian 2-fibration, we extend the notion of $\infty$-operads to the $(\infty,2)$-categorical context. These $(\infty,2)$-operads are used to define and study 2-rings in the $\infty$-categorical context.
We study a variant of algebraic K-theory and prove that it is stable and preserves module structures.