Related papers: $\mathrm{RO}(\mathbb T)$-graded $\mathrm{TF}$ of p…

We refine several results of Bhatt-Morrow-Scholze on THH to THR. In particular, we compute THR of perfectoid rings. This will be useful for establishing motivic filtrations on real topological Hochschild and cyclic homology of quasisyntomic…

In this work we study the $E_{\infty}$-ring $\text{THH}(\mathbb{F}_p)$ as a graded spectrum. Following an identification at the level of $E_2$-algebras with $\mathbb{F}_p[\Omega S^3]$, the group ring of the $E_1$-group $\Omega S^3$ over…

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 B\"okstedt generator $\sigma$,…

The main result of this paper is the computation of TR^n_{\alpha}(F_p;p) for \alpha in R(S^1). These R(S^1)-graded TR-groups are the equivariant homotopy groups naturally associated to the S^1-spectrum THH(F_p), the topological Hochschild…

We give an algorithm for calculating the RO(S^1)-graded TR-groups of F_p, completing the calculation started by the second author. We also calculate the RO(S^1)-graded TR-groups of Z with mod p coefficients and of the Adams summand \ell of…

We study the algebraic $K$-theory of rings of the form $R[x]/x^e$. We do this via trace methods and filtrations on topological Hochschild homology and related theories by quasisyntomic sheaves. We produce computations for $R$ a perfectoid…

Let $R$ be a ring and $\mathsf S$ be a class of strongly finitely presented (FP${}_\infty$) $R$-modules closed under extensions, direct summands, and syzygies. Let $(\mathsf A,\mathsf B)$ be the (hereditary complete) cotorsion pair…

We define the notion of an $\mathcal{RO}(G)$-graded Tambara functor and prove that any $G$-spectrum with norm multiplication gives rise to such an $\mathcal{RO}(G)$-graded Tambara functor.

Tate-Hochschild cohomology of an algebra is a generalization of ordinary Hochschild cohomology, which is defined on positive and negative degrees and has a ring structure. Our purpose of this paper is to study the eventual periodicity of an…

In this note we compute the topological Hochschild homology of quotients of DVRs. Along the way we give a short argument for B\"okstedt periodicity and generalizations over various other bases. Our strategy also gives a very efficient way…

In this paper, we use trace methods to study the algebraic $K$-theory of rings of the form $R[x_1,\ldots, x_d]/(x_1,\ldots, x_d)^2$. We compute the relative $p$-adic $K$ groups for $R$ a perfectoid ring. In particular, we get the integral…

We completely calculate the $RO(\mathbb{Z}/p)$-graded coefficients $H\underline{\mathbb{Z}/p}_\star H\underline{\mathbb{Z}/p}$ for the constant Mackey functor $\underline{\mathbb{Z}/p}$.

The topological Hochschild homology THH(R) of a commutative S-algebra (E_infty ring spectrum) R naturally has the structure of a commutative R-algebra in the strict sense, and of a Hopf algebra over R in the homotopy category. We show,…

Given a $1$-tilting cotorsion pair over a commutative ring, we characterise the rings over which the $1$-tilting class is an enveloping class. To do so, we consider the faithful finitely generated Gabriel topology $\mathcal{G}$ associated…

For a "genuine" equivariant commutative ring spectrum $R$, $\pi_0(R)$ admits a rich algebraic structure known as a Tambara functor. This algebraic structure mirrors the structure on $R$ arising from the existence of multiplicative norm…

In this paper we show that for a torsion-free abelian group $G$, $\operatorname{rank}_\mathbb{Z}G<\infty$ if and only if there exists a Noetherian $G$-graded ring $R$ such that the set $\{R_g \neq 0\}$ generates the group $G$. For every $G$…

We consider finitely generated normal algebras over an algebraically closed field of characteristic zero that come with a complexity one grading by a finitely generated abelian group such that the conditions of a UFD are satisfied for…

As an extension of previous ungraded work, we define a graded $p$-polar ring to be an analog of a graded commutative ring where multiplication is only allowed on $p$-tuples (instead of pairs) of elements of equal degree. We show that the…

We introduce and study graded perfectoid rings as graded analogues of Scholze's (integral) perfectoid rings. We establish a categorical equivalence between graded perfectoid rings and graded perfect prisms, extending the Bhatt-Scholze's…

We show that the algebraic K-theory space of stable infinity-categories is canonically functorial in polynomial functors. As a consequence, we obtain a new proof of B\"okstedt's calculation of $\mathrm{THH}(\mathbb{F}_p)$.