Bernhard Keller
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.
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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.
We generalize the notion of Külshammer ideals to the setting of a graded category. This allows us to define and study some properties of Külshammer type ideals in the graded center of a triangulated category and in the Hochschild cohomology of an algebra, providing new derived invariants. Further properties of Külshammer ideals are studied in the case where the category is $d$-Calabi-Yau.
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.
In this paper we investigate the functoriality properties of map-graded Hochschild complexes. We show that the category MAP of map-graded categories is naturally a stack over the category of small categories endowed with a certain Grothendieck topology of 3-covers. For a related topology of infinity-covers on the cartesian morphisms in MAP, we prove that taking map-graded Hochschild complexes defines a sheaf. From the functoriality related to "injections" between map-graded categories, we obtain Hochschild complexes "with support". We revisit Keller's arrow category argument from this perspect…
We show that the functor from curved differential graded algebras to differential graded categories, defined by the second author in [B], sends Cartesian diagrams to homotopy Cartesian diagrams, under certain reasonable hypotheses. This is an extension to the arena of dg categories of a construction of projective modules due to Milnor. As an example, we show that the functor satisfies descent for certain partitions of a complex manifold.
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。
Regular and higher regular graded algebras (in simplest case satisfying Von Neumann regularity $Θ_{1}Θ_{2}Θ_{1}=Θ_{1}$ instead of anticommutativity) are introduced and their properties are studied. They are described in terms of obstructed categories with nonclosed invertible and noninvertible morphisms for which generalized (obstructed) functors and natural transformations are given. Regular algebras and bialgebras are considered with examples. Corresponding regularizations of the cross product and Wick algebras are made.
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.
In this paper we describe an inductive machinery to investigate asymptotic behaviors of homology groups and related invariants of representations of certain graded combinatorial categories over a commutative Noetherian ring $k$, via introducing inductive functors which generalize important properties of shift functors of $\mathrm{FI}$-modules. In particular, a sufficient criterion for finiteness of Castelnuovo-Mumford regularity of finitely generated representations of these categories is obtained. As applications, we show that a few important infinite combinatorial categories appearing in rep…
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].
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 $n>1$.
Inductive graded rings, hyperfields and quadratic forms
本文详细分析归纳分次环范畴，并利用该分析深化特殊超域范畴与归纳分次环范畴之间的联系，服务于二次型代数理论中Marshall符号差猜想的研究。
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。
Keller introduced a notion of quotient of a differential graded category modulo a full differential graded subcategory which agrees with Verdier's notion of quotient of a triangulated category modulo a triangulated subcategory. This work is an attempt to further develop his theory. More than a half of the text is devoted to an overview of "well known" definitions and results. As a result, the e-print is essentially self-contained.
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。
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 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。
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 …
We construct an A_infinity-category D(C|B) from a given A_infinity-category C and its full subcategory B. The construction is similar to a particular case of Drinfeld's quotient of differential graded categories. We use D(C|B) to construct an A_infinity-functor of K-injective resolutions of a complex. The conventional derived category is obtained as the 0-th cohomology of the quotient of differential graded category of complexes over acyclic complexes.
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