Gorenstein AC-projective dimension of unbounded complexes

Recently, the notion of Gorenstein AC-projective (resp., Gorenstein AC-injective) modules was introduced in [3 Bravo, D., Gillespie, J., Hovey, M. The stable module category of a general ring. http://arxiv.org/abs/1405.5768. [Google Scholar]] by which the so-called “Gorenstein AC-homological algebra” was established. Here, we define and study a notion of Gorenstein AC-projective dimension for complexes (not necessarily bounded) over associative rings, which is inspired by Veliche’s construction of defining Gorenstein projective dimension. In particular, we show that such a dimension can be closely related to the “proper” Gorenstein AC-projective resolutions of complexes induced by a complete and hereditary cotorsion pair in the category of complexes of modules. This enables us to interpret this dimension of a complex in terms of vanishing of the derived functor RHom_R(?,?). As applications, some characterizations of the Gorenstein AC-projective dimension of a module are also obtained.     

Gorenstein projective dimensions of complexes

We show that over a right coherent left perfect ring R, a complex C of left R-modules is Gorenstein projective if and only if C ^m is Gorenstein projective in R-Mod for all m ∈ ℤ. Basing on this we show that if R is a right coherent left perfect ring then Gpd(C) = sup{Gpd(C ^m )|m ∈ ℤ} where Gpd(−) denotes Gorenstein projective dimension.     

Weak dimensions and Gorenstein weak dimensions of group rings

Czechoslovak Mathematical Journal - Let K be a field, and let G be a group. In the present paper, we investigate when the group ring K[G] has finite weak dimension and finite Gorenstein weak...     

Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov)

In this paper we develop a general, systematic, microlocal framework for the Fredholm analysis of non-elliptic problems, including high energy (or semiclassical) estimates, which is stable under perturbations. This framework, described in Sect. 2, resides on a compact manifold without boundary, hence in the standard setting of microlocal analysis. Many natural applications arise in the setting of non-Riemannian b-metrics in the context of Melrose’s b-structures. These include asymptotically de Sitter-type metrics on a blow-up of the natural compactification, Kerr-de Sitter-type metrics, as well as asymptotically Minkowski metrics. The simplest application is a new approach to analysis on Riemannian or Lorentzian (or indeed, possibly of other signature) conformally compact spaces (such as asymptotically hyperbolic or de Sitter spaces), including a new construction of the meromorphic extension of the resolvent of the Laplacian in the Riemannian case, as well as high energy estimates for the spectral parameter in strips of the complex plane. These results are also available in a follow-up paper which is more expository in nature (Vasy in Uhlmann, G. (ed.) Inverse Problems and Applications. Inside Out II, 2012). The appendix written by Dyatlov relates his analysis of resonances on exact Kerr-de Sitter space (which then was used to analyze the wave equation in that setting) to the more general method described here.     

Virtually Gorenstein rings and relative homology of complexes

We extend the notion of virtually Gorenstein rings to the setting of arbitrary rings, and prove that all rings R of finite Gorenstein weak global dimension are virtually Gorenstein such that all Gorenstein projective R-modules are Gorenstein flat. For such a ring R, we introduce the notion of relative homology functors of complexes with respect to Gorenstein projective (resp., flat) modules, and establish a balanced and a vanishing result for the homology functor.     

Thick subcategories of modules over commutative noetherian rings (with an appendix by Srikanth Iyengar)

For a commutative noetherian ring A, we compare the support of a complex of A-modules with the support of its cohomology. This leads to a classification of all full subcategories of A-modules which are thick (that is, closed under taking kernels, cokernels, and extensions) and closed under taking arbitrary direct sums. In addition, subcategories of A-modules that are closed under taking submodules, extensions, and direct unions are classified via associated prime ideals.     

Slice filtration on motives and the Hodge conjecture (with an appendix by J. Ayoub)

We clarify the expected properties of the slice filtration on triangulated motives from the point of view of the generalized Hodge conjecture. In the appendix, J. Ayoub proves unconditionally that the slice filtration does not respect geometric motives. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Suslin homology with integral coefficients in characteristic zero (with an appendix by Bruno Kahn)

We show that the Suslin homology group with integral coefficients of a scheme X separated of finite type over an algebraically closed field of characteristic 0 is a direct sum of a uniquely divisible group, finite copies of $\mathbb{Q}/\mathbb{Z}$, and a finitely generated group. We also study the possible type of homomorphisms between such groups induced by the morphisms of schemes. An appendix written by Bruno Kahn is included, which simplifies the proofs and generalizes the results.     

