arXiv subjects
Publications and source records attributed to Akhil Mathew.
V. Drinfeld and E. Lau introduced a ``decompletion'' of the ring of $p$-typical Witt vectors, following earlier work of T. Zink. The goal of this paper is to offer an exposition of this construction, which we call the sheared Witt vectors, on the category of rings $R$ whose reduction is a perfect $\mathbb{F}_p$-algebra.
The mod $p$ Riemann-Hilbert correspondence (in covariant and contravariant forms) relates $\mathbb{F}_p$-\'etale sheaves on the spectrum of an $\mathbb{F}_p$-algebra $R$ and Frobenius modules over $R$. We give an exposition of these correspondences using Breen's vanishing results on the perfect site.
We establish various properties of the p-adic algebraic K-theory of smooth algebras over perfectoid rings living over perfectoid valuation rings. In particular, the p-adic K-theory of such rings is homotopy invariant, and coincides with the p-adic K-theory of the p-adic generic fibre in high degrees. In the case of smooth algebras over perfectoid valuation rings of mixed characteristic the latter isomorphism holds in all degrees and generalises a result of Nizio{\l}.
The primary goal of this paper is to identify syntomic complexes with the $p$-adic \'etale Tate twists of Geisser--Schneider--Sato on regular $p$-torsionfree schemes. Our methods apply naturally to a broader class of schemes that we call "$F$-smooth". The $F$-smoothness of regular schemes leads to new results on the absolute prismatic cohomology of regular schemes.
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 $\mathrm{THH}(\mathbb{F}_p)$.
We give an account of the construction of the Bhatt--Morrow--Scholze motivic filtration on topological cyclic homology and related invariants, focusing on the case of equal characteristic $p$ and the connections to crystalline and de Rham--Witt theory.
We prove some $K$-theoretic descent results for finite group actions on stable $\infty$-categories, including the $p$-group case of the Galois descent conjecture of Ausoni-Rognes. We also prove vanishing results in accordance with Ausoni-Rognes's redshift philosophy: in particular, we show that if $R$ is an $\mathbb{E}_\infty$-ring spectrum with $L_{T(n)}R=0$, then $L_{T(n+1)}K(R)=0$. Our key observation is that descent and vanishing are logically interrelated, permitting to establish them simultaneously by induction on the height.
We discuss some general properties of $\mathrm{TR}$ and its $K(1)$-localization. We prove that after $K(1)$-localization, $\mathrm{TR}$ of $H\mathbb{Z}$-algebras is a truncating invariant in the sense of Land--Tamme, and deduce $h$-descent results. We show that for regular rings in mixed characteristic, $\mathrm{TR}$ is asymptotically $K(1)$-local, extending results of Hesselholt--Madsen. As an application of these methods and recent advances in the theory of cyclotomic spectra, we construct an analog of Thomason's spectral sequence relating $K(1)$-local $K$-theory and \'etale cohomology for $K(1)$-local $\mathrm{TR}$.
We prove two basic structural properties of the algebraic $K$-theory of rings after $K(1)$-localization at an implicit prime $p$. Our first result (also recently obtained by Land--Meier--Tamme by different methods) states that $L_{K(1)} K(R)$ is insensitive to inverting $p$ on $R$; we deduce this from recent advances in prismatic cohomology and $\mathrm{TC}$. Our second result yields a Künneth formula in $K(1)$-local $K$-theory for adding $p$-power roots of unity to $R$.
Using topological cyclic homology, we give a refinement of Beilinson's $p$-adic Goodwillie isomorphism between relative continuous $K$-theory and cyclic homology. As a result, we generalize results of Bloch-Esnault-Kerz and Beilinson on the $p$-adic deformations of $K$-theory classes. Furthermore, we prove structural results for the Bhatt-Morrow-Scholze filtration on $TC$ and identify the graded pieces with the syntomic cohomology of Fontaine-Messing.
The goal of this paper is to offer a new construction of the de Rham-Witt complex of smooth varieties over perfect fields of characteristic $p>0$. We introduce a category of cochain complexes equipped with an endomorphism $F$ of underlying graded abelian groups satisfying $dF = pFd$, whose homological algebra we study in detail. To any such object satisfying an abstract analog of the Cartier isomorphism, an elementary homological process associates a generalization of the de Rham-Witt construction. Abstractly, the homological algebra can be viewed as a calculation of the fixed points of the Berthelot-Ogus operator $L η_p$ on the $p$-complete derived category. We give various applications of this approach, including a simplification of the crystalline comparison for the $A Ω$-cohomology theory introduced in [BMS18].
