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We compute the MU-based syntomic cohomologies, mod $(p,v_1,\cdots,v_{n+1})$, of all $\mathbb{E}_1$-MU-algebra forms of connective Morava K-theory k(n). As qualitative consequences, we deduce the Lichtenbaum--Quillen conjecture, telescope conjecture, and redshift conjecture for the algebraic K-theories of all $\mathbb{E}_{1}$-$\mathbb{S}$-algebra forms of $(2p^n-2)$-periodic Morava K-theory. Notably, the motivic spectral sequence computing $\pi_*TC(k(n))_p$ is concentrated on at most three lines, independently of $n$., Comment: 63 pages, 5 figures, Comments welcome!

We provide a unifying framework for studying variants of topological Hochschild homology such as Real topological Hochschild homology. Associated to a crossed simplicial group $\Delta G$, a category that generalizes Connes' cyclic category, we introduce an invariant of rings with twisted $G$-action, which we call topological $\Delta G$-homology. Here a ring with twisted $G$-action is a common generalization of a ring with involution and a ring with $G$-action. Our construction provides homotopical analogues of quaternionic homology, symmetric homology, braid homology, and hyperoctahedral homology defined by Fiedorowicz--Loday. In the quaternionic case our construction, called quaternionic topological Hochschild homology, is equipped with a left Pin(2)-action and we compute it for loop spaces with a twisted $C_4$-action. As an important step along the way, we introduce a new family of crossed simplicial groups called the twisted symmetric crossed simplicial groups and we prove that they corresponds to operads for algebras with twisted $G$-action., Comment: 61 pages. Comments welcome!

We determine the A(1)-homotopy of the topological cyclic homology of the connective real K-theory spectrum ko. The answer has an associated graded that is a free F_2[v_2^4]-module of rank 52, on explicit generators in stems -1 \le * \le 30. The calculation is achieved by using prismatic and syntomic cohomology of ko as introduced by Hahn-Raksit-Wilson, extending work of Bhatt-Morrow-Scholze from the case of classical commutative rings to E_\infty rings. A new feature in our case is that there are nonzero differentials in the motivic spectral sequence from syntomic cohomology to topological cyclic homology., Comment: 52 pages, 12 figures, 1 table. Conjecture B from v1 is proved in v2, along with other improvements

We describe a deformation of the $\infty$-category of Borel $G$-spectra for a finite group $G$. This provides a new presentation of the $a$-complete real Artin--Tate motivic stable homotopy category when $G=C_2$ and gives a new interpretation of the $a$-completed $C_2$-effective slice spectral sequence. As a new computational tool, we present a modified Adams--Novikov spectral sequence which computes the $RO(G)$-graded Mackey functor valued homotopy of Borel $G$-spectra., Comment: updated introduction and references; 49 pages, comments welcome!

We calculate the mod (p, v_1, v_2) homotopy V(2)_* TC(BP<2>) of the topological cyclic homology of the truncated Brown--Peterson spectrum BP<2>, at all primes p\ge7, and show that it is a finitely generated and free F_p[v_3]-module on 12p+4 generators in explicit degrees within the range -1 \le * \le 2p^3+2p^2+2p-3. At these primes BP<2> is a form of elliptic cohomology, and our result also determines the mod (p, v_1, v_2) homotopy of its algebraic K-theory. Our computation is the first that exhibits chromatic redshift from pure v_2-periodicity to pure v_3-periodicity in a precise quantitative manner., Comment: New Figure 7.1. Sections 11 and 12 rewritten as per referee's advice

We prove that Real topological Hochschild homology can be characterized as the norm from the cyclic group of order $2$ to the orthogonal group $O(2)$. From this perspective, we then prove a multiplicative double coset formula for the restriction of this norm to dihedral groups of order $2m$. This informs our new definition of Real Hochschild homology of rings with anti-involution, which we show is the algebraic analogue of Real topological Hochschild homology. Using extra structure on Real Hochschild homology, we define a new theory of $p$-typical Witt vectors of rings with anti-involution. We end with an explicit computation of the degree zero $D_{2m}$-Mackey functor homotopy groups of $\operatorname{THR}(\underline{\mathbb{Z}})$ for $m$ odd. This uses a Tambara reciprocity formula for sums for general finite groups, which may be of independent interest., Comment: 52 pages including references. Comments welcome

