Dualizing Complex Research Articles

Sphere fibrations over highly connected manifolds

AbstractWe construct sphere fibrations over ‐connected ‐manifolds such that the total space is a connected sum of sphere products. More precisely, for even, we construct fibrations , where is a ‐connected ‐dimensional Poincaré duality complex that satisfies , in a localised category of spaces. The construction of the fibration is proved for , where the prime 2, and the primes that occur as torsion in are inverted. In specific cases, by either assuming is small, or assuming is large we can reduce the number of primes that need to be inverted. Integral results are obtained for or 4, and if is bigger than the number of cyclic summands in the stable stem , we obtain results after inverting 2. Finally, we prove some applications for fibrations over , and for looped configuration spaces.

Homotopy Fibrations with a Section After Looping

We analyze a general family of fibrations which, after looping, have sections. Methods are developed to determine the homotopy type of the fibre and the homotopy classes of the map from the fibre to the base. The methods are driven by applications to two-cones, Poincaré Duality complexes, the connected sum operation, and polyhedral products.

TILTING COMPLEXES AND CODIMENSION FUNCTIONS OVER COMMUTATIVE NOETHERIAN RINGS

AbstractIn the derived category of a commutative noetherian ring, we explicitly construct a silting object associated with each sp-filtration of the Zariski spectrum satisfying the “slice” condition. Our new construction is based on local cohomology and it allows us to study when the silting object is tilting. For a ring admitting a dualizing complex, this occurs precisely when the sp-filtration arises from a codimension function on the spectrum. In the absence of a dualizing complex, the situation is more delicate and the tilting property is closely related to the condition that the ring is a homomorphic image of a Cohen–Macaulay ring. We also provide dual versions of our results in the cosilting case.

Moore’s conjecture for connected sums

AbstractWe show that under mild conditions, the connected sum $M\# N$ of simply connected, closed, orientable n-dimensional Poincaré Duality complexes M and N is hyperbolic and has no homotopy exponent at all but finitely many primes, verifying a weak version of Moore’s conjecture. This is derived from an elementary framework involving $CW$ -complexes satisfying certain conditions.

The Finitistic Dimension and Global Dimension of an Artinian Ring with a Dualizing Complex

The Finitistic Dimension and Global Dimension of an Artinian Ring with a Dualizing Complex

Loop space decompositions of (2n-2)-connected (4n-1)-dimensional Poincaré Duality complexes

Beben and Wu showed that if M is a (2n-2)-connected (4n-1)-dimensional Poincaré Duality complex such that nge 3 and H^{2n}(M;{{mathbb {Z}}}) consists only of odd torsion, then Omega M can be decomposed up to homotopy as a product of simpler, well-studied spaces. We use a result from Beben and Theriault (Doc Math 27:183-211, 2022) to greatly simplify and enhance Beben and Wu’s work and to extend it in various directions.

Characterizing 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 and We show that C is quasi-isomorphic to a shift of R if and only if the nth Betti number of C is one. Also, we show that C is a dualizing complex for R if and only if the dth Bass number of C is one.

Homotopy groups of highly connected Poincaré duality complexes

Methods are developed to relate the action of a principal fibration to relative Whitehead products in order to determine the homotopy type of certain spaces. The methods are applied to thoroughly analyze the homotopy type of the based loops on certain cell attachments. Key examples are (n-1) -connected Poincaré Duality complexes of dimension 2n or 2n+1 with minor cohomological conditions.

A generalization of Gorenstein injective modules

Let R be an associative ring with identity and M be a left R-module. In this paper, we introduce M-Gorenstein injective modules as a generalization of Gorenstein injective modules. We verify some properties of M-Gorenstein injective modules analogous to those holding for Gorenstein injective modules. There is an interesting theorem in classical homological algebra which asserts that R is a Noetherian ring if and only if the class of injective modules over R is closed under arbitrary direct sum. Our goal in this paper is to investigate the M-Gorenstein injective counterpart of this fact. If the class of M-Gorenstein injective modules over R is closed under arbitrary direct sum, then R will be a Noetherian ring. Also, it has been proved that in the special case M=R, when R is a commutative Noetherian ring with a dualizing complex, then the class of R-Gorenstein injective modules is closed under arbitrary direct sum. In the main theorem of this paper, we prove the general case of this result. More precisely, we show that for any left R-module M over a Noetherian ring R in which every R-module has finite M-Gorenstein injective dimension, the class of M-Gorenstein injective modules is closed under arbitrary direct sum.

