We relate the variance theory for Cousin complexes − developed by Lipman, Nayak and the author to Grothendieck duality for Cousin complexes. Specifically for a Cousin complex F on (Y, ∆)—with ∆ a codimension function on a formal scheme Y (noetherian, universally catenary)—and a pseudo-finite type map f : (X, ∆) → (Y, ∆) of such pairs of schemes with codimension functions, we show there is a der...

Müller generalized in [12] the notion of a Frobenius extension to left (right) quasi-Frobenius extension and proved the endomorphism ring theorem for these extensions. Recently, Guo observed in [9] that for a ring homomorphism φ : R → S, the restriction of scalars functor has to induction functor S ⊗R − : RM → SM as right ”quasi” adjoint if and only if φ is a left quasi-Frobenius extension. In ...

The Grothendieck group of the tower of symmetric group algebras has a self-dual graded Hopf algebra structure. Inspired by this, we introduce by way of axioms, a general notion of a tower of algebras and study two Grothendieck groups on this tower linked by a natural paring. Using representation theory, we show that our axioms give a structure of graded Hopf algebras on each Grothendieck groups...

The subring of the Grothendieck ring of varieties generated by the graph hypersurfaces of quantum field theory maps to the monoid ring of stable birational equivalence classes of varieties. We show that the image of this map is the copy of Z generated by the class of a point. Thus, the span of the graph hypersurfaces in the Grothendieck ring is nearly killed by setting the Lefschetz motive L to...

2.1 Basic notations and constructions First recall the following notations and constructions • KD(X) the Grothendieck group of the category of D-equivariant locally free sheaves on X. • K(X) the Grothendieck group of the category of D-equivariant coherent sheaves on X. • KD(X) the Grothendieck group of D-equivariant locally free sheaves on the fixed locus X. • K(X) the Grothendieck group of cat...

Let E be a cocomplete topos. We show that if the exact completion of E is a topos then every indecomposable object in E is an atom. As a corollary we characterize the locally connected Grothendieck toposes whose exact completions are toposes. This result strengthens both the Lawvere–Schanuel characterization of Boolean presheaf toposes and Hofstra’s characterization of the locally connected Gro...

the date of receipt and acceptance should be inserted later Abstract Inclusion systems have been introduced in algebraic specification theory as a categorical structure supporting the development of a general abstract logic-independent approach to the algebra of specification (or programming) modules. Here we extend the concept of indexed categories and their Grothendieck flattenings to inclusi...

This is an overview and a preview of the theory of mixed motives of level ≤ 1 explaining some results, projects, ideas and indicating a bunch of problems. Dedicated to Jacob Murre Let k be an algebraically closed field of characteristic zero to start with and let S = Spec(k) denote our base scheme. Recall that Murre [42] associates to a smooth n-dimensional projective variety X over S a Chow co...