Peter Jorgensen
A recollement of triangulated categories describes one such category as being "glued together" from two others. This paper gives a precise criterion for the existence of a recollement of the derived category of a Differential Graded Algebra in terms of two other such categories.
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Ore Localisation for differential graded rings; Towards Goldie's theorem for differential graded algebras
本文研究微分分次代数的 Ore 局部化，定义 dg-素环、dg-半素环、dg-幂零根、dg-本质子模与 dg-一致维数，并推广 Goldie 定理至素 dg-环。
Differential Graded Algebras can be studied through their Differential Graded modules. Among these, the compact ones attract particular attention. This paper proves that over a suitable chain Differential Graded Algebra R, each compact Differential Graded module M satisfies amp M >= amp R, where amp denotes amplitude which is defined in a straightforward way in terms of the homology of a DG module. In other words, the homology of each compact DG module M is at least as long as the homology of R itself. Conversely, DG modules with shorter homology than R are not compact, and so in general, ther…
Consider a local chain Differential Graded algebra, such as the singular chain complex of a pathwise connected topological group. In two previous papers, a number of homological results were proved for such an algebra: An Amplitude Inequality, an Auslander-Buchsbaum Equality, and a Gap Theorem. These were inspired by homological ring theory. By the so-called looking glass principle, one would expect that analogous results exist for simply connected cochain Differential Graded algebras, such as the singular cochain complex of a simply connected topological space. Indeed, this paper establishes …
Differential graded orders, their class groups and idèles
该研究针对半单微分分次$K$-代数定义了微分分次$R$-序，证明其含于极大微分分次序，发展了微分分次环论，定义了相应类群与伊代尔并建立满射关系，还构造了同调类群同态与类Mayer-Vietoris序列。
In this note, we interpret Leibniz algebras as differential graded Lie algebras. Namely, we consider two functors from the category of Leibniz algebras to that of differential graded Lie algebras and show that they naturally give rise to the Leibniz cohomology and the Chevalley-Eilenberg cohomology. As an application, we prove a conjecture stated by Pirashvili in arXiv:1904.00121 [math.KT].
Cohomology of Lie coalgebras
本文建立了李余代数与相应微分分次代数之间的Koszul对偶型对应，将李余代数上同调解释为导出函子，并推广至交换情形。
Differential graded categories enhance our understanding of triangulated categories appearing in algebra and geometry. In this survey, we review their foundations and report on recent work by Drinfeld, Dugger-Shipley, ..., Toen and Toen-Vaquie.
Reflecting Perfection for Finite Dimensional Differential Graded Algebras
本文将有限维代数的中山引理与单模检测有限投射维数两个事实推广到有限维微分分次代数，证明两个分别关联Gorenstein微分分次代数与Koszul对偶的对偶版本，并给出余可表示性应用。
The concept of Koszul differential graded algebra (Koszul DG algebra) is introduced. Koszul DG algebras exist extensively, and have nice properties similar to the classic Koszul algebras. A DG version of the Koszul duality is proved. When the Koszul DG algebra $A$ is AS-regular, the Ext-algebra $E$ of $A$ is Frobenius. In this case, similar to the classical BGG correspondence, there is an equivalence between the stable category of finitely generated left $E$-modules, and the quotient triangulated category of the full triangulated subcategory of the derived category of right DG $A$-modules cons…
We construct a zig-zag of Quillen adjunctions between the homotopy theories of differential graded and simplicial categories. In an intermediate step we generalize Shipley-Schwede's work on connective DG algebras by extending the Dold-Kan correspondence to a Quillen equivalence between categories enriched over positive graded chain complexes and simplicial k-modules. As an application we obtain a conceptual explanation of Simpson's homotopy fiber construction.
Differential graded enhancements of singularity categories
针对环的奇点范畴，本文构造其Vogel dg范畴、奇异Yoneda dg范畴两种微分分次增强，证明二者拟等价，并汇报了奇异预倾斜猜想的相关研究进展。
On the cyclic homology of certain universal differential graded algebras
该研究针对奇素数$p$和无$p$-挠交换$\bZ_{(p)}$-代数$R$，计算了由$R$泛零化$p$得到的泛微分分次代数$R//p$的周期循环同调，以及其无穷多个次数的循环同调与负循环同调。
In this paper, we give a concrete description of the higher homotopy groups (n>0) of the mapping space Map_{Alg}(R,S) for R and S unbounded differential graded algebras (DGA) over a commutative ring k. In the connective case, we describe the relation between the higher (negative) Hochschild cohomology $HH^{-n+1}(R,S)$ and higher homotopy groups $π_{n} Map_{Alg}(R,S)$, when $n>1$.
We define a topological Hochschild (THH) and cyclic (TC) homology theory for differential graded (dg) categories and construct several non-trivial natural transformations from algebraic K-theory to THH(-). In an intermediate step, we prove that the homotopy theory of dg categories is Quillen equivalent, through a four step zig-zag of Quillen equivalences, to the homotopy theory of Eilenberg-Mac Lane spectral categories. Finally, we show that over the rationals two dg categories are topological equivalent if and only if they are quasi-equivalent.
The Chern character of $\vartheta$-summable $\mathcal{C}_q$-Fredholm modules over locally convex differential graded algebras
本文构造局部凸微分分次代数上 $\vartheta$-可和 $\mathcal{C}_q$-Fredholm 模的 Chern 特征标，统一奇偶情形并建立其与谱流的联系。
In this paper one considers three homotopy functors on the category of manifolds, $hH^\ast, cH^\ast, sH^\ast,$ and parallel them with other three homotopy functors on the category of connected commutative differential graded algebras, $HH^\ast, CH^\ast, SH^\ast.$ If $P$ is a smooth 1-connected manifold and the algebra is the de-Rham algebra of $P$ the two pairs of functors agree but in general do not. The functors $ HH^\ast $ and $CH^\ast$ can be also derived as Hochschild resp. cyclic homology of commutative differential graded algebra, but this is not the way they are introduced here. The th…
Intrinsic homological algebra for triangulated categories
本文提出了一种新的框架，用于研究三角范畴的同调性质，包括正则性、有限维数、戈林斯滕性质等，并推广了回溯理论中的结果。
Spectral sequences for the cyclic cohomology of differential graded algebras
本文为微分分次代数构造了新的谱序列以计算其循环上同调，证明了低维结果及相关映射的存在性，并将这些方法推广到 dg-范畴情形。
Proper connective differential graded algebras and their geometric realizations
本文证明每个正常连通DG-代数可由带完全例外集合的光滑射影概形几何实现，并由此得到有限维拟同构及非交换Chow motive的计算，同时探讨光滑性与正则性的关系。
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