Homological identities and dualizing complexes of commutative differential graded algebras

In this paper we study a connective commutative differential graded algebra (CDGA) which is piecewise Noetherian. The principal aim is to analyze a dualizing complex of CDGA’s and investigate a Gorenstein CDGA. We also establish many other results which can be expected to play basic roles in the study of a CDGA, such as the Auslaner-Buchsbaum formula and Bass formula without any unnecessary assumptions. The key notion is the sup-projective (sppj) and inf-injective (ifij) resolutions introduced by the author, which are DG-versions of the projective and injective resolutions for ordinary modules. In the paper, we show that sppj and ifij resolutions are powerful tools to study DG-modules.

Pseudo-dualizing complexes and pseudo-derived categories

The definition of a pseudo-dualizing complex is obtained from that of a dualizing complex by dropping the injective dimension condition, while retaining the finite generatedness and homothety isomorphism conditions. In the specific setting of a pair of associative rings, we show that the datum of a pseudo-dualizing complex induces a triangulated equivalence between a pseudo-coderived category and a pseudo-contraderived category. The latter terms mean triangulated categories standing “in between” the conventional derived category and the coderived or the contraderived category. The constructions of these triangulated categories use appropriate versions of the Auslander and Bass classes of modules. The constructions of derived functors providing the triangulated equivalence are based on a generalization of a technique developed in our previous paper [45].

On the Gorenstein defect categories

The main theme of this paper is to study different “Gorenstein defect categories” and their connections. This is done by studying rings for which 𝕂 a c ( Prj- R ) = 𝕂 t a c ( Prj- R ) , that is, rings enjoying the property that every acyclic complex of projectives is totally acyclic. Such studies have been started by Iyengar and Krause over commutative Noetherian rings with a dualizing complex. We show that a virtually Gorenstein Artin algebra is Gorenstein if and only if it satisfies the above mentioned property. Then, we introduce recollements connecting several categories which help in providing categorical characterizations of Gorenstein rings. Finally, we study relative singularity categories that lead us to some more “Gorenstein defect categories”.

Smash products of Calabi–Yau algebras by Hopf algebras

Let H be a Hopf algebra and A be an H -module algebra. This article investigates when the smash product A\sharp H is (skew) Calabi–Yau, has Van den Bergh duality or is Artin–Schelter regular or Gorenstein. In particular, if A and H are skew Calabi–Yau, then so is A\sharp H and its Nakayama automorphism is expressed using the ones of A and H . This is based on a description of the inverse dualising complex of A\sharp H when A is a homologically smooth dg algebra and H is homologically smooth and with invertible antipode. This description is also used to explain the compatibility of standard constructions of Calabi–Yau dg algebras with taking smash products.

On the Gorenstein dimensions over local homomorphisms

In this paper, we further study Gorenstein dimension over local homomorphisms developed by Iyengar and Sather-Wagstaff. For a local homomorphism and a homologically finite S-complex X, we show that , which improves the known result by removing the assumption that R should have a dualizing complex. As an application, we prove that . This is analogous to a theorem of Iyengar and Sather-Wagstaff on projective dimension over local homomorphisms.

Poincaré duality complexes with highly connected universal cover

Turaev conjectured that the classification, realization and splitting results for Poincar\'e duality complexes of dimension $3$ (PD$_{3}$-complexes) generalize to PD$_{n}$-complexes with $(n-2)$-connected universal cover for $n \ge 3$. Baues and Bleile showed that such complexes are classified, up to oriented homotopy equivalence, by the triple consisting of their fundamental group, orientation class and the image of their fundamental class in the homology of the fundamental group, verifying Turaev's conjecture on classification. We prove Turaev's conjectures on realization and splitting. We show that a triple $(G, \omega, \mu)$, comprising a group, $G$, a cohomology class $\omega \in H^{1}\left(G, \mathbb{Z}/2\mathbb{Z}\right)$ and a homology class $\mu \in H_{n}(G, \mathbb{Z}^{\omega})$, can be realized by a PD$_{n}$-complex with $(n-2)$-connected universal cover if and only if the Turaev map applied to $\mu$ yields is an equivalence. We show that such a PD$_{n}$-complex is a connected sum of two such complexes if and only if its fundamental group is a free product of groups. We then consider the indecomposable $PD_n$-complexes of this type. When $n$ is odd the results are similar to those for the case $n=3$. The indecomposables are either aspherical or have virtually free fundamental group. When $n$ is even the indecomposables include manifolds which are neither aspherical nor have virtually free fundamental group, but if the group is virtually free and has no dihedral subgroup of order $>2$ then it has two ends.

