Research

Abstract: We introduce and develop the notion of "unipotent spectra." This is defined to be the stabilization of Toën's category of affine stacks, and is related to recent work of Mondal--Reinecke. Unipotent spectra give rise to unipotent stable homotopy groups and unipotent homology, which are new invariants for schemes valued in unipotent group schemes. As applications, we recover the Artin--Mazur formal groups associated to schemes without any vanishing assumptions. Further, we show that syntomic cohomology admits a natural refinement to a perfect unipotent spectrum. Finally, we extend Milne's work on arithmetic duality theorems to the category of perfect unipotent spectra and apply it to refine Poincaré duality in syntomic cohomology.

Abstract: In this paper, we use the formalism of Poincaré ∞-categories, as developed by Calmès-Dotto-Harpaz-Hebestreit-Land-Moi-Nardin-Nikolaus-Steimle, to define and study moduli stacks of line bundles with $λ$-hermitian pairings and of Azumaya algebras equipped with an involution. Our moduli spaces give rise to enhancements of the ordinary Picard and Brauer groups which incorporate the data of an involution on the base; we will refer to these new invariants as the Poincaré Picard group and the Poincaré Brauer group. We show that we can recover the involutive Brauer group of Parimala-Srinivas from the Poincaré Brauer group when the former is defined; however, they no longer agree even for closed points due to the existence of shifted perfect pairings. We also define the Poincaré Picard and Brauer groups for Poincaré rings in spectra, and compute these invariants for the sphere spectrum and other examples. As a consequence, we deduce a derived enhancement of a classical theorem of Saltman.

Abstract: In this paper, we introduce a notion of derived involutive algebras in $ C_2 $-Mackey functors which simultaneously generalize commutative rings with involution and the (non-equivariant) derived algebras of Bhatt--Mathew and Raksit. We show that the $ \infty $-category of derived involutive algebras admits involutive enhancements of the cotangent complexes, de Rham complex, and de Rham cohomology functors; furthermore, their real Hochschild homology is defined. We identify a filtration on the real Hochschild homology of these derived involutive algebras via a universal property and show that its associated graded may be identified with the involutive de Rham complex. Using $ C_2 $-$ \infty $-categories of Barwick--Dotto--Glasman--Nardin--Shah, we show that our filtered real Hochschild homology specializes to the HKR-filtered Hochschild homology considered by Raksit.

Abstract: Let A be a finite connected graded cocommutative Hopf algebra over a field k. There is an endofunctor tw on the stable module category StMod_A of A which twists the grading by 1. We show the categorical entropy of tw is zero. We provide a lower bound for the categorical polynomial entropy of tw in terms of the Krull dimension of the cohomology of A, and an upper bound in terms of the existence of finite resolutions of A-modules of a particular form. We employ these tools to compute the categorical polynomial entropy of the twist functor for examples of finite graded Hopf algebras over $\mathbb{F}$₂.

Abstract: The topological Hochschild homology of a ring (or ring spectrum) R is an $S^1$-spectrum, and the fixed points of THH(R) for subgroups $C_n \subset S^1$ have been widely studied due to their use in algebraic K-theory computations. Hesselholt and Madsen proved that the fixed points of topological Hochschild homology are closely related to Witt vectors. Further, they defined the notion of a Witt complex, and showed that it captures the algebraic structure of the homotopy groups of the fixed points of THH. Recent work of Angeltveit, Blumberg, Gerhardt, Hill, Lawson and Mandell defines a theory of twisted topological Hochschild homology for equivariant rings (or ring spectra) that builds upon Hill, Hopkins and Ravenel's work on equivariant norms. In this paper, we study the algebraic structure of the equivariant homotopy groups of twisted THH. In particular, we define an equivariant Witt complex and prove that the equivariant homotopy of twisted THH has this structure. Our definition of equivariant Witt complexes contributes to a growing body of research in the subject of equivariant algebra.

Abstract: Genuine equivariant homotopy theory is equipped with a multitude of coherently commutative multiplication structures generalizing the classical notion of an $ \mathbb{E}_\infty $-algebra. In this paper we study the $ C_p $-$ \mathbb{E}_\infty $-algebras of Nardin--Shah with respect to a cyclic group $ C_p $ of prime order. We show that many of the higher coherences inherent to the definition of parametrized algebras collapse; in particular, they may be described more simply and conceptually in terms of ordinary $ \mathbb{E}_\infty $-algebras as a diagram category which we call normed algebras. Our main result provides a relatively straightforward criterion for identifying $ C_p $-$ \mathbb{E}_\infty $-algebra structures. We visit some applications of our result to real motivic invariants.

Abstract: In this short note, we given a new proof of Mitchell's theorem that $L_{T(n)}K(\mathbb{Z}) \simeq 0$ for $n \geq 2$. Instead of reducing the problem to delicate representation theory, we use recently established hyperdescent technology for chromatically-localized algebraic K-theory.