LECTURES ON ERGODIC THEORY OF GROUP ACTIONS (A VON
... σ morphism σ : Γ → Aut(X, µ). We’ll often use the notation Γ y (X, µ) to emphasize an action σ, or simply Γ y X if no confusion is possible. We’ll sometimes consider topological groups G other than discrete (typically locally compact or Polish), in which case an action of G on (X, µ) will be a morph ...
... σ morphism σ : Γ → Aut(X, µ). We’ll often use the notation Γ y (X, µ) to emphasize an action σ, or simply Γ y X if no confusion is possible. We’ll sometimes consider topological groups G other than discrete (typically locally compact or Polish), in which case an action of G on (X, µ) will be a morph ...
Lectures on Modules over Principal Ideal Domains
... 1. If f : M → M 0 is an isomorphism of R−modules, prove that f −1 : M 0 → M is also a homomorphism of R- modules. 2. Prove that the natural map η : M → M/N is an R- module homomorphism. 3. Define the following terms in the context of modules over a commutative ring: Set of generators, linearly indep ...
... 1. If f : M → M 0 is an isomorphism of R−modules, prove that f −1 : M 0 → M is also a homomorphism of R- modules. 2. Prove that the natural map η : M → M/N is an R- module homomorphism. 3. Define the following terms in the context of modules over a commutative ring: Set of generators, linearly indep ...
Locally ringed spaces and affine schemes
... We have a (contravariant) functor Spec from the category of rings to the category of locally ringed spaces defined as follows. Let ϕ : A −→ B be a morphism of rings. We already know how this induces a morphism f : Spec B −→ Spec A of topological spaces. Let X = Spec B and Y = Spec A. There is a can ...
... We have a (contravariant) functor Spec from the category of rings to the category of locally ringed spaces defined as follows. Let ϕ : A −→ B be a morphism of rings. We already know how this induces a morphism f : Spec B −→ Spec A of topological spaces. Let X = Spec B and Y = Spec A. There is a can ...
