arXiv subjects
Publications and source records attributed to Piotr Pstrągowski.
We prove the $K(n)$-local analogue of the Hahn-Wilson conjecture on fp-spectra, which states that the truncated Brown-Peterson spectra generate the category of fp-spectra as a thick subcategory. As a corollary, we deduce the original conjecture at height $1$. Along the way, we prove the existence of $K(n)$-local finite complexes with particularly regular rings of homotopy groups.
Using an argument of Jacob Lurie, we prove the existence of a E_2-lax monoidal structure on the Tate filtration coming from the standard cell structure on the infinite complex projective space. We then compare it to the filtration induced by the standard t-structure on spectra. In the last part of the paper, we construct a synthetic variant of the cellular Tate filtration and use it to compare Antieau's, Bhatt-Lurie's and Raksit's HKR filtrations on negative cyclic and periodic homology.
We show that $\kgl$-linear cohomology theories over an affine Dedekind scheme $S$ admit a canonical weight filtration on resolvable motives without inverting residual characteristics. Combined with upcoming work of Annala--Hoyois--Iwasa, this endows essentially all known logarithmic cohomology theories with weight filtrations when evaluated on projective sncd pairs $(X,D)$ over $S$. Furthermore, the weight-filtered cohomology is an invariant of the open part $U = X-D$. On variants of de Rham cohomology, we show that our weight filtration recovers the décalaged pole-order filtration defined by Deligne. One interpretation of this is that the spectral sequence associated to the pole-order filtration is an invariant of $U$ from the $E_2$-page onwards, which generalizes a result of Deligne from characteristic 0 to positive and mixed characteristic, and suggests that ``mixed Hodge theory'' is a useful invariant of $S$-schemes. Finally, we compute explicit examples of weight filtered pieces of cohomology theories. One of the computations reproves a slight weakening of a result of Thuillier stating that the singular cohomology of the dual complex associated to the boundary divisor of a good projective compactification does not depend on the chosen compactification. In the appendix, we prove the folklore results that the Whitehead tower functor is fully faithful and that perfect bivariant pairings with respect to the twisted arrow category correspond to duality.
This paper is the first in a series of articles devoted to the construction and study of synthetic deformations of $\infty$-categories in the unstable context: that is, deformations of $\infty$-categories that categorify spectral sequence or obstruction-theoretic information. This paper sets up the foundations of our study. We introduce and study various classes of $\infty$-categorical and infinitary algebraic theories. We establish many basic properties of the $\infty$-categories of the models of different classes of theories, as well as recognition theorems identifying the $\infty$-categories that arise this way. We give an intrinsic definition of a Malcev theory in higher universal algebra. We establish that the $\infty$-category of models of a Malcev theory may be characterized as freely adjoining geometric realizations to the theory. This leads to the notion of a derived functor between $\infty$-categories of models of Malcev theories, and we study the behavior of these derived functors with respect to connectivity and limits. We recall the notion of a loop theory and study in detail the interaction between functors and derived functors of $\infty$-categories of loop models and models, establishing that a large class of comonads on the $\infty$-category of loop models deform canonically to the $\infty$-category of all models. In the last part of the paper, we show that by considering the coalgebras for these deformed comonads over $\infty$-categories of models, one can recover various stable deformations considered in the literature, such as filtered models or Postnikov-complete synthetic spectra. We then expand on these results by constructing $\infty$-categories of synthetic spaces and synthetic $\mathbf{E}_k$-rings.
This paper is the second in a series devoted to the study of unstable synthetic deformations through the lens of Malcev theories: certain $\infty$-categorical algebraic theories $\mathcal{P}$ with well-behaved $\infty$-categories $\mathrm{Model}_{\mathcal{P}}$ of models. In this paper, we show that Malcev theories and their models admit a well-behaved deformation theory, generalizing the classical deformation theory of rings and modules. As our main example, we prove that the Postnikov tower of a Malcev theory $\mathcal{P}$ is a tower of square-zero extensions, and that all of this structure is preserved by passage to $\infty$-categories of models. This allows us to control the difference between the $\infty$-categories $\mathrm{Model}_{h_{n+r}\mathcal{P}}$ and $\mathrm{Model}_{h_n\mathcal{P}}$ for $r \leq n$, and forms the basis of a ``cofibre of $τ$'' formalism in our approach to unstable synthetic homotopy theory. As an application, we derive from this a variety of new Blanc--Dwyer--Goerss style decompositions of moduli spaces of lifts along the tower $\mathrm{Model}_{\mathcal{P}}\to\cdots\to\mathrm{Model}_{h\mathcal{P}}$.
We provide a description of Voevodsky's $\infty$-category of motivic spectra in terms of the subcategory of motives of smooth proper varieties. As applications, we construct weight filtrations on the Betti and étale cohomologies of algebraic varieties with coefficients in any complex oriented ring spectrum. We show that these filtrations satisfy $\ell\mathrm{dh}$-descent, giving an effective way of calculating them in positive characteristic. In the complex motivic case, we further refine the weight filtration to one defined at the level of stable homotopy types.
