相关论文: Adams spectral sequences and Franke's algebraicity…

Classical homological algebra takes place in additive categories. In homotopy theory such additive categories arise as homotopy categories of ``additive groupoid enriched categories'', in which a secondary analog of homological algebra can…

In 1996, Jens Franke proved the equivalence of certain triangulated categories possessing an Adams spectral sequence. One particular application of this theorem is that the K_(p)-local stable homotopy category at an odd prime can be…

A colleague asked about the Adams filtrations of the homotopy classes in the homotopy of the fiber of a particular map between GEMs. The theorem proved in arXiv:2105.02601v3 [math.AT] proves to be effective in answering this (Theorem 4.4).…

Using Patchkoria--Pstr\k{a}gowski's version of Franke's algebraicity theorem, we prove that the category of $K_p(n)$-local spectra is exotically equivalent to the category of derived $I_n$-complete periodic comodules over the Adams Hopf…

The theory of secondary chomology operations leads to a conjecture concerning the algebra of higher cohomology operations in general. This conjecture is discussed here in detail and its connection with homotopy groups of spheres and the…

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…

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…

We explain how the approach of Andre and Quillen to defining cohomology and homology as suitable derived functors extends to generalized (co)homology theories, and how this identification may be used to study the relationship between them.…

We establish abstract Adams isomorphisms in an arbitrary equivariantly presentable equivariantly semiadditive global category. This encompasses the well-known Adams isomorphism in equivariant stable homotopy theory, and applies more…

Let $R$ be a commutative ring with unit. We consider the homotopy theory of the category of spectral sequences of $R$-modules with the class of weak equivalences given by those morphisms inducing a quasi-isomorphism at a certain fixed page.…

The $E_2$ term of the Adams spectral sequence may be identified with certain derived functors, and this also holds for a number of other spectral sequences. Our goal is to show how the higher terms of such spectral sequences are determined…

Frank Adams introduced the notion of a complex oriented cohomology theory represented by a commutative ring-spectrum and proved the Poincar\'e Duality theorem for this general case. In the current paper we consider oriented cohomology…

Mahowald's conjecture arose as part of a program attempting to view chromatic phenomena in stable homotopy theory through the lens of the classical Adams spectral sequence. The conjecture predicts the existence of nonzero classes in the…

In this paper we elaborate a general homotopy-theoretic framework in which to study problems of descent and completion and of their duals, codescent and cocompletion. Our approach to homotopic (co)descent and to derived (co)completion can…

We prove the "Gluing Conjecture" on the spectral side of the categorical geometric Langlands correspondence. The key tool is the structure of crystal on the category of singularities, which allows to reduce the conjecture to the question of…

Classical homological algebra considers chain complexes, resolutions, and derived functors in additive categories. We describe "track algebras in dimension n", which generalize additive categories, and we define higher order chain…

We revisit methods of proof of the Adams Conjecture in order to correct and supplement earlier efforts to prove analogous conjectures in the stable homotopy category. We utilize simplicial schemes over an algebraically closed field of…

We use obstruction theory based on the unstable Adams spectral sequence to construct self maps of finite quaternionic projective spaces. As a result, a conjecture of Feder and Gitler regarding the classification of self maps up to homology…

We decompose the K-theory space of a Waldhausen category in terms of its Dwyer-Kan simplicial localization. This leads to a criterion for functors to induce equivalences of K-theory spectra that generalizes and explains many of the criteria…

The Adams spectral sequence was invented by J.F.Adams fifty years ago for calculations of stable homotopy groups of topological spaces and in particular of spheres. The calculation of differentials of this spectral sequence is one of the…