Let A be a reduced affine K-algebra. Let X = mSpec be the set of all maximal ideals of A. For each f e A, let Uf = {m EXf&m}. a) Show that the sets Uf form a basis of a topology on X, and give a charac- terization of its closed sets in terms of prime ideals of A. b) For each open set U CX, let Ox(U) be the set of all functions U K mapping that are locally of the form m + g(m) for some f, g A.1 Show that this defines a sheaf Ox of K-algebras on X. This makes X into a ringed space. c) Let Y = mSpecB for a reduced affine K-algebra B. Given an algebra homo- morphism 6: A + B, show that n 16-1(n) defines a morphism of ringed spaces (Y, Oy) + (X, Ox). This morphism will be denoted mSpec(4). Thus we get a functor Spec from the category of reduced affine K-algebras to the category of ringed spaces. (You do not need to show functoriality.) d) Show that mSpec(A) is an affine variety, and that the functor mSpec is quasi-inverse to the functor F of Problem 6 from Problem Sheet 1. Let A be a reduced affine K-algebra. Let X = mSpec be the set of all maximal ideals of A. For each f e A, let Uf = {m EXf&m}. a) Show that the sets Uf form a basis of a topology on X, and give a charac- terization of its closed sets in terms of prime ideals of A. b) For each open set U CX, let Ox(U) be the set of all functions U K mapping that are locally of the form m + g(m) for some f, g A.1 Show that this defines a sheaf Ox of K-algebras on X. This makes X into a ringed space. c) Let Y = mSpecB for a reduced affine K-algebra B. Given an algebra homo- morphism 6: A + B, show that n 16-1(n) defines a morphism of ringed spaces (Y, Oy) + (X, Ox). This morphism will be denoted mSpec(4). Thus we get a functor Spec from the category of reduced affine K-algebras to the category of ringed spaces. (You do not need to show functoriality.) d) Show that mSpec(A) is an affine variety, and that the functor mSpec is quasi-inverse to the functor F of Problem 6 from Problem Sheet 1