Universal deformation rings of modules over Frobenius algebras

Modules, comodules and cotensor products over Frobenius algebras

We characterize noncommutative Frobenius algebras A in terms of the existence of a coproduct which is a map of left A-modules. We show that the category of right (left) comodules over A, relative to this coproduct, is isomorphic to the category of right (left) modules. This isomorphism enables a reformulation of the cotensor product of Eilenberg and Moore as a functor of modules rather than com...

Associated Graphs of Modules Over Commutative Rings

Let $R$ be a commutative ring with identity and let $M$ be an $R$-module. In this paper we introduce a new graph associated to modules over commutative rings. We study the relationship between the algebraic properties of modules and their associated graphs. A topological characterization for the completeness of the special subgraphs is presented. Also modules whose associated graph is complete...

Triangulaizability of Algebras Over Division Rings

Presentations of universal deformation rings

Let F be a finite field of characteristic ` > 0, F a number field, GF the absolute Galois group of F and let ρ̄ : GF → GLN (F) be an absolutely irreducible continuous representation. Suppose S is a finite set of places containing all places above ` and above ∞ and all those at which ρ̄ ramifies. Let O be a complete discrete valuation ring of characteristic zero with residue field F. In such a sit...

Duality and Rational Modules in Hopf Algebras over Commutative Rings

Let A be an algebra over a commutative ring R. If R is noetherian and A◦ is pure in R, then the categories of rational left A-modules and right A◦-comodules are isomorphic. In the Hopf algebra case, we can also strengthen the Blattner– Montgomery duality theorem. Finally, we give sufficient conditions to get the purity of A◦ in R. © 2001 Academic Press