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A note on the equivalence of m-periodic derived categories

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Science China Mathematics 2014年第11期57卷 2329-2334页

作者： ZHAO XiaoJuan Department of Mathematics Sichuan University

Let A and B be finite-dimensional algebras over a field k of finite global dimension. Using some results of Gorsky in "Semi-derived Hall algebras and tilting invariance of Bridgeland-Hall algebras",we prove that if A ... 详细信息

关键词： cyclic complex derived category root category triangulated category

On duality preservability of Auslander-Reiten quivers of derived categories and cluster categories

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Frontiers of Mathematics in China 2016年第4期11卷 957-984页

作者： Fang LI Lingyu WAN Department of Mathematics Zhejiang University (Yuquan Campus) Hangzhou 310027 China

We give sufficient conditions and necessary conditions on duality preservability of Auslander-Reiten quivers of derived categories and cluster categories over hereditary algebras. of generalized path algebras as cleft... 详细信息

关键词： Dual quiver Auslander-Reiten quiver derived category cluster category

derived Equivalences and Recollements of Differential Graded Algebras and Schemes

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Algebra Colloquium 2016年第3期23卷 385-408页

作者： Xinhong Chen Ming Lu Department of Mathematics Siehuan University Chengdu 610064 China

In this paper, we first prove for two differential graded algebras （DGAs） A, B which are derived equivalent to k-algebras A, F, respectively, that ：D（Ak B） ≈D（Ak Г）. In particular, Hp^b（Ak B） ≈ Hb（proj-A k Г）. Secondly, for two quasi-compact and sepa- rated schemes X, Y and two algebras A, B over k which satisfy ：D（Qcoh（X）） ≈：D（A） and ：D（Qcoh（Y）） ≈D（B）, we show that ：D（Qcoh（X × Y）） ≈ 79（AB） and ：Db（Coh（X × Y）） ≈Db（mod-（A B））. Finally, we prove that if X is a quasi-compact and separated scheme over k, then ：D（Qcoh（X ~ pl）） admits a recollement relative to D（Qcoh（X））, and we de- scribe the functors in the recollement explicitly. This recollement induces a recollement of bounded derived categories of coherent sheaves and a recollement of singularity categories. When the scheme X is derived equivalent to a DGA or algebra, then the recollement which we get corresponds to the recollement of DGAs in [14] or the recollement of upper triangular algebras in [7].

关键词： derived category differential graded algebra perfect complex recollement,singularity category

Frobenius morphisms and stable module categories of repetitive algebras

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Science China Mathematics 2008年第2期51卷 169-184页

作者： DENG BangMing WAN JinKui School of Mathematical Sciences Beijing Normal UniversityBeijing 100875China

Let k be the algebraic closure of a finite field $\mathbb{F}_q $ and A be a finite dimensional k-algebra with a Frobenius morphism F. In the present paper we establish a relation between the stable module category of the repetitive algebra ? of A and that of the repetitive algebra of the fixed-point algebra A F. As an application, it is shown that the derived category of A F is equivalent to the subcategory of F-stable objects in the derived category of A when A has a finite global dimension.

关键词： Frobenius morphism repetitive algebra stable module category derived category 16G10 18E30

Bounded derived categories of very simple manifolds

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Science China Mathematics 2011年第8期54卷 1741-1750页

作者： OGUISO Keiji Department of Mathematics Graduate School of ScienceOsaka University Korea Institute for Advanced Study

A non-representable cohomological functor of finite type of the bounded derived category of coherent sheaves of a compact complex manifold of dimension greater than one with no proper closed subvariety is given explic... 详细信息

关键词： derived category cohomological functor very simple manifold

Koszul differential graded modules

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Science China Mathematics 2009年第9期52卷 2027-2035页

作者： HE JiWei WU QuanShui Department of Mathematics Shaoxing College of Arts and SciencesShaoxing 312000China School of Mathematical Sciences Fudan UniversityShanghai 200433China

The concept of Koszulity for differential graded (DG, for short) modules is introduced. It is shown that any bounded below DG module with bounded Ext-group to the trivial module over a Koszul DG algebra has a Koszul DG submodule (up to a shift and truncation), moreover such a DG module can be approximated by Koszul DG modules (Theorem 3.6). Let A be a Koszul DG algebra, and Dc(A) be the full triangulated subcategory of the derived category of DG A-modules generated by the object AA. If the trivial DG module kA lies in Dc(A), then the heart of the standard t-structure on Dc(A) is anti-equivalent to the category of finitely generated modules over some finite dimensional algebra. As a corollary, Dc(A) is equivalent to the bounded derived category of its heart as triangulated categories.

关键词： Koszul DG module derived category heart of t-structure 16E05 16E40 16W50

The tale Homology and the Cycle Maps in Adic Coefficients

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Communications in Mathematical Research 2013年第1期29卷 68-87页

作者： Li Ting Du Xian-kun Mathematical College Sichuan University

In this article, we define the l-adic homology for a morphism of schemes satisfying certain finiteness conditions. This homology has these functors similar to the Chow groups： proper push-forward, fiat pull-back, base change, cap-product, etc. In particular, on singular varieties, this kind of l-adic homology behaves much better than the classical l-adie cohomology. As an application, we give a much easier approach to construct the cycle maps for arbitrary algebraic schemes over fields. And we prove that these cycle maps kill the algebraic equivalences and commute with the Chern action of locally free sheaves.

关键词： l-adic cohomology cycle map derived category