In this paper, we describe a novel way of identifying Adams spectral sequence $E_2$-terms in terms of homological algebra of quiver representations. Our method applies much more broadly than the standard techniques based on descent-flatness, bearing on a varied array of ring spectra. In the particular case of $p$-local integral homology, we are able to give a decomposition of the $E_2$-term, describing it completely in terms of the classical Adams spectral sequence. In the appendix, which can be read independently from the main body of the text, we develop functoriality of deformations of $\infty$-categories of the second author and Patchkoria.
Inspired by the work of Hahn-Raksit-Wilson, we introduce a variant of the even filtration which is naturally defined on $\mathbf{E}_{1}$-rings and their modules. We show that our variant satisfies flat descent and so agrees with the Hahn-Raksit-Wilson filtration on ring spectra of arithmetic interest, showing that various "motivic" filtrations are in fact invariants of the $\mathbf{E}_{1}$-structure alone. We prove that our filtration can be calculated via appropriate resolutions in modules and apply it to the study of even cohomology of connective $\mathbf{E}_{1}$-rings, proving vanishing above the Milnor line, base-change formulas, and explicitly calculating cohomology in low weights.
To any well-behaved homology theory we associate a derived $\infty$-category which encodes its Adams spectral sequence. As applications, we prove a conjecture of Franke on algebraicity of certain homotopy categories and establish homotopy-coherent monoidality of the Adams filtration.
We describe a homotopy-theoretic approach to the theory of moduli of realizations of Blanc-Dwyer-Goerss, reproducing their obstructions to realizing a given $Π$-algebra as homotopy groups of a pointed space. Our techniques are based on the $\infty$-category $\mathcal{P}_Σ(\mathcal{S}ph)$ of product-preserving presheaves on finite wedges of positive-dimensional spheres, leading to more conceptual and streamlined arguments.
We extend the construction of the normal cone of a closed embedding of schemes to any locally of finite type morphism of higher Artin stacks and show that in the Deligne-Mumford case our construction recovers the relative intrinsic normal cone of Behrend and Fantechi. We characterize our extension as the unique one satisfying a short list of axioms, and use it to construct the deformation to the normal cone. As an application of our methods, we associate to any morphism of Artin stacks equipped with a choice of a global perfect obstruction theory a relative virtual fundamental class in the Chow group of Kresch.
To an Adams-type homology theory we associate a notion of a synthetic spectrum, this is a product-preserving sheaf on the site of finite spectra with projective $E$-homology. We prove that the $\infty$-category $Syn_{E}$ of synthetic spectra based on $E$ is in a precise sense a deformation of the $\infty$-category of spectra into quasi-coherent sheaves over a certain algebraic stack, and show that this deformation encodes the $E$-based Adams spectral sequence. We describe a symmetric monoidal functor from cellular motivic spectra over the complex numbers into an even variant of synthetic spectra based on $MU$ and show that it induces an equivalence between the $\infty$-categories of $p$-complete objects for all primes $p$. In particular, it follows that the $p$-complete cellular motivic category can be described purely in terms of chromatic homotopy theory.
We give a simple counterexample to the plausible conjecture that Adams-type maps of ring spectra are stable under composition. We then show that over a field, this failure is quite extreme, as any map of $\mathbb{E}_{\infty}$-algebras is a transfinite composition of Adams-type maps.
As a consequence of the algebraicity of chromatic homotopy at large primes, we show that the Hopkins' Picard group of the $K(n)$-local category coincides with the algebraic one when $2p-2 > n^{2}+n$.
We prove the convergence of the Adams spectral sequence based on Morava K-theory and relate it to the filtration by powers of the maximal ideal in the Lubin-Tate ring through a Miller square. We use the filtration by powers to construct a spectral sequence relating the homology of the K-local sphere to derived functors of completion and express the latter as cohomology of the Morava stabilizer group. As an application, we compute the zeroth limit at all primes and heights.
We present an abstract version of Goerss-Hopkins theory in the setting of a prestable $\infty$-category equipped with a suitable periodicity operator. In the case of the $\infty$-category of synthetic spectra, this yields obstructions to realizing a comodule algebra as a homology of a commutative ring spectrum, recovering the results of Goerss and Hopkins.
We show that if $E$ is a $p$-local Landweber exact homology theory of height $n$ and $p > n^2+n+1$, then there exists an equivalence $h \mathcal{S}p_{E} \simeq h\mathcal{D}(E_{*}E)$ between homotopy categories of $E$-local spectra and differential $E_{*}E$-comodules, generalizing Bousfield's and Franke's results to heights $n > 1$.
We prove coherence theorems for dualizable objects in monoidal bicategories and for fully dualizable objects in symmetric monoidal bicategories, describing coherent dual pairs and coherent fully dual pairs. These are property-like structures one can attach to an object that are equivalent to the properties of dualizability and full dualizability. We extend diagrammatic calculus of surfaces of Christopher Schommer-Pries to the case of surfaces equipped with a framing. We present two equivalence relations on so obtained framed planar diagrams, one which can be used to model isotopy classes of framings on a fixed surface and one modelling diffeomorphism-isotopy classes of surfaces. We use the language of framed planar diagrams to derive a presentation of the framed bordism bicategory, completely classifying all two-dimensional framed topological field theories with arbitrary target. We then use it to show that the framed bordism bicategory is equivalent to the free symmetric monoidal bicategory on a coherent fully dual pair. In lieu of our coherence theorems, this gives a new proof of the Cobordism Hypothesis in dimension two.