Integral geometry for \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\mathcal {D}}$\end{document}‐modules on dual flag manifolds and generalized Verma modules

Given a pair of dual generalized flag manifolds of a semisimple algebraic group, we show that the integral transform between them given by the open orbit in their product is an equivalence. We also describe the links of this problem with the structure of generalized Verma modules, and how the above construction can be applied to the representation theory of real forms of the group.     

Über Die Assoziierten Primideale des Bidualen

Let (R, 𝔪) be a commutative, noetherian, local ring, E the injective hull of the residue field R/𝔪, and M ^○○ = Hom _R (Hom _R (M, E), E) the bidual of an R-module M. We investigate the elements of Ass(M ^○○) as well as those of Coatt(M) = {𝔭 ∈ Spec(R)|𝔭 = Ann _R (Ann _M (𝔭))} and provide criteria for equality in one of the two inclusions Ass(M) ? Ass(M ^○○) ? Coatt(M). If R is a Nagata ring and M a minimax module, i.e., an extension of a finitely generated R-module by an artinian R-module, we show that Ass(M ^○○) = Ass(M) ∪ {𝔭 ∈ Coatt(M)| R/𝔭 is incomplete}.     

Cohen–Macaulay Modules in Dimension >s and Results on Local Cohomology

Let (R,𝔪) be a local ring and s ≥ ?1. Using the notion of M-sequence in dimension > s, we introduce Cohen–Macaulay modules in dimension > s. Among other things concerning Cohen–Macaulay modules in dimension > s, some finiteness results of the support and the associated primes of local cohomology modules are investigated.     

On Strongly Clean Modules

An element of a ring R is called “strongly clean” if it is the sum of an idempotent and a unit that commute, and R is called “strongly clean” if every element of R is strongly clean. A module M is called “strongly clean” if its endomorphism ring End(M) is a strongly clean ring. In this article, strongly clean modules are characterized by direct sum decompositions, that is, M is a strongly clean module if and only if whenever M′⊕ B = A ₁⊕ A ₂ with M′? M, there are decompositions M′ = M ₁⊕ M ₂, B = B ₁⊕ B ₂, and A _i  = C _i ⊕ D _i (i = 1,2) such that M ₁⊕ B ₁ = C ₁⊕ D ₂ = M ₁⊕ C ₁ and M ₂⊕ B ₂ = D ₁⊕ C ₂ = M ₂⊕ C ₂.     

Lifting to Cluster-Tilting Objects in 2-Calabi–Yau Triangulated Categories

We show that a tilting module over the endomorphism algebra of a cluster-tilting object in a 2-Calabi–Yau triangulated category lifts to a cluster-tilting object in this 2-Calabi–Yau triangulated category. This generalizes a recent work of Smith for cluster categories.     

Self-Orthogonal Modules of Finite Projective Dimension

Let R be a ring and _R ω a self-orthogonal module. We introduce the notion of the right orthogonal dimension (relative to _R ω) of modules. We give a criterion for computing this relative right orthogonal dimension of modules. For a left coherent and semilocal ring R and a finitely presented self-orthogonal module _R ω, we show that the projective dimension of _R ω and the right orthogonal dimension (relative to _R ω) of R/J are identical, where J is the Jacobson radical of R. As a consequence, we get that _R ω has finite projective dimension if and only if every left (finitely presented) R-module has finite right orthogonal dimension (relative to _R ω). If ω is a tilting module, we then prove that a left R-module has finite right orthogonal dimension (relative to _R ω) if and only if it has a special ω^⊥-preenvelope.     

On the Asymptotic Depth of Multigraded Modules

The aim of this article is to establish, among other results, the asymptotic stability of the depth of the graded pieces of a nonstandard multigraded module. As a corollary, we get the asymptotic stability of the depth of the graded pieces of the multigraded Rees algebra defined by a finite set of ideals and their associated multigraded rings.     

Quasi-Piecewise-Koszul Modules

In this article we first study an equivariant cyclic cohomology for weak H-module agebras over a weak Hopf algebra H with a bijective antipode. Then we define an equivariant K-theory for weak quantum Yetter–Drinfeld algebras over H and establish a generalized Connes' pairing between the equivariant cyclic cohomology and the equivariant K-theory. As an application we consider our theory for groupoids.     

Gorenstein Flat Covers over GF-Closed Rings

We prove that all modules over a left GF-closed ring have Gorenstein flat covers.