Affine Varieties and Morphisms: Sheaf Properties and Irreducibility, Assignments of Algebra

Download Affine Varieties and Morphisms: Sheaf Properties and Irreducibility and more Assignments Algebra in PDF only on Docsity! Due Thursday, September 24. 1. Show that the structure sheaf of an affine variety is indeed a sheaf. 2. Exercise 2.6.12 in Gathmanns notes. 3. Let (X,OX) be a ringed space such that OX is a sheaf of k-valued functions and such that there is an open cover {U} of X such that (U,OX |U ) is an affine variety. Let X = U1 ∪ · · · ∪ Ur be a finite open cover. A topological space is called irreducible if it is not the union of two proper closed subsets. Show that X is irreducible if and only if the following two conditions hold • Ui is irreducible for 1 ≤ i ≤ r. • Ui ∩ Uj 6= ∅ for 1 ≤ i, j ≤ r. 4. If X is an affine algebraic variety, and if f ∈ OX(X), then we denote by Uf the open subset Uf = {x ∈ X | f(x) 6= 0}. Suppose that φ : X → Y is a morphism of affine algebraic varieties, and denote by φ# : OY (Y ) → OX(X) the induced ring homomorphism, that takes a ∈ OY (Y ) to a ◦ φ. Show that if f ∈ OY (Y ), then • We have φ−1(Uf ) = Ug, where g = φ#(f). • The induced ring homomorphism OY (Uf ) → OX(Ug) can be identified with the homomorphism OY (Y )f → OX(X)g induced by φ# by localization. Note thatOY (Y )f is the ring of fractions of OY (Y ) with respect to the {fn | n ≥ 0} (see Atiyah-Macdonald, Chapter 3 for example). 5. Let φ : X → Y be a morphism of affine varieties. Show that the closure of the image is the closed subset of Y defined by Ker(φ# : OY (Y ) → OX(X)). 1