Classification theorems for central simple algebras with involution (with an appendix by R. Parimala)

The involutions in this paper are algebra anti-automorphisms of period two. Involutions on endomorphism algebras of finite-dimensional vector spaces are adjoint to symmetric or skew-symmetric bilinear forms, or to hermitian forms. Analogues of the classical invariants of quadratic forms (discriminant, Clifford algebra, signature) have been defined for arbitrary central simple algebras with involution. In this paper it is shown that over certain fields these invariants are sufficient to classify involutions up to conjugation. For algebras of low degree a classification is obtained over an arbitrary field. Received: 29 April 1999     

Injective Envelopes and (Gorenstein) Flat Covers

We characterize left Noetherian rings in terms of the duality property of injective preenvelopes and flat precovers. For a left and right Noetherian ring R, we prove that the flat dimension of the injective envelope of any (Gorenstein) flat left R-module is at most the flat dimension of the injective envelope of _R R. Then we get that the injective envelope of _R R is (Gorenstein) flat if and only if the injective envelope of every Gorenstein flat left R-module is (Gorenstein) flat, if and only if the injective envelope of every flat left R-module is (Gorenstein) flat, if and only if the (Gorenstein) flat cover of every injective left R-module is injective, and if and only if the opposite version of one of these conditions is satisfied.     

Elliptic operators, conormal derivatives and positive parts of functions (with an appendix by Haïm Brezis)

Haïm Brezis and Augusto Ponce introduced and studied several extensions of Kato's inequality, in particular Kato's inequalities up to the boundary involving the Laplacian and the normal derivative of the positive part of a W1,1 function in a smooth domain [H. Brezis, A.C. Ponce, Kato's inequality when Δu is a measure, C. R. Acad. Sci. Paris Sér. I 338 (2004) 599-604; H. Brezis, A.C. Ponce, Kato's inequality up to the boundary, Commun. Contemp. Math. 10 (2008) 1217-1241]. Using potential theoretic methods we answer here some questions raised in [H. Brezis, A.C. Ponce, Kato's inequality up to the boundary, Commun. Contemp. Math. 10 (2008) 1217-1241] about the relations between the normal derivative of a function u and the normal derivative of its positive part u₊. The results apply to a large class of domains and elliptic operators in divergence form and finally an expression of the normal derivative of a function of u is given. In the final appendix, H. Brezis solves an old question of J. Serrin about pathological solutions of certain elliptic equations [J. Serrin, Pathological solutions of elliptic differential equations, Ann. Sc. Norm. Super. Pisa (3) 18 (1964) 385-387]. This is used in the paper to extend the first version of our main result.     

HJB equations in infinite dimensions with locally Lipschitz Hamiltonian and unbounded terminal condition

We study Hamilton Jacobi Bellman equations in an infinite dimensional Hilbert space, with Lipschitz coefficients, where the Hamiltonian has superquadratic growth with respect to the derivative of the value function, and the final condition is not bounded. This allows to study stochastic optimal control problems for suitable controlled state equations with unbounded control processes. The results are applied to a controlled wave equation.     

First,Second, and Third Change of Rings Theorems for Gorenstein Homological dimensions

In this article, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, second, and third change of rings theorems for the classical projective and injective dimensions to the Gorenstein projective and injective dimensions, respectively. Each of the results established in this article for the Gorenstein projective dimension is a generalization of a G-dimension of a finitely generated module M over a noetherian ring R.     

On the equivariant structure of ideals in abelian extensions of local fields (with an appendix by W. Bley)

Let p be an odd rational prime and K a finite extension of \Bbb Q_p {\Bbb Q}_p . We give a complete classification of those finite abelian extensions L/K L/K in which any ideal of the valuation ring of L is free over its associated order in \Bbb Q_p[Gal(L/K)] {\Bbb Q}_p[Gal(L/K)] . In an appendix W. Bley describes an algorithm which can be used to determine the structure of Galois stable ideals in abelian extensions of number fields. The algorithm is applied to give several new and interesting examples.     

Analysis of the scattering by an unbounded rough surface

This paper is concerned with the mathematical analysis of the solution for the wave propagation from the scattering by an unbounded penetrable rough surface. Throughout, the wavenumber is assumed to have a nonzero imaginary part that accounts for the energy absorption. The scattering problem is modeled as a boundary value problem governed by the Helmholtz equation with transparent boundary conditions proposed on plane surfaces confining the scattering surface. The existence and uniqueness of the weak solution for the model problem are established by using a variational approach. Furthermore, the scattering problem is investigated for the case when the scattering profile is a sufficiently small and smooth deformation of a plane surface. Under this assumption, the problem is equivalently formulated into a set of two‐point boundary value problems in the frequency domain, and the analytical solution, in the form of an infinite series, is deduced by using a boundary perturbation technique combined with the transformed field expansion approach. Copyright © 2012 John Wiley & Sons, Ltd.