Mathematics G4261x Commutative Algebra Assignment #10 Due Nov. 12, 2014 From Atiyah-Macdonald: *1. Let R be a subring of S such that S n R is closed under multiplication. Show that R is integrally closed in S. *2. If R ⊂ S ⊂ T are rings with R integrally closed in S and S integrally closed in T , show that R is integrally closed in T . *3. Let A be a subring of a domain B, and let C be the integral closure of A in B. If f; g 2 B[x] are monic polynomials such that fg 2 C[x], show that f; g 2 C[x]. Hint: factor them in the algebraic closure of the fraction field of B. *4. Let G be a finite group of automorphisms of a ring R, and let RG ⊂ R be the ring of invariants, that is, the set of r 2 R such that σ(r) = r for all σ 2 G. Prove that R is integral G Q over R . Hint: consider σ2G(x − σ(r)). From Hartshorne: For any point x in a scheme X, the residue field k(x) is defined to be Ox=mx, where Ox is the local ring at x. As discussed in class, for p 2 Spec R this coincides with Rp=pp. 5. Describe the residue fields at every point of the affine schemes discussed in class. 6. Describe the prime spectrum of the zero ring, and show that it is an initial object for the category of schemes. *7. (a) Let Spec R be an affine scheme, N(R) ⊂ R the nilradical. Show that the closed subscheme V (N(R)) is homeomorphic to Spec R, with each point having the same residue fields in both schemes. (b) Let X be an arbitrary scheme, and let Xred be the ringed space homeomorphic to X but with sheaf of rings Γ(Xred; O) := Γ(X; O)=N(Γ(X; O)). Show that Xred is a scheme with the same residue field at each point. It is called the reduced induced subscheme. *8. Let K be any field, X any scheme. Show that a morphism Spec K ! X determines, and is determined by, a point x 2 X and an inclusion k(x) ! K. 9. If a scheme X admits a morphism to Spec K, K a field, then for any open affine Spec R ⊂ X, R is a K-algebra, and every residue field is an extension of K. 2 *10. For K a field, the ring of dual numbers is DK := K[]=() . If X is a scheme over K (that is, equipped with a morphism to Spec K), show that a morphism Spec DK ! X determines, and is determined by, a K-rational point of X (that is, a point with residue field K) and an 2 ∗ element of the Zariski tangent space TxX = (mx=mx) . *11. (a) State and prove necessary and sufficient conditions on R for Spec R to be irreducible as a topological space. (b) Do the same for a scheme X with an open cover by Spec Ri. (c) Show that every (nonempty) irreducible scheme has an unique generic point, meaning a point whose closure is the whole scheme..