Uniqueness of differential characters and differential K-theory via homological algebra

Homological algebra in characteristic one

This article develops several main results for a general theory of homological algebra in categories such as the category of sheaves of idempotent modules over a topos. In the analogy with the development of homological algebra for abelian categories the present paper should be viewed as the analogue of the development of homological algebra for abelian groups. Our selected prototype, the category Bmod of modules over the Boolean semifield B is the replacement for the category of abelian groups. We show that the semi-additive category Bmod fulfills analogues of the axioms AB1 and AB2 for abeli…

Homological epimorphisms and homotopy epimorphisms

We show that the notions of homotopy epimorphism and homological epimorphism in the category of differential graded algebras are equivalent. As an application we obtain a characterization of acyclic maps of topological spaces in terms of induced maps of their chain algebras of based loop spaces. In the case of a universal acyclic map we obtain, for a wide class of spaces, an explicit algebraic description for these induced maps in terms of derived localization.

具有消失André-Quillen同调的单纯交换代数的刻画

Characterizing Simplicial Commutative Algebras with Vanishing André-Quillen Homology

使用同调和同伦的方法，如Tor和André-Quillen同调，在描述交换代数时有很大的用处。主要的范畴设置是微分分次代数和模，但最近单纯范畴也被证明是有用的设置。在本文中，我们把这个观点上升了一个档次，通过扩展一些最近使用的同调代数的特点诺特交换代数的特点，通过使用单纯同伦理论的有限诺特同伦的单纯交换代数。这些特征涉及扩展的概念，局部完全交叉和局部Gorenstein代数的单纯同伦设置。

同调代数中的另一点：不连续群作用的对偶性

Another point in homological algebra: Duality for discontinuous group actions

本文考虑可压缩$n$维流形$X$上群$G$的不连续运算.令$E$是特征为0的域$k$上的$G$的有限维表示。令$\mathcal{E}$是与$E$相关联的商空间$Y=G \setminus X$上的层。设$H^{\bullet}_{\textbf{!}}(Y;\mathcal{E})$是紧支集上同调在$H^{\bullet}(Y;\mathcal{E})$中的像.在$H^{\bullet}_{\textbf{!}}(Y;\mathcal{E})$和$H^{\bullet}_{\textbf{!}}(Y;\mathcal{E}^*)$（$\mathcal{E}^*$是与$E$对偶的表示相关联的层）都是有限维的情况下，我们建立了$H^{m}_{\textbf{!}}(Y;\mathcal{E})$和$H^{n-m}_{\textbf{!}}(Y;\mathcal{E}^{\ast})$之间的非退化对偶。我们还表明，这种对偶是兼容的Hecke运营商。

Garland方法的同调代数与偏序集版本

Homological algebra and poset versions of the Garland method

本文将Garland的消失准则从单纯复形推广到称为Garland偏序集的组合结构所对应的非单纯（上）链复形，并深入研究了立方复形情形。

Quillen model structures for relative homological algebra

An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is encompassed by Quillen's homotopical algebra. The goal of this paper is to show that more general forms of homological algebra also fit into Quillen's framework. Specifically, a projective class on a complete and cocomplete abelian category A is exactly the information needed to do homological algebra in A. The main result is that, under weak hypotheses, the category …