Some remarks on growth of algebras over commutative noetherian rings

Using a growth function,GK defined for algebras over integral domains, we construct a generalization of Gelfand Kirillov dimensionGGK. GGK coincides with the classical no-tion of GK for algebras over a field, but is defined for algebras over arbitrary commutative rings. It is proved that GGK exceeds the Krull dimension for affine Noetherian PI algebras. The main result is that algebras of GGK at most one are PI for a large class of commutative Noetherian base rings including the ring of integers, Z. This extends the well-known result of Small, Stafford, and Warfield found in [11].     

Hopf algebras over hopf algebras

Summary In any category with products and a terminal object one may define the notions of group, module over a group etc. if f: R′→R is a homomorphism of groups, and M an R-module, then one has an induced R′-module f*(M). If one is working in the category of sets, one may define a functor left adjoint to f* by N→R⊗_R′ N, where N is an R′-module. In this paper we show that f* has a left adjoint when one is working in the category of graded connected coalgebras over a field.     

Self-dual hopf algebras

We introduce the notions of self-dual (graded) Hopf algebras and of structurally simple (graded) Hopf algebras. We prove that the self-dual Hopf algebras are structurally simple and provide a construction of self-dual Hopf algebras. Finally, we classify the self-dual quotients of the form T_B (M)/I, where T_B (M) is a path algebra with a graded Hopf algebra structure, and I is a graded admissible Hopf ideal.     

Twisted hopf comodule algebras

For k a commutative ring, H a k‐bialgebra and A a right H‐comodule k‐algebra, we define a new multiplication on the H‐comodule A to obtain a twisted algebra” A^T, T sumHom(H,End (A)). If ^T is convolution invertible, the categories of relative right Hopf modules over A and A^Tare isomorphic. Similarly a convolution invertible left twisting gives an isomorphism of the categories of relative left Hopf modules. We show that crossed products are invertible twistings of the tensor product, and obtain, as a corollary, a duality theorem for crossed products