Related papers: Duality and symmetry of complexity over complete i…
We show that every thick subcategory of the singularity category of a complete intersection ring is self dual. We also prove the analogous statement for thick subcategories of the bounded derived category and give applications to the…
A general method for establishing results over a commutative complete intersection local ring by passing to differential graded modules over a graded exterior algebra is described. It is used to deduce, in a uniform way, results on the…
This thesis is comprised of three chapters. The first chapter deals with bounded complexes of Gorenstein projective and Gorenstein injective modules. Deploying methods of relative homological algebra, we approximate such complexes with…
In this paper, we answer a question of Dwyer, Greenlees, and Iyengar by proving a local ring $R$ is a complete intersection if and only if every complex of $R$-modules with finitely generated homology is proxy small. Moreover, we establish…
A duality theorem for the singularity category of a finite dimensional Gorenstein algebra is proved. It complements a duality on the category of perfect complexes, discovered by Happel. One of its consequences is an analogue of Serre…
Let R be a commutative, noetherian, local ring. Topological Q-vector spaces modelled on full subcategories of the derived category of R are constructed in order to study intersection multiplicities.
We study complexes of finite complete intersection dimension in the derived category of a local ring. Given such a complex, we prove that the thick subcategory it generates contains complexes of all possible complexities. In particular, we…
In this paper we treat Grothendieck Duality for noetherian rings via rigid dualizing complexes. In particular, we prove that every ring, essentially finite type over a regular base ring, has a unique rigid dualizing complex. The rigid…
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…
We prove a Lefschetz duality result for intersection homology. Usually, this result applies to pseudomanifolds with boundary which are assumed to have a "collared neighborhood of their boundary". Our duality does not need this assumption…
We prove that the $0$-th local cohomology of the jacobian ring of a projective hypersurface with isolated singularities has a nice interpretation it in the context of linkage theory. Roughly speaking, it represents a measure of the failure…
In this article a higher order support theory, called the cohomological jump loci, is introduced and studied for dg modules over a Koszul extension of a local dg algebra. The generality of this setting applies to dg modules over local…
In this paper we study homological dimensions of finitely generated modules over commutative Noetherian local rings, called reducing homological dimensions. We obtain new characterizations of Gorenstein and complete intersection local rings…
Regularity, complete intersection and Gorenstein properties of a local ring can be characterized by homological conditions on the canonical homomorphism into its residue field (Serre, Avramov, Auslander). It is also known that in positive…
A method is provided for computing an upper bound of the complexity of a module over a local ring, in terms of vanishing of certain cohomology modules. We then specialize to complete intersections, which are precisely the rings over which…
We study local equivalence of bounded complexes over a polynomial ring $R[w]$, where $R$ is a noetherian ring. We provide a homological algebra approach to the results, the variants of which have been proved in many places in the…
New homotopy invariant finiteness conditions on modules over commutative rings are introduced, and their properties are studied systematically. A number of finiteness results for classical homological invariants like flat dimension,…
We show that a derived bi-duality dg-module is quasi-isomorphic to the homotopy limit of a certain tautological functor. This is a simple observation, which seems to be true in wider context. From the view point of derived Gabriel topology,…
We prove the Lefschetz duality for intersection (co)homology in the framework of $\partial$-pesudomanifolds. We work with general perversities and without restriction on the coefficient ring.
A duality theorem of the bounded derived category of quasi-finite comodules over an artinian coalgebra is established. Let $A$ be a noetherian complete basic semiperfect algebra over an algebraically closed field, and $C$ be its dual…