arXiv subjects
Publications and source records attributed to Yuri J. F. Sulyma.
Transfer systems as studied in equivariant algebra admit minimal generating sets, analogous to bases in linear algebra. It is natural to wonder if minimal generating sets form a matroid. We show that this happens only for lattices which are linear orders, or for cyclic groups of prime power order. In this case, we compute some invariants of the resulting matroid.
We construct a faithful and conservative functor from prisms to $C_{p^\infty}$-Tambara functors; in appropriate situations, this gives an algebraic description of $\underlineπ_0{\mathrm{TC}^-}$. We also present two integral variants using the generalized $n$-series of Devalapurkar-Misterka. The construction is based on the "twisted $I$-adic" or "$(p^\bullet)_q$" filtration, and is closely related to $q$-divided powers. To verify the axioms, we introduce a new technique for constructing Tambara functors, inspired by transversal prisms. We apply this to give a conceptual construction of Molokov's de Rham-Witt comparison map, and generalize it to a triangle sandwiching prismatic theory between theories built from Witt vectors and adjunction of $p$-power roots.
We calculate $\mathrm K_*(k[x]/x^e;\mathbf Z_p)$ by evaluating the syntomic cohomology $\mathbf Z_p(i)(k[x]/x^e)$ introduced by Bhatt-Morrow-Scholze and Bhatt-Scholze. This recovers calculations of Hesselholt-Madsen and Speirs, and generalizes an example of Mathew treating the case $e=2$ and $p>2$. Our main innovation is systematic use of the floor and ceiling functions, which clarifies matters substantially even for $e=2$. We furthermore observe a persistent phenomenon of slopes. As an application, we answer some questions of Hesselholt.
For a perfectoid ring $R$, we compute the full $\mathrm{RO}(\mathbb T)$-graded ring $\mathrm{TF}_\bigstar(R;\mathbf Z_p)$. This extends and simplifies work of Gerhardt and Angeltveit-Gerhardt. In even degrees, we find an $\mathrm{RU}(\mathbb T)$-graded version of Bökstedt periodicity, with some additional classes in the case of perfect $\mathbf F_p$-algebras. In odd degrees, we find extremely intricate and rather mysterious torsion. We also discuss the $\mathrm{RO}(\mathbb T)$-graded homotopy Tambara functors $\underlineπ_\bigstar\mathrm{THH}(R;\mathbf Z_p)$.
Let $R$ be a perfectoid ring. Hesselholt and Bhatt-Morrow-Scholze have identified the Postnikov filtration on $\mathrm{THH}(R;\mathbb Z_p)$: it is concentrated in even degrees, generated by powers of the Bökstedt generator $σ$, generalizing classical Bökstedt periodicity for $R=\mathbb F_p$. We study an equivariant generalization of the Postnikov filtration, the *regular slice filtration*, on $\mathrm{THH}(R;\mathbb Z_p)$. The slice filtration is again concentrated in even degrees, generated by $RO(\mathbb T)$-graded classes which can loosely be thought of as the *norms* of $σ$. The slices are expressible as $RO(\mathbb T)$-graded suspensions of Mackey functors obtained from the Witt Mackey functor. We obtain a sort of filtration by $q$-factorials. A key ingredient, which may be of independent interest, is a close connection between the Hill-Yarnall characterization of the slice filtration and Anschütz-le Bras' $q$-deformation of Legendre's formula.
Riehl and Verity have introduced an "$\infty$-cosmic" framework in which they redevelop the category theory of $\infty$-categories using 2-categorical arguments. In this paper, we begin with a self-contained review of the parts of their theory needed to discuss adjunctions and monadicity. This is applied in order to extend to the $\infty$-categorical context the classical criterion for fully faithfulness of the comparison functor induced by an adjunction. We discuss the relation with previous work in the literature---which primarily uses model-categorical techniques---and indicate applications to descent theory.