Peter Jorgensen
Let A be a noetherian local commutative ring and let M be a suitable complex of A-modules. This paper proves that M is a dualizing complex for A if and only if the trivial extension A \ltimes M is a Gorenstein Differential Graded Algebra. As a corollary follows that A has a dualizing complex if and only if it is a quotient of a Gorenstein local Differential Graded Algebra.
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In this article we survey recent results on rigid dualizing complexes over commutative algebras. We begin by recalling what are dualizing complexes. Next we define rigid complexes, and explain their functorial properties. Due to the possible presence of torsion, we must use differential graded algebras in the constructions. We then discuss rigid dualizing complexes. Finally we show how rigid complexes can be used to understand Cohen-Macaulay homomorphisms and relative dualizing sheaves.
For the Cousin complex of certain modules, we investigate finiteness of cohomology modules, local duality property and injectivity of its terms. The existence of canonical modules of Noetherian non-local rings and the Cousin complexes of them with respect to the height filtration are discussed .
Dualizing and canonical complexes on finite posets
本文发展了有限偏序集上的对偶复形理论及科恩-麦考利性理论，为相关代数几何研究奠定基础。
We investigate how one can detect the dualizing property for a chain complex over a commutative local noetherian ring R. Our focus is on homological properties of contracting endomorphisms of R, e.g., the Frobenius endomorphism when R contains a field of positive characteristic.
Let C be a semidualizing complex over a noetherian local ring A. If there exists a local homomorphism with source A satisfying some homological properties, then C is dualizing.
We characterize the seminormality of an affine semigroup ring in terms of the dualizing complex, and the normality of a Cohen-Macaulay semigroup ring by the "shape" of the canonical module. We also characterize the seminormality of a toric face ring in terms of the dualizing complex. A toric face ring is a simultaneous generalization of Stanley-Reisner rings and affine semigroups.
Let $k$ be a field, and let $X,Y$ be two locally noetherian $k$-schemes (respectively $k$-formal schemes) with dualizing complexes $R_X$ and $R_Y$ respectively. We show that $R_X \boxtimes_{k} R_Y$ (respectively its derived completion) is a dualizing complex over $X\times_{k} Y$ if and only if $X\times_{k} Y$ is locally noetherian of finite Krull dimension.
Gorenstein injective filtrations over rings with dualizing complexes
本文将Gorenstein环上Gorenstein内射模的有限Gorenstein内射子模滤过结论，推广至带对偶复形的交换诺特环，并用Auslander范畴和截面函子实现了这类滤过。
Formal lifting of dualizing complexes and consequences
本文证明了 $I$-进完备 Noetherian 环上对偶复形的形式提升定理，并据此给出了完备局部环和 excellent Henselian 局部环容许对偶复形的替代证明，同时将对偶复形概念推广至无穷维环。
Rigid Dualizing Complexes over Commutative Rings and their Functorial Properties
本文通过刚性对偶复形处理noetherian环的Grothendieck Duality，证明了正则基环上本质有限型环具有唯一的刚性对偶复形，并利用其函子性质构造了扭曲诱导伪函子，为后续在Deligne-Mumford stacks上建立全局对偶性奠定基础。
In this paper we present a new approach to Grothendieck duality over commutative rings. Our approach is based on the idea of rigid dualizing complexes, which was introduced by Van den Bergh in the context of noncommutative algebraic geometry. The method of rigidity was modified to work over general commutative base rings in our paper [YZ5]. In the present paper we obtain many of the important local features of Grothendieck duality, yet manage to avoid lengthy and difficult compatibility verifications. Our results apply to essentially finite type algebras over a regular noetherian finite dimens…
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。
We prove basic facts about reflexivity in derived categories over noetherian schemes; and about related notions such as semidualizing complexes, invertible complexes, and Gorenstein-perfect maps. Also, we study a notion of rigidity with respect to semidualizing complexes, in particular, relative dualizing complexes for Gorenstein-perfect maps. Our results include theorems of Yekutieli and Zhang concerning rigid dualizing complexes on schemes. This work is a continuation of part I, which dealt with commutative rings.
A commutative noetherian ring with a dualizing complex is Gorenstein if and only if every acyclic complex of injective modules is totally acyclic. We extend this characterization, which is due to Iyengar and Krause, to arbitrary commutative noetherian rings, i.e. we remove the assumption about a dualizing complex. In this context Gorenstein, of course, means locally Gorenstein at every prime.
We use the anti-equivalence between Cohen-Macaulay complexes and coherent sheaves on formal schemes to shed light on some older results and prove new results. We bring out the relations between a coherent sheaf M satisfying an S_2 condition and the lowest cohomology N of its "dual" complex. We show that if a scheme has a Gorenstein complex satisfying certain coherence conditions, then in a finite étale neighborhood of each point, it has a dualizing complex. If the scheme already has a dualizing complex, then we show that the Gorenstein complex must be a tensor product of a dualizing complex an…
Some characterizations of dualizing complexes in terms of $G_{C}$-dimension
本文研究了局部环上对偶复形的刻画问题，基于 $G_{C}$-维数，证明了若干关于半对偶化复形成为对偶复形或同构于环本身的充要条件。
We introduce and study a class of objects that encompasses Christensen and Foxby's semidualizing modules and complexes and Kubik's quasi-dualizing modules: the class of $\mathfrak{a}$-adic semidualizing modules and complexes. We give examples and equivalent characterizations of these objects, including a characterization in terms of the more familiar semidualizing property. As an application, we give a proof of the existence of dualizing complexes over complete local rings that does not use the Cohen Structure Theorem.
We prove that a local ring $R$ of embedding codepth at most 3 has at most two semidualizing complexes up to shift-isomorphism, namely, $R$ itself and a dualizing $R$-complex if one exists.