We prove a purity property in telescopically localized algebraic $K$-theory of ring spectra: For $n\geq 1$, the $T(n)$-localization of $K(R)$ only depends on the $T(0)\oplus \dots \oplus T(n)$-localization of $R$. This complements a classical result of Waldhausen in rational $K$-theory. Combining our result with work of Clausen--Mathew--Naumann--Noel, one finds that $L_{T(n)}K(R)$ in fact only depends on the $T(n-1)\oplus T(n)$-localization of $R$, again for $n \geq 1$. As consequences, we deduce several vanishing results for telescopically localized $K$-theory, as well as an equivalence between $K(R)$ and $\mathrm{TC}(\tau_{\geq 0} R)$ after $T(n)$-localization for $n\geq 2$.
We prove a version of faithfully flat descent in rigid analytic geometry, for almost perfect complexes and without finiteness assumptions on the rings involved. This extends results of Drinfeld for vector bundles.
We give counterexamples to the degeneration of the HKR spectral sequence in characteristic $p$, both in the untwisted and twisted settings. We also prove that the de Rham--$\mathrm{HP}$ and crystalline--$\mathrm{TP}$ spectral sequences need not degenerate.
We study the \'etale sheafification of algebraic K-theory, called \'etale K-theory. Our main results show that \'etale K-theory is very close to a noncommutative invariant called Selmer K-theory, which is defined at the level of categories. Consequently, we show that \'etale K-theory has surprisingly well-behaved properties, integrally and without finiteness assumptions. A key theoretical ingredient is the distinction, which we investigate in detail, between sheaves and hypersheaves of spectra on \'etale sites.
A theorem of Lurie and Pridham establishes a correspondence between formal moduli problems and differential graded Lie algebras in characteristic zero, thereby formalising a well-known principle in deformation theory. We introduce a variant of differential graded Lie algebras, called partition Lie algebras, in arbitrary characteristic. We then explicitly compute the homotopy groups of free algebras, which parametrise operations. Finally, we prove generalisations of the Lurie-Pridham correspondence classifying formal moduli problems via partition Lie algebras over an arbitrary field, as well as over a complete local base.
Let $G$ be a finite group. To any family $\mathscr{F}$ of subgroups of $G$, we associate a thick $\otimes$-ideal $\mathscr{F}^{\mathrm{Nil}}$ of the category of $G$-spectra with the property that every $G$-spectrum in $\mathscr{F}^{\mathrm{Nil}}$ (which we call $\mathscr{F}$-nilpotent) can be reconstructed from its underlying $H$-spectra as $H$ varies over $\mathscr{F}$. A similar result holds for calculating $G$-equivariant homotopy classes of maps into such spectra via an appropriate homotopy limit spectral sequence. In general, the condition $E\in \mathscr{F}^{\mathrm{Nil}}$ implies strong collapse results for this spectral sequence as well as its dual homotopy colimit spectral sequence. As applications, we obtain Artin and Brauer type induction theorems for $G$-equivariant $E$-homology and cohomology, and generalizations of Quillen's $\mathcal{F}_p$-isomorphism theorem when $E$ is a homotopy commutative $G$-ring spectrum. We show that the subcategory $\mathscr{F}^{\mathrm{Nil}}$ contains many $G$-spectra of interest for relatively small families $\mathscr{F}$. These include $G$-equivariant real and complex $K$-theory as well as the Borel-equivariant cohomology theories associated to complex oriented ring spectra, any $L_n$-local spectrum, the classical bordism theories, connective real $K$-theory, and any of the standard variants of topological modular forms. In each of these cases we identify the minimal family such that these results hold.
We give a new proof of the recent Künneth theorem for periodic topological cyclic homology (TP) of smooth and proper dg categories over perfect fields of characteristic p>0 due to Blumberg and Mandell. Our result is slightly stronger and implies a finiteness theorem for topological cyclic homology (TC) of such categories.