We compute the cardinalities of the integral algebraic K-theory groups of connective Morava K-theory $k(n)$ in all degrees that are not congruent to $0$ or $1$ modulo $2p-2$. After base-changing the coefficients of $k(n)$ to the algebraic closure $\overline{\mathbb{F}}_p$, we determine the corresponding cardinalities in all degrees and show that the groups vanish in even degrees. Our approach uses what we call the orbit filtration on topological cyclic homology arising from the May filtration on topological Hochschild homology. Combining this with an analysis of the topological cyclic homology of formal DGAs of the form $\mathbb{F}[x_{2m}]$, where $\mathbb{F}$ is a perfect field of characteristic $p$, we prove strong cardinality results for the topological cyclic homology of $\mathbb{E}_1$-rings with homotopy $\mathbb{F}[x_{2m}]$. For Morava K-theories over finite fields, we further use the motivic filtration of Hahn--Raksit--Wilson. As an application of our methods, we compute the cardinalities of the algebraic K-theory groups of the truncation $W(\overline{\mathbb{F}}_p)/p^n$ of the $p$-typical Witt vectors of $\overline{\mathbb{F}}_p$; in particular, they vanish in even degrees.
We develop a spectrum-level graded $K$-theory for real super Banach algebras. Our construction is categorical and homotopy theoretic, in the style of algebraic $K$-theory: the graded $K$-theory spectrum is obtained by a (co)fiber sequence from the $(\infty,1)$-categorical group completion of topological groupoids of fi...
One of the most effective approaches to computations in algebraic $K$-theory is trace methods, which compare algebraic $K$-theory with topological Hochschild homology and topological cyclic homology. In recent work, two of the authors, together with Gerhardt, construct an equivariant refinement of topological Hochschil...
David Chan, Marc Gotliboym, Inbar Klang et al.· 1 citation
Let $\mathfrak{X}_k\subseteq{\sf Mat}_{m, n}$ denote the variety of $m\times n$ complex matrices with rank at most $k$. The power series and rational expressions for the Hilbert series of $\mathfrak{X}_k$ are known by geometric arguments, and equating these expressions yields a family of formulas generalizing the class...
Let $K$ be a number field with ring of integers $\mathcal{O}_K$, and let $A,B,C \in \mathcal{O}_K \setminus\{0\}$. Denote by $S_K'$ the set of prime ideals of $\mathcal{O}_K$ dividing $2ABC$. Assuming two standard conjectures concerning the modularity of mod-$p$ Galois representations and the Eichler-Shimura correspond...
Let $\mathbb{K}$ be an algebraically closed field of characteristic zero and let $A_n(\mathbb{K})$ be the $n$-th Weyl algebra. We prove that for every complex reflection group $G$ and every $n \geq 2$ the ring of invariant differential operators $A_n(\mathbb{K})^G$ has irreducible non-holonomic modules of Gelfand-Kiril...
Let $K$ be an imaginary quadratic field and let $p \ge 5$ be a prime that is unramified in $K$. Let $\mathcal{A}_f/\mathbb{Q}$ be an abelian variety of $\mathrm{GL}_2$-type associated with a weight-two modular form $f$, with good non-ordinary reduction at $p$, and suppose that $(f,K)$ satisfies the generalized Heegner...
Antonio Lei, Luo-Chen Zhao· 0 citations
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