Universal deformation rings and dihedral blocks with two simple modules

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...

Explicit Construction of Universal Deformation Rings

Let V be an absolutely irreducible representation of a profinite group G over the residue field k of a noetherian local ring O. For local complete O-algebras A with residue field k the representations of G over A that reduce to V over k are given by O-algebra homomorphisms R → A, where R is the universal deformation ring of V . We show this with an explicit construction of R. The ring R is noet...

Biendomorphism rings of nitely presented simple modules

Public Key Cryptography Based on Simple Modules over Simple Rings

The Diffie Hellman key exchange and the ElGamal oneway trapdoor function are the basic ingredients of public key cryptography. Both these protocols are based on the hardness of the discrete logarithm problem in a finite ring. In this paper we show how the action of a ring on a module gives rise to a generalized Diffie-Hellman and ElGamal protocol. This leads naturally to a cryptographic protoco...

Rings and modules

Proof. Consider the map g:A/I → C , a+I 7→ f (a). It is well defined: a+I = a′ +I implies a− a′ ∈ I implies f (a) = f (a′). The element a + I belongs to the kernel of g iff g(a + I) = f (a) = 0, i.e. a ∈ I , i.e. a + I = I is the zero element of A/I . Thus, ker(g) = 0. The image of g is g(A/I) = {f (a) : a ∈ A} = C . Thus, g is an isomorphism. The inverse morphism to g is given by f (a) 7→ a + I .