On Four-Dimensional Poincaré Duality Cobordism Groups

This paper continues the study of four-dimensional Poincare duality cobordism theory from our previous work Cavicchioli et al. (Homol. Homotopy Appl. 18(2):267–281, 2016). Let P be an oriented finite Poincare duality complex of dimension 4. Then, we calculate the Poincare duality cobordism group $$\Omega _{4}^{{\text {PD}}}(P)$$ . The main result states the existence of the exact sequence $$0 \rightarrow L_4 (\pi _1 (P))/A_4 (H_2 (B\pi _1 (P), L_2)) \rightarrow {{\widetilde{\Omega }}}_{4}^{\mathrm{PD}}(P) \rightarrow \mathbb Z_8 \rightarrow 0$$ , where $${{\widetilde{\Omega }}}_{4}^{\mathrm{PD}}(P)$$ is the kernel of the canonical map $${\Omega }_{4}^{\mathrm{PD}}(P) \rightarrow H_4 (P, \mathbb Z) \cong \mathbb Z$$ and $$A_4 : H_4 (B\pi _1, \mathbb L) \rightarrow L_4 (\pi _1 (P))$$ is the assembly map. It turns out that $${\Omega }_{4}^{\mathrm{PD}}(P)$$ depends only on $$\pi _1 (P)$$ and the assembly map $$A_4$$ . This does not hold in higher dimensions. Then, we discuss several examples. The cases in which the canonical map $$\Omega _{4}^{{\text {TOP}}}(P) \rightarrow \Omega _{4}^{{\text {PD}}}(P)$$ is not surjective are of particular interest. Its image coincides with the kernel of the total surgery obstruction map. In fact, we establish an exact sequence where s is Ranicki’s total surgery obtruction map. In the above cases, there are $${\text {PD}}_4$$ -complexes X which cannot be homotopy equivalent to manifolds.

The relationship of generalized manifolds to Poincaré duality complexes and topological manifolds

The primary purpose of this paper concerns the relation of (compact) generalized manifolds to finite Poincaré duality complexes (PD complexes). The problem is that an arbitrary generalized manifold X is always an ENR space, but it is not necessarily a complex. Moreover, finite PD complexes require the Poincaré duality with coefficients in the group ring Λ (Λ-complexes). Standard homology theory implies that X is a Z-PD complex. Therefore by Browder's theorem, X has a Spivak normal fibration which in turn, determines a Thom class of the pair (N,∂N) of a mapping cylinder neighborhood of X in some Euclidean space. Then X satisfies the Λ-Poincaré duality if this class induces an isomorphism with Λ-coefficients. Unfortunately, the proof of Browder's theorem gives only isomorphisms with Z-coefficients. It is also not very helpful that X is homotopy equivalent to a finite complex K, because K is not automatically a Λ-PD complex. Therefore it is convenient to introduce Λ-PD structures. To prove their existence on X, we use the construction of 2-patch spaces and some fundamental results of Bryant, Ferry, Mio, and Weinberger. Since the class of all Λ-PD complexes does not contain all generalized manifolds, we appropriately enlarge this class and then describe (i.e. recognize) generalized manifolds within this enlarged class in terms of the Gromov–Hausdorff metric.

Totally acyclic complexes and locally Gorenstein 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.

Triangulated Equivalences Involving Gorenstein Projective Modules

AbstractFor any ringR, we show that, in the bounded derived categoryDb(ModR) of leftR-modules, the subcategory of complexes with finite Gorenstein projective (resp. injective) dimension modulo the subcategory of complexes with finite projective (resp. injective) dimension is equivalent to the stable category(resp.) of Gorenstein projective (resp. injective) modules. As a consequence, we get that ifRis a left and right noetherian ring admitting a dualizing complex, thenandare equivalent.

Homological unimodularity and Calabi–Yau condition for Poisson algebras

In this paper, we show that the twisted Poincar\'e duality between Poisson homology and cohomology can be derived from the Serre invertible bimodule. This gives another definition of a unimodular Poisson algebra in terms of its Poisson Picard group. We also achieve twisted Poincar\'e duality for Hochschild (co)homology of Poisson bimodules using rigid dualizing complex. For a smooth Poisson affine variety with the trivial canonical bundle, we prove that its enveloping algebra is a Calabi-Yau algebra if the Poisson structure is unimodular.