Suspension splittings of 5-dimensional Poincaré duality complexes and their applications

Let A be a noetherian commutative ring. A complex R ∈ D(A) is called a dualizing complex (DC) if it has bounded finitely generated cohomology, finite injective dimension, and the derived Morita property, which says that the derived homothety morphism : A → RHomA(R, R) in D(A) is an isomorphism. We prove uniqueness of DCs and existence when A is essentially finite type over a regular noetherian ring. A residue complex is a DC that consists of injective modules of the correct multiplicity in each degree. There is a stronger uniqueness property for residue complexes. To understand residue complexes, we review the Matlis classification of injective A-modules. In the last two sections we talk about Van den Bergh rigidity. We prove that if A has a rigid DC R, then it is unique up to a unique rigid isomorphism. Existence of a rigid DC is harder to prove, and we just give a reference to it. Rigid residue complexes always exist, and they are unique in a very strong sense. We end this chapter with remarks that explain how rigid residue complexes allow a new approach to residues and duality on schemes and Deligne--Mumford stacks.

Dualizing complexes and systems of parameters

A graded Gersten–Witt complex for schemes with a dualizing complex and the Chow group

Stabilization of manifolds by a product of spheres or a projective space is important in geometry. There has been considerable recent work that studies the homotopy theory of stabilization for connected manifolds. This paper generalizes that work by developing new methods that allow for a generalization to stabilization of Poincaré duality complexes. This includes the systematic study of a homotopy theoretic generalization of a gyration, obtained from a type of surgery in the manifold case. In particular, for a fixed Poincaré duality complex, a criterion is given for the possible homotopy types of gyrations and shows there are only finitely many.

One of the important motivating problems for ring theory is to describe the rings which have some of the properties of commutative rings. In this talk we consider this problem for graded domains of dimension 3. The conjectures we present are based on ideas of my friends, especially of Toby Stafford, Michel Van den Bergh, and James Zhang. However, they may not be willing to risk making them, because only fragments of a theory exist at present. Everything here should be taken with a grain of salt. I am especially indebted to Toby Stafford for showing me some rings constructed from differential operators which I had overlooked in earlier versions of this manuscript.

A generalization of the dualizing complex structure and its applications

Given a field \(\Bbbk \), Noetherian \(\Bbbk \)-schemes (resp., \(\Bbbk \)-formal schemes) X, Y, and dualizing complexes \(R_X, R_Y\) over X, Y, we show that \(R_X \boxtimes _{\Bbbk } R_Y\) (resp., its derived completion) is a dualizing complex over \(X\times _{\Bbbk } Y\) if and only if \(X\times _{\Bbbk } Y\) is Noetherian of finite Krull dimension.

A Conjecture of Sharp — The Case of Local Rings with dim nonCM ≤ 1 or dim ≤ 5

In this article, we show that if X is an excellent surface with rational singularities, the constant sheaf ℚl is a dualizing complex. In coefficient ℤl, we also prove that the obstruction for ℤl to become a dualizing complex, lies on the divisor class groups at singular points. As applications, we study the perverse sheaves and the weights of l-adic cohomology groups on such surfaces.

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.

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.

In this chapter we present a version of Grothendieck local duality for a Noetherian local ring admitting a dualizing complex. We derive it from a more general result involving the local cohomology with respect to an arbitrary ideal of a Noetherian ring admitting a dualizing complex, originally proved by Hartshorne for a regular ring of finite Krull dimension. We also extend Hartshorne’s affine duality stated for regular rings of finite Krull dimension to any Noetherian ring with a dualizing complex and provide a counterpart in local homology. Among other things we provide duality results involving both local homology and local cohomology, a recurrent theme in this monograph. We also investigate the local homology of a complex with Artinian homology, more generally with mini-max homology. We end the chapter with a short approach to Greenlees’ Warwick duality.

We show that for a Noetherian ring A that is I-adically complete for an ideal I, if $A/I$ admits a dualizing complex, so does A. This gives an alternative proof of the fact that a Noetherian complete local ring admits a dualizing complex. We discuss several consequences of this result. We also consider a generalization of the notion of dualizing complexes to infinite-dimensional rings and prove the results in this generality. In addition, we give an alternative proof of the fact that every excellent Henselian local ring admits a dualizing complex, using ultrapower.

Applications of duality theory to Cousin complexes

As the 20th century rushes to a speedy conclusion, African Americans' political sensibilities remain entangled in the duality complex that Du Bois spoke of nearly 100 years ago. That America's residents of African descent exhibit this complex in their politics is not surprising, considering that the subject of their national identity has yet to be reconciled. In its operative modes, the twoness condition yields paradoxical political urgings from African American communities seeking freedom and justice. One part finds expression most notably through assimilative and integrative-oriented activities; the other part is primarily embodied in a range of nationalist projects. On closer examination, we discover that the manifest political urgings of African Americans are byproducts of competing core values (mainly between those rooted in European and Euro-American conceptions of individualism) and those rooted in collective consciousness (an African-centered value). The troublesome aspect of African Americans' enigmatic political consciousness is that it impedes the development of an unfettered, proactive posture. More appropriately put, this precarious politics results from lacking a collective consciousness toward