For a finitely generated A-module M we define the dimension filtration M = {Mi}0≤i≤d, d = dimA M, where Mi denotes the largest submodule of M of dimension ≤ i. Several properties of this filtration are investigated. In particular, in case the local ring (A,m) possesses a dualizing complex, then this filtration occurs as the filtration of a spectral sequence related to duality. Furthermore, we c...

The paper presents a further step within a comprehensive program of “Dualizing Universal Algebra” [4, 7, 10, 11, 12]. We analyze the construction of subalgebras generated by a given subset (of generators) of the carrier of an algebra and reformulate it in a systematic categorical way. Dualizing the new categorical description, we obtain then the construction of quotient coalgebras “cogenerated”...

Consider a ring K, a topological space X and a sheaf A on X of K[[~]]-algebras. Assuming A ~-adically complete and without ~torsion, we first show how to deduce a coherency theorem for complexes of A -modules from the corresponding property for complexes of A /~A -modules. We apply this result to prove that, under a natural properness condition, the convolution of two coherent kernels over defo...

To a topological group G, we assign a naive G-spectrum DG, called the dualizing spectrum of G. When the classifying space BG is finitely dominated, we show that DG detects Poincaré duality in the sense that BG is a Poincaré duality space if and only ifDG is a homotopy finite spectrum. Secondly, we show that the dualizing spectrum behavesmultiplicatively on certain topological group extensions. ...

Consider a ring K, a topological space X and a sheaf A on X of K[[~]]-algebras. Assuming that A is ~-complete and without ~torsion, we first show how to deduce a coherency theorem for complexes of A -modules from a corresponding property for complexes of A /~A -modules. We apply this result to prove that, under a natural properness condition, the convolution of two coherent kernels over deforma...

1.1. For a finite type separated Deligne-Mumford stack X/k and integer i, the i-th `-adic Borel-Moore homology group of X, denoted Hi(X), is defined to be H (X,ΩX), where ΩX ∈ D c(X,Q`) is the `-adic dualizing complex of X. These groups were considered already by Laumon in [17] (and, we have been informed, by Grothendieck in unpublished work), where he showed they enjoyed a number of good prope...