arXiv subjects

Shaul Ragimov

Publications and source records attributed to Shaul Ragimov.

4 recordsLinked to original sources

arXiv2026

Higher Semiadditive Character Theory

We introduce and develop the theory of semiadditive characters in the higher semiadditive setting, generalizing both the $T(n)$-local monoidal character and the $K(t)$-local transchromatic character. These are natural transformations compatible with restriction and transfer maps along $π$-finite spaces, with an $(n-t)$-fold $p$-typical free loop space correction built into the target. We show that every $\infty$-commutative monoid admits a universal $(n-t)$-fold character. This universal character has several strong structural properties: it exhibits blue shift, satisfies higher cyclotomic descent, and is compatible with the semiadditive Fourier transform. We compute it for an arbitrary $K(n)$-local object and show that, for Morava $E$-theory, it recovers the $K(t)$-local transchromatic character. By functoriality, the universal character carries a natural action of the profinite group $\mathrm{GL}_{n-t}(\mathbb{Z}_p)$. When $t=0$, the fixed points of this action recover rationalization. As a consequence, we derive an explicit description of $L_{\mathbb{Q}}(S^A_{K(n)})$ for every $π$-finite space $A$ and compute the ring of rational $K(n)$-local power operations.

math.AT↗

arXiv2026

The $\infty$-Categorical Reflection Theorem and Applications

We prove an $\infty$-categorical version of the reflection theorem of Adámek-Rosický. Namely, that a full subcategory of a presentable $\infty$-category which is closed under limits and $κ$-filtered colimits is a presentable $\infty$-category. We then use this theorem in order to classify subcategories of a symmetric monoidal $\infty$-category which are equivalent to a category of modules over an idempotent algebra.

math.AT↗

arXiv2025

Semiadditive Alternating Powers and Twisted Power Operation

We study a class of representations of symmetric groups in higher semiadditive categories. For these representations in $\mathrm{Mod}^{\wedge}_{E_n}$, the transchromatic character of Hopkins--Kuhn--Ravenel and Stapleton is recovered as a sequence of monoidal characters on suitable categorifications, giving an explicit algorithm for its computation, and relating it to the iterated monoidal character in $(\infty,n)$-categories. These representations also give rise to notions of alternating powers and power operations in semiadditive categories, extending the classical alternating powers and $λ$-operations in $\mathrm{K}$-theory. We provide explicit computations in both the chromatic and higher categorical settings at low heights.

math.AT↗

arXiv2025

Twisted Graded Categories

Given a presentably symmetric monoidal $\infty$-category $\mathcal{C}$ and an $\mathbb{E}_{\infty}$-monoid $M$, we introduce and classify twisted graded categories, which generalize the Day convolution structure on $\mathrm{Fun}(M, \mathcal{C})$. These are characterized by a braiding encoded in symmetric group actions on tensor powers, whose character we show depends only on the $\mathbb{T}$-equivariant monoidal dimension. We analyze the $\mathbb{T}$-action on the dimension of invertible objects and identify it with the $\mathbb{T}$-transfer map. Finally, we compute braiding characters in examples arising from higher cyclotomic extensions, such as the $(\mathbb{S}, n+1)$-oriented extension of $\mathrm{Mod}_{En}^{\wedge}$ at all primes and heights, and of the cyclotomic closure of $\mathrm{Vect}^n$ at low heights.

math.AT↗