Comodules and Landweber exact homology theories

Franke 余模范畴中的可逆对象

Invertible objects in Franke's comodule categories

本文计算了 Franke 准周期余模范畴的 Picard 群，证明在特定条件下其为无限循环群，并与已知稳定同伦范畴的 Picard 群计算结果进行比较。

阿贝尔紧李群的等变Landweber正合函子定理

An Equivariant Landweber Exact Functor Theorem for Abelian Compact Lie Groups

本文为阿贝尔紧李群证明了等变Landweber正合函子定理，给出分次模成为同调理论的充要条件，并应用证明Wisdom猜想。

$L$-complete Hopf algebroids and their comodules

We investigate Hopf algebroids in the category of $L$-complete modules over a commutative Noetherian regular complete local ring. The main examples are provided by the Hopf algebroids associated to Lubin-Tate spectra in the K(n)-local stable homotopy category and we show that these have Landweber filtrations for all finitely generated discrete modules. Along the way we investigate the canonical Hopf algebras associated to Hopf algebroids over fields and introduce a notion of unipotent Hopf algebroid generalising that for Hopf algebras. In two appendices we continue the discussion of the connec…

Homotopy theory of comodules over a Hopf algebroid

Given a good homology theory E and a topological space X, the E-homology of X is not just an E_{*}-module but also a comodule over the Hopf algebroid (E_{*}, E_{*}E). We establish a framework for studying the homological algebra of comodules over a well-behaved Hopf algebroid (A, Gamma). That is, we construct the derived category Stable(Gamma) of (A, Gamma) as the homotopy category of a Quillen model structure on the category of unbounded chain complexes of Gamma-comodules. This derived category is obtained by inverting the homotopy isomorphisms, NOT the homology isomorphisms. We establish the…

Derived completion for comodules

The objective of this paper is to introduce and study completions and local homology of comodules over Hopf algebroids, extending previous work of Greenlees and May in the discrete case. In particular, we relate module-theoretic to comodule-theoretic completion, construct various local homology spectral sequences, and derive a tilting-theoretic interpretation of local duality for modules. Our results translate to quasi-coherent sheaves over global quotient stacks and feed into a novel approach to the chromatic splitting conjecture.

关于拓扑余Hochschild同调和余张量积

On topological coHochschild homology and cotensor products

本文研究了拓扑余Hochschild同调的余张量积及其余煤格结构，建立了其Morita-Takeuchi不变性。

Waldhausen K-theory of spaces via comodules

Let $X$ be a simplicial set. We construct a novel adjunction between the categories of retractive spaces over $X$ and of $X_{+}$-comodules, then apply recent work on left-induced model category structures (arXiv:1401.3651v2 [math.AT],arXiv:1509.08154 [math.AT]) to establish the existence of a left proper, simplicial model category structure on the category of $X_+$-comodules, with respect to which the adjunction is a Quillen equivalence after localization with respect to some generalized homology theory. We show moreover that this model category structure stabilizes, giving rise to a model cat…

Cotensor products of modules

Let C be a coalgebra over a field k and A its dual algebra. The category of C-comodules is equivalent to a category of A-modules. We use this to interpret the cotensor product M \square N of two comodules in terms of the appropriate Hochschild cohomology of the A-bimodule M \otimes N, when A is finite-dimensional, profinite, graded or differential-graded. The main applications are to Galois cohomology, comodules over the Steenrod algebra, and the homology of induced fibrations.

关于高阶实K理论连通模型的MU同调

On MU-homology of connective models of higher Real K-theories

本研究利用切片过滤分析高阶实K理论连通模型不动点的MU同调，揭示其与周期版本的差异，明确偶无挠性成立的高度条件，并完成tmf₀(3)对应余模代数的完整计算。

Flat replacements of homology theories

To a homology theory one can associate an additive site and a new homological functor with values in the category of additive sheaves on that site. If this category of sheaves can be shown to be equivalent to a category of comodules of a Hopf algebroid, then we obtain a new homology theory by composing with the underlying module functor. This new homology theory is always flat and we call it a flat replacement of the original theory. For example, Pstrągowski has shown that complex cobordism is a flat replacement of singular homology. In this article we study the basic properties of the sites a…

On p-local Topological Automorphic Forms for $U(1,1;\mathbb{Z}[i])$

We present a new flavor of TAF-type (co)homology theories, which are p-local of height two and based on the isometry group of the odd unimodular hermitian lattice of signature (1,1) over the Gaussian integers. Using a suitable family of hyperelliptic curves, we explicitly construct a genus to automorphic forms, prove an integrality statement and verify Landweber's criterion.

An Unstable Change of Rings for Morava E-Theory

The Bousfield-Kan (or unstable Adams) spectral sequence can be constructed for various homology theories such as Brown-Peterson homology theory BP, Johnson-Wilson theory $E(n)$, or Morava $E$-theory $E_n$. For nice spaces the $E_2$-term is given by Ext in a category of unstable comodules. We establish an unstable Morava change of rings isomorphism between $\text{Ext}_{\mathcal{U}_{Γ_B}}(B,M)$ and $\text{Ext}_{\mathcal{U}_{E_{n*}E_n}/I_{n}}(E_{n*}/I_{n},E_{n*}\otimes_{BP_*} M)$ where $(B,Γ_B)$ denotes the Hopf Algebroid $(v_n^{-1}BP_*/I_{n}, v_n^{-1}BP_*BP/I_{n} )$. We show that the latter grou…

对偶 Steenrod 代数上余模范畴中积与极限的导出函子

Derived functors of product and limit in the category of comodules over the dual Steenrod algebra

本文将对偶 Steenrod 代数上余模范畴中序列极限的导出函子识别为 Steenrod 代数的局部上同调，从而为 Sadofsky 谱序列提供了可计算的输入项。