Dualizing Complexes and Perverse Modules over Differential Algebras

Change of grading, injective dimension and dualizing complexes

Let $G,H$ be groups, $ϕ: G \rightarrow H$ a group morphism, and $A$ a $G$-graded algebra. The morphism $ϕ$ induces an $H$-grading on $A$, and on any $G$-graded $A$-module, which thus becomes an $H$-graded $A$-module. Given an injective $G$-graded $A$-module, we give bounds for its injective dimension when seen as $H$-graded $A$-module. Following ideas by Van den Bergh, we give an application of our results to the stability of dualizing complexes through change of grading.

仿射Hecke代数的刚性对偶复形

Rigid dualizing complexes of affine Hecke algebras

本文确定了约化根系关联的仿射Hecke代数$H_q$的刚性对偶复形，并推导出经典Hecke代数在特定条件下在其中心上为Frobenius的，且Nakayama自同构由显式对合给出。

Coherent rings, fp-injective modules, dualizing complexes, and covariant Serre-Grothendieck duality

For a left coherent ring A with every left ideal having a countable set of generators, we show that the coderived category of left A-modules is compactly generated by the bounded derived category of finitely presented left A-modules (reproducing a particular case of a recent result of Stovicek with our methods). Furthermore, we present the definition of a dualizing complex of fp-injective modules over a pair of noncommutative coherent rings A and B, and construct an equivalence between the coderived category of A-modules and the contraderived category of B-modules. Finally, we define the notio…

Dualizing Complexes, Morita Equivalence and the Derived Picard Group of a Ring

Two rings A and B are said to be derived Morita equivalent if their derived categories of modules are equivalent. By results of Rickard, if A and B are derived Morita equivalent algebras over a field k, then there is a complex of bimodules T s.t. the derived tensor product with T is an equivalence. The complex T is called a tilting complex. When B = A the isomorphism classes of tilting complexes T form the derived Picard group DPic(A). We prove that when the algebra A is either local or commutative, then any derived Morita equivalent algebra B is actually Morita equivalent. So we can compute D…

Pseudo-dualizing complexes of bicomodules and pairs of t-structures

This paper is a coalgebra version of arXiv:1703.04266 and a sequel to arXiv:1607.03066. We present the definition of a pseudo-dualizing complex of bicomodules over a pair of coassociative coalgebras $\mathcal C$ and $\mathcal D$. For any such complex $\mathcal L^\bullet$, we construct a triangulated category endowed with a pair of (possibly degenerate) t-structures of the derived type, whose hearts are the abelian categories of left $\mathcal C$-comodules and left $\mathcal D$-contramodules. A weak version of pseudo-derived categories arising out of (co)resolving subcategories in abelian/exact…

伪对偶化复形与伪导出范畴

Pseudo-dualizing complexes and pseudo-derived categories

本文通过去掉内射维数条件定义伪对偶化复形，并证明在一对结合环上它诱导伪余导出范畴与伪反导出范畴之间的三角等价，推广了先前导出函子构造技术。

Homological invariants associated to semi-dualizing bimodules

Cohen-Macaulay dimension for modules over a commutative noetherian local ring has been defined by A. A. Gerko. That is a homological invariant sharing many properties with projective dimension and Gorenstein dimension. The main purpose of this paper is to extend the notion of Cohen-Macaulay dimension for modules over commutative noetherian local rings to that for bounded complexes over non-commutative noetherian rings.

Recollement for perverse sheaves on real hyperplane arrangements

We consider a hyperplane arrangement in $\mathbb{C}^n$ defined over $\mathbb{R}$, and the associated natural stratification of $\mathbb{C}^n$. The category of perverse sheaves smooth with respect to this stratification was described by Kapranov and Schechtman in terms of quiver representations. Using work of Weissman, we reinterpret this category as the category of finite-dimensional modules over an explicit algebra. We also describe recollement (open-closed decomposition) of perverse sheaves in terms of this module category. As an application, we identify the modules associated to all interse…

Tilting 复形与交换诺特环上的余维数函数

Tilting complexes and codimension functions over commutative noetherian rings

本文在交换诺特环的导出范畴中，利用局部上同调构造与 sp-滤过关联的 silting 对象，并刻画其成为 tilting 的条件，特别与余维数函数及 Cohen-Macaulay 环同态像相关，同时给出 cosilting 对偶结果。

Acyclicity versus total acyclicity for complexes over noetherian rings

It is proved that for a commutative noetherian ring with dualizing complex the homotopy category of projective modules is equivalent, as a triangulated category, to the homotopy category of injective modules. Restricted to compact objects, this statement is a reinterpretation of Grothendieck's duality theorem. Using this equivalence it is proved that the (Verdier) quotient of the category of acyclic complexes of projectives by its subcategory of totally acyclic complexes and the corresponding category consisting of injective modules are equivalent. A new characterization is provided for comple…

Gorenstein 无环复形与有限维数

Gorenstein acyclic complexes and finitistic dimensions

本文研究带对偶复形的双边Noether环的大有限维数有限性问题，给出其与Gorenstein投射-无环上链复形可缩性的等价刻画，并对Artin代数证明Rickard定理的Gorenstein版本。

Acyclic complexes of injectives and finitistic dimensions

For a ring $A$, we consider the question whether every bounded above cochain complex of injective $A$-modules which is acyclic is null-homotopic. We show that if $A$ is left and right noetherian and has a dualizing complex, then this implies that the finitistic dimension of $A$ is finite. In the appendix, Nakamura and Thompson show that the opposite holds over any ring. Our results give several new necessary and sufficient conditions for a ring to have finite finitistic dimension in a very general setting. Applications include a generalization of a recent result of Rickard about relations betw…

极小Cohen-Macaulay DG复形

Maximal Cohen-Macaulay DG-complexes

本文提出极小Cohen-Macaulay DG复形的定义，扩展了Iyengar等人关于局部环的极小Cohen-Macaulay复形的研究，并建立了具有恒定振幅的DG环的导出改进新交集定理。

Quivers with analytic potentials

Given a quiver $Q$, a formal potential is called analytic if its coefficients are bounded by the terms of a geometric series. As shown by Toda, the potentials appearing in the deformation theory of complexes of coherent sheaves on complex projective Calabi-Yau threefolds are analytic. Our paper consists of two parts. In the first part, we establish the foundations of the differential calculus of quivers with analytic potentials and prove two fundamental results: the inverse function theorem and Moser's trick. As an application, we prove finite determinacy of analytic potentials with finite-dim…

正分次代数上的数值同调正则性

Numerical homological regularities over positively graded algebras

本文研究正分次代数上复形的多种数值同调正则性，证明其与导出 Hom 复形极值次数的一致性，建立若干不等式及其成为等式的条件，并利用数值 AS-正则性刻画 AS-正则性质和有限维性质。