It is known that the numerical invariants Betti numbers and Bass numbers are worthwhile tools for decoding a large amount of information about modules over commutative rings. We highlight this fact, further, by establishing some criteria for certain semidualizing complexes via their Betti and Bass numbers. Two distinguished types of semidualizing complexes are the shifts of the underlying rings and dualizing complexes. Let $C$ be a semidualizing complex for an analytically irreducible local ring $R$ and set $n:=\sup C$ and $d:=\dim_RC$. We show that $C$ is quasi-isomorphic to a shift of $R$ if…
A notion of rigidity with respect to an arbitrary semidualizing complex C over a commutative noetherian ring R is introduced and studied. One of the main result characterizes C-rigid complexes. Specialized to the case when C is the relative dualizing complex of a homomorphism of rings of finite Gorenstein dimension, it leads to broad generalizations of theorems of Yekutieli and Zhang concerning rigid dualizing complexes, in the sense of Van den Bergh. Along the way, a number of new results concerning derived reflexivity with respect to C are established. Noteworthy is the statement that derive…
Pseudo-dualizing complexes of torsion modules and semi-infinite MGM duality
本文研究了交换环中 torsion 模的半无限 MGM 对偶性，构造了伪双化复叠并探讨了其在不同环结构下的等价关系。
For a given class of modules $\A$, we denote by $\widetilde{\A}$ the class of exact complexes $X$ having all cycles in $\A$, and by $dw(\A)$ the class of complexes $Y$ with all components $Y_j$ in $\A$. We consider a two sided noetherian ring $R$ and we use the notations $\mathcal{GI}$ $(\mathcal{GF}, \mathcal{GP})$ for the class of Gorenstein injective (flat, projective respectively) $R$-modules. We prove (Theorem 1) that the following are equivalent: 1. Every exact complex of injective modules is totally acyclic. 2. Every exact complex of Gorenstein injective modules is in $\widetilde{\mathc…
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…
We investigate the set S(R) of shift-isomorphism classes of semidualizing R-complexes, ordered via the reflexivity relation, where R is a commutative noetherian local ring. Specifically, we study the question of whether S(R$ has cardinality 2^n for some n. We show that, if there is a chain of length n in S(R) and if the reflexivity ordering on S(R) is transitive, then S(R) has cardinality at least 2^n, and we explicitly describe some of its order-structure. We also show that, given a local ring homomorphism f: R\to S of finite flat dimension, if R and S admit dualizing complexes and if f is no…
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。
Let $R$ be a commutative noetherian ring with a dualizing complex. By recent work of Iyengar and Krause, the difference between the category of acyclic complexes and its subcategory of totally acyclic complexes measures how far $R$ is from being Gorenstein. In particular, $R$ is Gorenstein if and only if every acyclic complex is totally acyclic. In this note we exhibit a specific acyclic complex with the property that it is totally acyclic if and only if $R$ is Gorenstein.
Let A be a commutative ring, B a commutative A-algebra and M a complex of B-modules. We begin by constructing the square Sq_{B/A} M, which is also a complex of B-modules. The squaring operation is a quadratic functor, and its construction requires differential graded (DG) algebras. If there exists an isomorphism ρ: M \iso Sq_{B/A} M then the pair (M,ρ) is called a rigid complex over B relative to A (there are some finiteness conditions). There is an obvious notion of rigid morphism between rigid complexes. We establish several properties of rigid complexes, including their uniqueness, existenc…
We give a sufficient condition for the class of Gorenstein injective modules be precovering: if $R$ is right noetherian and if the class of Gorenstein injective modules, $\mathcal{GI}$, is closed under filtrations, then $\mathcal{GI}$ is precovering in $R-Mod$. The converse is also true when we assume that $\mathcal{GI}$ is covering. We extend our results to the category of complexes. We prove that if the class of Gorenstein injective modules is closed under filtrations then the class of Gorenstein injective complexes is precovering in $Ch(R)$. We also give a sufficient condition for the exist…
On a suitable category of formal schemes equipped with codimension functions we construct a canonical pseudofunctor (-)^# taking values in the corresponding categories of Cousin complexes. Cousin complexes on such a formal scheme X functorially represent derived-category objects F by the local cohomologies H_x^{codim(x)}(F) (x \in X) together with "residue maps" from the cohomology at x to that at each immediate specialization of x; this representation is faithful when restricted to F which are Cohen-Macaulay (CM), i.e., H_x^i(F)=0 unless i=codim(x). Formal schemes provide a framework for trea…
Let R be a commutative noetherian local ring with completion R^. We apply differential graded (DG) algebra techniques to study descent of modules and complexes from R^ to R' where R' is either the henselization of R or a pointed étale neighborhood of R: We extend a given R^-complex to a DG module over a Koszul complex; we describe this DG module equationally and apply Artin approximation to descend it to R. This descent result for Koszul extensions has several applications. When R is excellent, we use it to descend the dualizing complex from R^ to a pointed étale neighborhood of R; this yields…
Let $(R,\fm)$ be a local ring and $(-)^{\vee}$ denote the Matlis duality functor. We investigate the relationship between Foxby equivalence and local duality through generalized local cohomology modules. Assume that $R$ possesses a normalized dualizing complex $D$ and $X$ and $Y$ are two homologically bounded complexes of $R$-modules with finitely generated homology modules. We present several duality results for $\fm$-section complex ${\bf R}Γ_{\fm}({\bf R}\Hom_R(X,Y))$. In particular, if G-dimension of $X$ and injective dimension of $Y$ are finite, then we show that $${\bf R}Γ_{\fm}({\bf R}\…
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