We compute topological Hochschild homology of sufficiently structured forms of truncated Brown--Peterson spectra with coefficients. In particular, we compute $\mathrm{THH}_*(B\langle n\rangle ;H\mathbb{Z}_{(p)})$ for all $n$ and $\mathrm{THH}_*(B\langle 2\rangle ;M)$ for $M\in \{ k(1),k(2)\}$ where $B\langle n\rangle$ is an $E_3$ form of $BP\langle n\rangle$ for certain primes $p$. For example, this gives a computation of $\mathrm{THH}(\mathrm{taf}^{D};M)$ for $M\in \{H\mathbb{Z}_{(3)},k(1),k(1))$ where $\mathrm{taf}^{D}$ is the $E_{\infty}$ form of $BP\langle 2\rangle$ constructed by Hill--Lawson., Comment: 27 pages including references. Accepted version. To appear in Algebraic & Geometric Topology

Let $L$ be finite extension of $\mathbb{Q}_p$ with ring of integers $\mathcal{O}_L$. I show that periodic topological cyclic homology of $\mathcal{O}_L$, over the base $\mathbb{E}_{\infty}$-ring $\mathbb{S}_{W(\mathbb{F}_q)}[z]$ carries a $p$-height one formal group law mod $p$ that depends on the Eisenstein polynomial of $L$ over $\mathbb{Q}_p$ for a choice of uniformizer $\varpi\in \mathcal{O}_L$., Comment: 9 pages, including references. Accepted version

This paper provides conditions for Morava K-theory to commute with certain homotopy limits. These conditions extend previous work on this question by allowing for homotopy limits of sequences of spectra that are not uniformly bounded below. As an application, we prove the $K(n)$-local triviality (for sufficiently large $n$) of the algebraic K-theory of algebras over truncated Brown-Peterson spectra, building on work of Bruner and Rognes and extending a classical theorem of Mitchell on $K(n)$-local triviality of the algebraic K-theory spectrum of the integers for large enough n., Comment: 32 pages, V2: Streamlined a lemma in Section 2 and added two lemmas to Section 3 to clarify part of the argument in the proof of the main theorem. Comments welcome

The family of Thom spectra $y(n)$ interpolate between the sphere spectrum and the mod two Eilenberg-MacLane spectrum. Computations of Mahowald, Ravenel, and Shick and the authors show that the $E_1$ ring spectrum $y(n)$ has chromatic complexity $n$. We show that topological periodic cyclic homology of $y(n)$ has chromatic complexity $n+1$. This gives evidence that topological periodic cyclic homology shifts chromatic height at all chromatic heights, supporting a variant of the Ausoni--Rognes red-shift conjecture. We also show that relative algebraic K-theory, topological cyclic homology, and topological negative cyclic homology of $y(n)$ at least preserve chromatic complexity., Comment: 32 pages. Includes a streamlined introduction and a correction of an error in the previous draft. Replaces a conjecture with a reference to a recent preprint. Comments welcome!

The Lichtenbaum--Quillen conjecture (LQC) relates special values of zeta functions to algebraic K-theory groups. The Ausoni--Rognes red-shift conjectures generalize the LQC to higher chromatic heights in a precise sense. In this paper, we propose an alternate generalization of the LQC to higher chromatic heights and give evidence for it at height two. In particular, if the $n$-th Greek letter family is detected by a commutative ring spectrum $R$, then we conjecture that the $n+1$-st Greek letter family will be detected by the algebraic K-theory of $R$. We prove this in the case $n=1$ for $R=\text{K}(\mathbb{F}_q)$ modulo $(p,v_1)$ where $p\ge 5$ and $q=\ell^k$ is a prime power generator of the units in $\mathbb{Z}/p^2\mathbb{Z}$. In particular, we prove that the commutative ring spectrum $\text{K}(\text{K}(\mathbb{F}_q))$ detects the part of the $p$-primary $\beta$-family that survives mod $(p,v_1)$. The method of proof also implies that these $\beta$ elements are detected in iterated algebraic K-theory of the integers. Consequently, one may relate iterated algebraic K-theory groups of the integers to integral modular forms satisfying certain congruences., Comment: 33 pages including references. To appear in Trans. Am. Math. Soc. This version may differ slightly from the published version